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The Free Book of SymmetryNotation

ContentsArticles

List of spherical symmetry groups 1List of planar symmetry groups 4Dihedral symmetry in three dimensions 6Tetrahedral symmetry 8Octahedral symmetry 13Icosahedral symmetry 21Cyclic symmetries 27Reflection symmetry 28Inverse (mathematics) 29Point groups in three dimensions 30Cyclic group 40Dihedral group 44Lattice (group) 51Wallpaper group 54

ReferencesArticle Sources and Contributors 85Image Sources, Licenses and Contributors 86

Article LicensesLicense 93

List of spherical symmetry groups 1

List of spherical symmetry groupsSpherical symmetry groups are also called point groups in three dimensions, however this article is limitied to thefinite symmetries. This article lists the common name and associated Schoenflies notation, Coxeter notation,Orbifold notation, and order to describe three dimensional symmetries.

List of symmetry groups on the sphereThere are four fundamental symmetry classes which have triangular fundamental domains: dihedral, tetrahedral,octahedral, icosahedral. There are infinitely many dihedral symmetry groups.The final classes, under other have digonal or monogonal fundamental domains.

Dihedral symmetry [2,n]There are an infinite set of dihedral symmetries. n can be any positive integer 2 or greater (n = 1 is also possible, butthese three symmetries are equal to C2, C2v, and C2h).

Name Schönfliescrystallographic

notation

Coxeternotation

Orbifoldnotation

Order Fundamentaldomain

Polyditropic Dn [2,n]+ 22n 2n

Polydiscopic Dnh [2,n] *22n 4n

Polydigyros Dnd [2+,2n] 2*n 4n

Tetrahedral symmetry [3,3]

Name Schönfliescrystallographic

notation

Coxeternotation

Orbifoldnotation

Order Fundamentaldomain

Chiral tetrahedral T [3,3]+ 332 12

Achiral tetrahedral Td [3,3] *332 24

Pyritohedral Th [3+,4] 3*2 24

List of spherical symmetry groups 2

Octahedral symmetry [3,4]

Name Schönfliescrystallographic

notation

Coxeternotation

Orbifoldnotation

Order Fundamentaldomain

Chiral octahedral O [3,4]+ 432 24

Achiral octahedral Oh [3,4] *432 48

Icosahedral symmetry [3,5]

Name Schönfliescrystallographic

notation

Coxeternotation

Orbifoldnotation

Order Fundamentaldomain

Chiral icosahedral I [3,5]+ 532 60

Achiral icosahedral Ih [3,5] *532 120

OtherThese final forms have digonal or monogonal fundamental regions with Cyclic symmetries and reflection symmetry.There are four infinite sets with index n being any positive integer; for n=1 two cases are equal, so there are three;they are separately named.

Name Schönfliescrystallographic

notation

Coxeternotation

Orbifoldnotation

Order Fundamentaldomain

no symmetry (monotropic) C1 [1]+ 11 1

discrete rotational symmetry (polytropic) Cn [n]+ nn n

reflection symmetry (monoscopic) Cs = C1v = C1h [1] *11 2

List of spherical symmetry groups 3

Polyscopic Cnv [n] *nn 2n

Polygyros Cnh [2,n+] n* 2n

inversion symmetry (monodromic) Ci = S2 [2+,2+] 1× 2

Polydromic S2n [2+,2n+] n× 2n

Relation between orbifold notation and orderThe order of each group is 2 divided by the orbifold Euler characteristic; the latter is 2 minus the sum of the featurevalues, assigned as follows:• n without or before * counts as (n − 1)/n• n after * counts as (n − 1)/(2n)• * and x count as 1This can also be applied for wallpaper groups: for them, the sum of the feature values is 2, giving an infinite order;see orbifold Euler characteristic for wallpaper groups

References• Peter R. Cromwell, Polyhedra (1997), Appendix I• Finite spherical symmetry groups [1]

• Weisstein, Eric W., "Schoenflies symbol [2]" from MathWorld.• Simplest Canonical Polyhedra of Each Symmetry Type [3], by David I. McCooey

References[1] http:/ / www. geom. uiuc. edu/ ~math5337/ Orbifolds/ costs. html[2] http:/ / mathworld. wolfram. com/ SchoenfliesSymbol. html[3] http:/ / homepage. mac. com/ dmccooey/ polyhedra/ Simplest. html

List of planar symmetry groups 4

List of planar symmetry groupsThis article summarizes the classes of discrete planar symmetry groups:1. 1 simple symmetries (reflection)2. 2 infinite set of point groups3. 7 Frieze groups4. 17 wallpaper groups

Simple symmetryPoint groups:

Example Symbols

Example:Kite

(*)Reflection symmetry

Point groupsThere are two classes of point groups, rotational and reflectional.Point groups:

Example Symbols

Example:Flag of Hong Kong C5

Cn (n)Cyclic group

Example: Snowflake D6

Dn (*n)Dihedral group

Frieze groupsThere are also 7 Frieze groups in the plane which have a fundamental line of symmetry and infinite fundamentaldomains.

Examplepattern

orbifold notation

List of planar symmetry groups 5

1. (∞∞)2. (∞x)3. (∞*)4. (*∞∞)5. (22∞)6. (2*∞)7. (*22∞)

Wallpaper groupsThere are 17 wallpaper groups in the plane with finite fundamental domains.

Rotation

p2 (2222)parallelogrammetic

p4 (442)

p3 (333)

p6 (632)

Maximum symmetry per lattice type

pmm(*2222)rectangular

cmm (2*22)rhombic

p4m (*442)square

p6m (*632)hexagonal

Mixed

p3m1 (*333)

p31m (3*3)

p4g (4*2)

Other

pm (**)

p1 (o)

pg (xx)

pmg (22*)

cm (*x)

pgg (22x)

Note: with regard to the number of mirrors p4m is "more symmetry" than p4g, with regard to the size of thefundamental domain it is an "equal amount of symmetry".

List of planar symmetry groups 6

External references• "Conway's manuscript" on Orbifold notation [1]

• http:/ / www. xahlee. org/ Wallpaper_dir/ c5_17WallpaperGroups. html

References[1] http:/ / www. geom. uiuc. edu/ docs/ doyle/ mpls/ handouts/ node39. html

Dihedral symmetry in three dimensionsThis article deals with three infinite sequences of point groups in three dimensions which have a symmetry groupthat as abstract group is a dihedral group Dihn ( n ≥ 2 ).See also point groups in two dimensions.Chiral:

• Dn (22n) of order 2n – dihedral symmetry (abstract group Dn)Achiral:

• Dnh (*22n) of order 4n – prismatic symmetry (abstract group Dn × C2)• Dnd (or Dnv) (2*n) of order 4n – antiprismatic symmetry (abstract group D2n)For a given n, all three have n-fold rotational symmetry about one axis (rotation by an angle of 360°/n does notchange the object), and 2-fold about a perpendicular axis, hence about n of those. For n = ∞ they correspond to threefrieze groups. Schönflies notation is used, and, in parentheses, Orbifold notation. The term horizontal (h) is usedwith respect to a vertical axis of rotation.In 2D the symmetry group Dn includes reflections in lines. When the 2D plane is embedded horizontally in a 3Dspace, such a reflection can either be viewed as the restriction to that plane of a reflection in a vertical plane, or asthe restriction to the plane of a rotation about the reflection line, by 180°. In 3D the two operations are distinguished:the group Dn contains rotations only, not reflections. The other group is pyramidal symmetry Cnv of the same order.With reflection symmetry with respect to a plane perpendicular to the n-fold rotation axis we have Dnh (*22n).Dnd (or Dnv) has vertical mirror planes between the horizontal rotation axes, not through them. As a result thevertical axis is a 2n-fold rotoreflection axis.Dnh is the symmetry group for a regular n-sided prisms and also for a regular n-sided bipyramid. Dnd is the symmetrygroup for a regular n-sided antiprism, and also for a regular n-sided trapezohedron. Dn is the symmetry group of apartially rotated prism.n = 1 is not included because the three symmetries are equal to other ones:• D1 and C2: group of order 2 with a single 180° rotation• D1h and C2v: group of order 4 with a reflection in a plane and a 180° rotation through a line in that plane• D1d and C2h: group of order 4 with a reflection in a plane and a 180° rotation through a line perpendicular to that

planeFor n = 2 there is not one main axes and two additional axes, but there are three equivalent ones.• D2 (222) of order 4 is one of the three symmetry group types with the Klein four-group as abstract group. It has

three perpendicular 2-fold rotation axes. It is the symmetry group of a cuboid with an S written on two oppositefaces, in the same orientation.

• D2h (*222) of order 8 is the symmetry group of a cuboid• D2d (2*2) of order 8 is the symmetry group of e.g.:

• a square cuboid with a diagonal drawn on one square face, and a perpendicular diagonal on the other one

Dihedral symmetry in three dimensions 7

• a regular tetrahedron scaled in the direction of a line connecting the midpoints of two opposite edges (D2d is asubgroup of Td, by scaling we reduce the symmetry).

SubgroupsFor Dnh• Cnh• Cnv• DnFor Dnd• S2n• Cnv• DnDnd is also subgroup of D2nh.See also cyclic symmetries

ExamplesD

nh (*22n):

prisms

D5h

(*225):

Pentagrammic prismPentagrammic antiprism

D4d

(2*4):

Snub square antiprism

D5d

(2*5):

Pentagonal antiprism

Pentagrammic crossed-antiprism pentagonal trapezohedron

D17d

(*22(17)):

Dihedral symmetry in three dimensions 8

Heptadecagonal antiprism

Tetrahedral symmetry

A regular tetrahedron, an example of asolid with full tetrahedral symmetry

A regular tetrahedron has 12 rotational (or orientation-preserving)symmetries, and a symmetry order of 24 including transformations thatcombine a reflection and a rotation.

The group of all symmetries is isomorphic to the group S4 of permutations offour objects, since there is exactly one such symmetry for each permutation ofthe vertices of the tetrahedron. The set of orientation-preserving symmetriesforms a group referred to as the alternating subgroup A4 of S4.

Details

Chiral and full (or achiral) tetrahedral symmetry and pyritohedralsymmetry are discrete point symmetries (or equivalently, symmetries on thesphere). They are among the crystallographic point groups of the cubic crystal system.

Chiral tetrahedral symmetry

The tetrahedral rotation group T with fundamental domain; for the triakis tetrahedron, seebelow, the latter is one full face

T or 332 or 23, of order 12 - chiral orrotational tetrahedral symmetry.There are three orthogonal 2-foldrotation axes, like chiral dihedralsymmetry D2 or 222, with in additionfour 3-fold axes, centered between thethree orthogonal directions. This groupis isomorphic to A4, the alternatinggroup on 4 elements; in fact it is thegroup of even permutations of the four3-fold axes: e, (123), (132), (124),(142), (134), (143), (234), (243),(12)(34), (13)(24), (14)(23).

The conjugacy classes of T are:• identity• 4 × rotation by 120° clockwise

(seen from a vertex): (234), (143),(412), (321)

• 4 × rotation by 120° anti-clockwise(ditto)

• 3 × rotation by 180°

Tetrahedral symmetry 9

A tetrahedron can be placed in 12 distinct positions by rotation alone.These are illustrated above in the cycle graph format, along with the 180°

edge (blue arrows) and 120° vertex (reddish arrows) rotations that permutethe tetrahedron through those positions.

In the tetrakis hexahedron one full face is afundamental domain; other solids with the same

symmetry can be obtained by adjusting theorientation of the faces, e.g. flattening selected

subsets of faces to combine each subset into oneface, or replacing each face by multiple faces, or

a curved surface.

The rotations by 180°, together with the identity,form a normal subgroup of type Dih2, withquotient group of type Z3. The three elements ofthe latter are the identity, "clockwise rotation", and"anti-clockwise rotation", corresponding topermutations of the three orthogonal 2-fold axes,preserving orientation.

A4 is the smallest group demonstrating that theconverse of Lagrange's theorem is not true ingeneral: given a finite group G and a divisor d of|G|, there does not necessarily exist a subgroup ofG with order d: the group G = A4 has no subgroupof order 6. Although it is a property for the abstractgroup in general, it is clear from the isometrygroup of chiral tetrahedral symmetry: because ofthe chirality the subgroup would have to be C6 orD3, but neither applies.

Subgroups

• T• D2• C3 and C2• E

Achiral tetrahedral symmetry

Td

or *332 or , of order 24 - achiral or full

tetrahedral symmetry, also known as the (2,3,3)triangle group. This group has the same rotationaxes as T, but with six mirror planes, each throughtwo 3-fold axes. The 2-fold axes are now S4 ( )axes. Td and O are isomorphic as abstract groups:they both correspond to S4, the symmetric groupon 4 objects. Td is the union of T and the setobtained by combining each element of O \ T withinversion. See also the isometries of the regulartetrahedron.

The conjugacy classes of Td are:• identity• 8 × rotation by 120°• 3 × rotation by 180°• 6 × reflection in a plane through two rotation

axes• 6 × rotoreflection by 90°

Subgroups

Tetrahedral symmetry 10

• Td• T• D2d• D3 and D2• C3v and C2v• C3 and C2• S4 and S2=Ci• E and Cs

Pyritohedral symmetryT

h or 3*2 or , of order 24 - pyritohedral symmetry. This group has the same rotation axes as T, with

mirror planes through two of the orthogonal directions. The 3-fold axes are now S6 ( ) axes, and there is inversionsymmetry. Th is isomorphic to T × Z2: every element of Th is either an element of T, or one combined with inversion.Apart from these two normal subgroups, there is also a normal subgroup D2h (that of a cuboid), of type Dih2 × Z2 =Z2 × Z2 × Z2 . It is the direct product of the normal subgroup of T (see above) with Ci. The quotient group is the sameas above: of type Z3. The three elements of the latter are the identity, "clockwise rotation", and "anti-clockwiserotation", corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation.

The Gaelic football has pyritohedralsymmetry

It is the symmetry of a cube with on each face a line segment dividing theface into two equal rectangles, such that the line segments of adjacent facesdo not meet at the edge. The symmetries correspond to the even permutationsof the body diagonals and the same combined with inversion. It is also thesymmetry of a pyritohedron, which is extremely similar to the cube described,with each rectangle replaced by a pentagon with one symmetry axis and 4equal sides and 1 different side (the one corresponding to the line segmentdividing the cube's face); i.e., the cube's faces bulge out at the dividing lineand become narrower there. It is a subgroup of the full icosahedral symmetrygroup (as isometry group, not just as abstract group), with 4 of the 10 3-foldaxes.

The conjugacy classes of Th include those of T, with the two classes of 4combined, and each with inversion:

• identity• 8 × rotation by 120°• 3 × rotation by 180°• inversion• 8 × rotoreflection by 60°• 3 × reflection in a plane

Tetrahedral symmetry 11

The full tetrahedral group Td with fundamental domain

The pyritohedral group Th with fundamental domain

Subgroups

• Th• T• D2h• D3d• D3 and D2• C2h• C3v and C2v• C3 and C2• S6 and S2=Ci• E and Cs

Solids with chiraltetrahedral symmetry

The Icosahedroncolored as a snub tetrahedron haschiral symmetry.

Solids with fulltetrahedral symmetry

Tetrahedral symmetry 12

Platonic solid

Name Picture Faces Edges Vertices Edges per face Facesmeeting

at each vertex

tetrahedron

(Animation)

4 6 4 3 3

Archimedean solid(semi-regular: vertex-uniform)

Name picture Faces Edges Vertices Vertex configuration

truncated tetrahedron

(Video)

8 4 triangles4 hexagons

18 12 3,6,6

Catalan solid(semi-regular dual: face-uniform)

Name picture Dual Archimedean solid Faces Edges Vertices Face polygon

triakis tetrahedron

(Video)

truncated tetrahedron 12 18 8 isosceles triangle

Nonconvex uniform polyhedron

Tetrahemihexahedron Octahemioctahedron

Octahedral symmetry 13

Octahedral symmetry

The cube is the most common shape withoctahedral symmetry

A regular octahedron has 24 rotational (or orientation-preserving)symmetries, and a symmetry order of 48 including transformations thatcombine a reflection and a rotation. A cube has the same set of symmetries,since it is the dual of an octahedron.

The group of orientation-preserving symmetries is S4, or the group ofpermutations of four objects, since there is exactly one such symmetry foreach permutation of the four pairs of opposite sides of the octahedron.

Details

The octahedral rotation group O with fundamental domain

Chiral and full (or achiral)octahedral symmetry are the discretepoint symmetries (or equivalently,symmetries on the sphere) with thelargest symmetry groups compatiblewith translational symmetry. They areamong the crystallographic pointgroups of the cubic crystal system.

Octahedral symmetry 14

Chiral octahedral symmetryO, 432, or of order 24, is chiral octahedral symmetry or rotational octahedral symmetry . Thisgroup is like chiral tetrahedral symmetry T, but the C2 axes are now C4 axes, and additionally there are 6 C2 axes,through the midpoints of the edges of the cube. Td and O are isomorphic as abstract groups: they both correspond toS4, the symmetric group on 4 objects. Td is the union of T and the set obtained by combining each element of O \ Twith inversion. O is the rotation group of the cube and the regular octahedron.

Subgroups• O and T• D4, D3 and D2• C4, C3 and C2• E

Conjugacy classes• identity• 6 × rotation by 90°• 8 × rotation by 120°• 3 × rotation by 180° about a 4-fold axis• 6 × rotation by 180° about a 2-fold axis

Achiral octahedral symmetry• O

h (*432) of order 48 - achiral octahedral symmetry or full octahedral symmetry. This group has the same

rotation axes as O, but with mirror planes, comprising both the mirror planes of Td and Th. This group isisomorphic to S4 × C2, and is the full symmetry group of the cube and octahedron. It is the hyperoctahedral groupfor n = 3. See also the isometries of the cube.

A dual cube-octahedron.

In the disdyakis dodecahedron one full face is afundamental domain; other solids with the same

symmetry can be obtained by adjusting theorientation of the faces, e.g. flattening selected

subsets of faces to combine each subset into oneface, or replacing each face by multiple faces, or

a curved surface.

Octahedral symmetry 15

The octahedral group Oh with fundamental domain

With the 4-fold axes as coordinate axes, a fundamental domain of Oh

is given by 0 ≤ x ≤ y ≤ z. An object with thissymmetry is characterized by the part of the object in the fundamental domain, for example the cube is given by z =1, and the octahedron by x + y + z = 1 (or the corresponding inequalities, to get the solid instead of the surface). ax +by + cz = 1 gives a polyhedron with 48 faces, e.g. the disdyakis dodecahedron.Faces are 8-by-8 combined to larger faces for a = b = 0 (cube) and 6-by-6 for a = b = c (octahedron).

Subgroups• Oh• O, Th, Td and T• D4h and D2h• D3d and D2d• D4, D3 and D2• C4h, C3h and C2h• C4v, C3v and C2v• C4, C3 and C2• S6, S4 and S2=Ci• E and Cs

Octahedral symmetry 16

Conjugacy classes• inversion• 6 × rotoreflection by 90°• 8 × rotoreflection by 60°• 3 × reflection in a plane perpendicular to a 4-fold axis• 6 × reflection in a plane perpendicular to a 2-fold axis

The isometries of the cube(To be integrated in the rest of the text.)

The cube has 48 isometries, forming the symmetry group Oh, isomorphic to S4 × C2. They can be categorized asfollows:• O (the identity and 23 proper rotations) with the following conjugacy classes (in parentheses are given the

permutations of the body diagonals and the unit quaternion representation):• identity (identity; 1)• rotation about an axis from the center of a face to the center of the opposite face by an angle of 90°: 3 axes, 2

per axis, together 6 ((1 2 3 4), etc.; ((1±i)/√2, etc.)• ditto by an angle of 180°: 3 axes, 1 per axis, together 3 ((1 2)(3 4), etc.; i,j,k)• rotation about an axis from the center of an edge to the center of the opposite edge by an angle of 180°: 6 axes,

1 per axis, together 6 ((1 2), etc.; ((i±j)/√2, etc.)• rotation about a body diagonal by an angle of 120°: 4 axes, 2 per axis, together 8 ((1 2 3), etc.; (1±i±j±k)/2)

• The same with inversion (x is mapped to −x) (also 24 isometries). Note that rotation by an angle of 180° about anaxis combined with inversion is just reflection in the perpendicular plane. The combination of inversion androtation about a body diagonal by an angle of 120° is rotation about the body diagonal by an angle of 60°,combined with reflection in the perpendicular plane (the rotation itself does not map the cube to itself; theintersection of the reflection plane with the cube is a regular hexagon).

An isometry of the cube can be identified in various ways:• by the faces three given adjacent faces (say 1, 2, and 3 on a die) are mapped to• by the image of a cube with on one face a non-symmetric marking: the face with the marking, whether it is

normal or a mirror image, and the orientation• by a permutation of the four body diagonals (each of the 24 permutations is possible), combined with a toggle for

inversion of the cube, or notFor cubes with colors or markings (like dice have), the symmetry group is a subgroup of Oh. Examples:• C4v: if one face has a different color (or two opposite faces have colors different from each other and from the

other four), the cube has 8 isometries, like a square has in 2D.• D2h: if opposite faces have the same colors, different for each set of two, the cube has 8 isometries, like a cuboid.• D4h: if two opposite faces have the same color, and all other faces have one different color, the cube has 16

isometries, like a square prism (square box).• C2v:

• if two adjacent faces have the same color, and all other faces have one different color, the cube has 4isometries.

• if three faces, of which two opposite to each other, have one color and the other three one other color, the cubehas 4 isometries.

• if two opposite faces have the same color, and two other opposite faces also, and the last two have differentcolors, the cube has 4 isometries, like a piece of blank paper with a shape with a mirror symmetry.

• Cs:

Octahedral symmetry 17

• if two adjacent faces have colors different from each other, and the other four have a third color, the cube has 2isometries.

• if two opposite faces have the same color, and all other faces have different colors, the cube has 2 isometries,like an asymmetric piece of blank paper.

• C3v: if three faces, of which none opposite to each other, have one color and the other three one other color, thecube has 6 isometries.

For some larger subgroups a cube with that group as symmetry group is not possible with just coloring whole faces.One has to draw some pattern on the faces. Examples:• D2d: if one face has a line segment dividing the face into two equal rectangles, and the opposite has the same in

perpendicular direction, the cube has 8 isometries; there is a symmetry plane and 2-fold rotational symmetry withan axis at an angle of 45° to that plane, and, as a result, there is also another symmetry plane perpendicular to thefirst, and another axis of 2-fold rotational symmetry perpendicular to the first.

• Th: if each face has a line segment dividing the face into two equal rectangles, such that the line segments ofadjacent faces do not meet at the edge, the cube has 24 isometries: the even permutations of the body diagonalsand the same combined with inversion (x is mapped to −x).

• Td: if the cube consists of eight smaller cubes, four white and four black, put together alternatingly in all threestandard directions, the cube has again 24 isometries: this time the even permutations of the body diagonals andthe inverses of the other proper rotations.

• T: if each face has the same pattern with 2-fold rotational symmetry, say the letter S, such that at all edges a top ofone S meets a side of the other S, the cube has 12 isometries: the even permutations of the body diagonals.

The full symmetry of the cube (Oh) is preserved if and only if all faces have the same pattern such that the fullsymmetry of the square is preserved, with for the square a symmetry group of order 8.The full symmetry of the cube under proper rotations (O) is preserved if and only if all faces have the same patternwith 4-fold rotational symmetry.

Octahedral symmetry of the Bolza surfaceIn Riemann surface theory, the Bolza surface, sometimes called the Bolza curve, is obtained as the ramified doublecover of the Riemann sphere, with ramification locus at the set of vertices of the regular inscribed octahedron. Itsautomorphism group includes the hyperelliptic involution which flips the two sheets of the cover. The quotient bythe order 2 subgroup generated by the hyperelliptic involution yields precisely the group of symmetries of theoctahedron. Among the many remarkable properties of the Bolza surface is the fact that it maximizes the systoleamong all genus 2 hyperbolic surfaces.

Octahedral symmetry 18

Chiral solids with octahedral rotational symmetry

Snub hexahedron (Ccw) Pentagonal icositetrahedron

• Note to Pentagonal icositetrahedron: (Ccw) - note that, not very clear in the image, at some vertices 4 faces meet(in the edge of the image)

Archimedean solids

Name picture Faces Edges Vertices Vertex configuration

snub cubeor snub cuboctahedron (2 chiral forms)

(Video)

(Video)

38 32triangles6 squares

60 24 3,3,3,3,4

Catalan solids

Name picture Dual Archimedean solid Faces Edges Vertices Face Polygon

pentagonal icositetrahedron

(Video)(Video)

snub cube 24 60 38 irregular pentagon

Solids with full octahedral symmetry

Platonic solids

Octahedral symmetry 19

Name Picture Faces Edges Vertices Edges per face Facesmeeting

at each vertex

cube (hexahedron)

(Animation)

6 12 8 4 3

octahedron

(Animation)

8 12 6 3 4

Archimedean solids(semi-regular: vertex-uniform)

Name picture Faces Edges Vertices Vertex configuration

cuboctahedron(quasi-regular: vertex- and edge-uniform)

(Video)

14 8 triangles6 squares

24 12 3,4,3,4

truncated cubeor truncated hexahedron

(Video)

14 8 triangles6 octagons

36 24 3,8,8

truncated octahedron

(Video)

14 6 squares8 hexagons

36 24 4,6,6

rhombicuboctahedronor small rhombicuboctahedron

(Video)

26 8 triangles18 squares

48 24 3,4,4,4

truncated cuboctahedronor great rhombicuboctahedron

(Video)

26 12 squares8 hexagons6 octagons

72 48 4,6,8

Catalan solids(semi-regular duals: face-uniform)

Octahedral symmetry 20

Name picture Dual Archimedean solid Faces Edges Vertices Face polygon

rhombic dodecahedron(quasi-regular dual: face- and edge-uniform)

(Video)

cuboctahedron 12 24 14 rhombus

triakis octahedron

(Video)

truncated cube 24 36 14 isosceles triangle

tetrakis hexahedron

(Video)

truncated octahedron 24 36 14 isosceles triangle

deltoidal icositetrahedron

(Video)

rhombicuboctahedron 24 48 26 kite

disdyakis dodecahedronor hexakis octahedron

(Video)

truncated cuboctahedron 48 72 26 scalene triangle

Other

stella octangula

Icosahedral symmetry 21

Icosahedral symmetry

A Soccer ball, a common example of a sphericaltruncated icosahedron, has full icosahedral

symmetry.

A regular icosahedron has 60 rotational (or orientation-preserving)symmetries, and a symmetry order of 120 including transformationsthat combine a reflection and a rotation. A regular dodecahedron hasthe same set of symmetries, since it is the dual of the icosahedron.

The set of orientation-preserving symmetries forms a group referred toas A5 (the alternating group on 5 letters), and the full symmetry group(including reflections) is the product A5 × C2. The latter group is alsoknown as the Coxeter group H3.

As point group

The icosahedral rotation group I with fundamental domain

Apart from the two infinite series ofprismatic and antiprismatic symmetry,rotational icosahedral symmetry orchiral icosahedral symmetry of chiralobjects and full icosahedralsymmetry or achiral icosahedralsymmetry are the discrete pointsymmetries (or equivalently,symmetries on the sphere) with thelargest symmetry groups.

Icosahedral symmetry is notcompatible with translationalsymmetry, so there are no associatedcrystallographic point groups or spacegroups.

Icosahedral symmetry 22

Schönflies crystallographic notation Coxeter notation Orbifold notation Order

I [3,5]+ 532 60

Ih [3,5] *532 120

Presentations corresponding to the above are:

These correspond to the icosahedral groups (rotational and full) being the (2,3,5) triangle groups.The first presentation was given by William Rowan Hamilton in 1856, in his paper on Icosian Calculus.[1]

Note that other presentations are possible, for instance as an alternating group (for I).

Group structureThe icosahedral rotation group I is of order 60. The group I is isomorphic to A5, the alternating group of evenpermutations of five objects. This isomorphism can be realized by I acting on various compounds, notably thecompound of five cubes (which inscribe in the dodecahedron), the compound of five octahedra, or either of the twocompounds of five tetrahedra (which are enantiomorphs, and inscribe in the dodecahedron).The group contains 5 versions of Th with 20 versions of D3 (10 axes, 2 per axis), and 6 versions of D5.The full icosahedral group I

h has order 120. It has I as normal subgroup of index 2. The group Ih is isomorphic to I

× C2, or A5 × C2, with the inversion in the center corresponding to element (identity,-1), where C2 is writtenmultiplicatively.Ih acts on the compound of five cubes and the compound of five octahedra, but -1 acts as the identity (as cubes andoctahedra are centrally symmetric). It acts on the compound of ten tetrahedra: I acts on the two chiral halves(compounds of five tetrahedra), and -1 interchanges the two halves. Notably, it does not act as S5, and these groupsare not isomorphic; see below for details.The group contains 10 versions of D3d and 6 versions of D5d (symmetries like antiprisms).I is also isomorphic to PSL2(5), but Ih is not isomorphic to SL2(5).

Commonly confused groupsThe following groups all have order 120, but are not isomorphic:• S5, the symmetric group on 5 elements• Ih, the full icosahedral group (subject of this article, also known as H3)• 2I, the binary icosahedral groupThey correspond to the following short exact sequences (which do not split) and product

In words,• is a normal subgroup of • is a factor of , which is a direct product• is a quotient group of Note that has an exceptional irreducible 3-dimensional representation (as the icosahedral rotation group), but does not have an irreducible 3-dimensional representation, corresponding to the full icosahedral group not being the

Icosahedral symmetry 23

symmetric group.These can also be related to linear groups over the finite field with five elements, which exhibit the subgroups andcovering groups directly; none of these are the full icosahedral group:

• the projective special linear group;• the projective general linear group;• the special linear group.

Conjugacy classesThe conjugacy classes of I are:• identity• 12 × rotation by 72°, order 5• 12 × rotation by 144°, order 5• 20 × rotation by 120°, order 3• 15 × rotation by 180°, order 2Those of Ih include also each with inversion:• inversion• 12 × rotoreflection by 108°, order 10• 12 × rotoreflection by 36°, order 10• 20 × rotoreflection by 60°, order 6• 15 × reflection, order 2

Subgroups• Ih,I , Th and T• D2h• D5d, D3d• D5, D3 and D2• C2h• C5v, C3v and C2v• C5, C3 and C2• S10, S6 and S2=Ci• E and CsAll of these classes of subgroups are conjugate (i.e., all vertex stabilizers are conjugate), and admit geometricinterpretations.Note that the stabilizer of a vertex/edge/face/polyhedron and its opposite are equal, since is central.

Vertex stabilizers

Stabilizers of an opposite pair of vertices can be interpreted as stabilizers of the axis they generate.• vertex stabilizers in I give cyclic groups C3• vertex stabilizers in Ih give dihedral groups D3• stabilizers of an opposite pair of vertices in I give dihedral groups D3• stabilizers of an opposite pair of vertices in Ih give

Icosahedral symmetry 24

Edge stabilizers

Stabilizers of an opposite pair of edges can be interpreted as stabilizers of the rectangle they generate.• edges stabilizers in I give cyclic groups C2• edges stabilizers in Ih give Klein four-groups • stabilizers of a pair of edges in I give Klein four-groups ; there are 5 of these, given by rotation by

180° in 3 perpendicular axes.• stabilizers of a pair of edges in Ih give ; these are 5 of these, given by reflections in 3

perpendicular axes.

Face stabilizers

Stabilizers of an opposite pair of faces can be interpreted as stabilizers of the anti-prism they generate.• face stabilizers in I give cyclic groups C5• face stabilizers in Ih give dihedral groups D5• stabilizers of an opposite pair of faces in I give dihedral groups D5• stabilizers of an opposite pair of faces in Ih give

Polyhedron stabilizers

For each of these, there are 5 conjugate copies, and the conjugation action gives a map, indeed an isomorphism,.

• stabilizers of the inscribed tetrahedra in I are a copy of T• stabilizers of the inscribed tetrahedra in Ih are a copy of Th• stabilizers of the inscribed cubes (or opposite pair of tetrahedra, or octahedrons) in I are a copy of O• stabilizers of the inscribed cubes (or opposite pair of tetrahedra, or octahedrons) in Ih are a copy of Oh

Fundamental domainFundamental domains for the icosahedral rotation group and the full icosahedral group are given by:

The icosahedral rotation group I with fundamental domain The full icosahedral group Ih with fundamental domain

Icosahedral symmetry 25

Fundamental domain in the disdyakistriacontahedron

In the disdyakis triacontahedron one full face is a fundamental domain;other solids with the same symmetry can be obtained by adjusting theorientation of the faces, e.g. flattening selected subsets of faces tocombine each subset into one face, or replacing each face by multiplefaces, or a curved surface.

Solids with icosahedral symmetry

Full icosahedral symmetry

Platonic solids - regular polyhedra (all faces of the same type)

{5,3} {3,5}

Archimedean solids - polyhedra with more than one polygon face type.

3.10.10 4.6.10 5.6.6 3.4.5.4 3.5.3.5

Catalan solids - duals of the Archimedean solids.

V3.10.10 V4.6.10 V5.6.6 V3.4.5.4 V3.5.3.5

Other objects with icosahedral symmetry• Barth surfaces

Related geometriesIcosahedral symmetry is equivalently the projective special linear group PSL(2,5), and is the symmetry group of themodular curve X(5), and more generally PSL(2,p) is the symmetry group of the modular curve X(p). The modularcurve X(5) is geometrically a dodecahedron with a cusp at the center of each polygonal face, which demonstrates thesymmetry group.This geometry, and associated symmetry group, was studied by Felix Klein as the monodromy groups of a Belyi surface – a Riemann surface with a holomorphic map to the Riemann sphere, ramified only at 0, 1, and infinity (a

Icosahedral symmetry 26

Belyi function) – the cusps are the points lying over infinity, while the vertices and the centers of each edge lie over0 and 1; the degree of the covering (number of sheets) equals 5.This arose from his efforts to give a geometric setting for why icosahedral symmetry arose in the solution of thequintic equation, with the theory given in the famous (Klein 1888); a modern exposition is given in (Tóth 2002,Section 1.6, Additional Topic: Klein's Theory of the Icosahedron, p. 66 [2]).Klein's investigations continued with his discovery of order 7 and order 11 symmetries in (Klein 1878/79b) and(Klein 1879) (and associated coverings of degree 7 and 11) and dessins d'enfants, the first yielding the Klein quartic,whose associated geometry has a tiling by 24 heptagons (with a cusp at the center of each).Similar geometries occur for PSL(2,n) and more general groups for other modular curves.More exotically, there are special connections between the groups PSL(2,5) (order 60), PSL(2,7) (order 168) andPSL(2,11) (order 660), which also admit geometric interpretations – PSL(2,5) is the symmetries of the icosahedron(genus 0), PSL(2,7) of the Klein quartic (genus 3), and PSL(2,11) the buckyball surface (genus 70). These groupsform a "trinity" in the sense of Vladimir Arnold, which gives a framework for the various relationships; see trinitiesfor details.

See also• tetrahedral symmetry• octahedral symmetry• binary icosahedral group• Icosian Calculus

References[1] Sir William Rowan Hamilton (1856), "Memorandum respecting a new System of Roots of Unity" (http:/ / www. maths. tcd. ie/ pub/

HistMath/ People/ Hamilton/ Icosian/ NewSys. pdf), Philosophical Magazine 12: 446,[2] http:/ / books. google. com/ books?id=i76mmyvDHYUC& pg=PA66

• Klein, F. (1878). "Ueber die Transformation siebenter Ordnung der elliptischen Functionen (On the order-seventransformation of elliptic functions)" (http:/ / www. springerlink. com/ content/ j13026l720t560k8/ fulltext. pdf).Mathematische Annalen 14 (3): 428–471. doi:10.1007/BF01677143, English translation in The Eightfold Way,Silvio Levy, 1999 (Levy 1999).

• Klein, F. (1879), "Ueber die Transformation elfter Ordnung der elliptischen Functionen (On the eleventh ordertransformation of elliptic functions)", Mathematische Annalen 15: 533–555, doi:10.1007/BF02086276, collectedas pp. 140–165 in Oeuvres, Tome 3 (http:/ / mathdoc. emath. fr/ cgi-bin/ oetoc?id=OE_KLEIN__3)

• Klein, Felix (1888), Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree, Trübner &Co., ISBN 0486495280trans. George Gavin Morrice

• Tóth, Gábor (2002), Finite Möbius groups, minimal immersions of spheres, and moduli

Cyclic symmetries 27

Cyclic symmetriesThis article deals with the four infinite series of point groups in three dimensions (n≥1) with n-fold rotationalsymmetry about one axis (rotation by an angle of 360°/n does not change the object), and no other rotationalsymmetry (n=1 covers the cases of no rotational symmetry at all):Chiral:

• Cn

(nn) of order n - n-fold rotational symmetry (abstract group Cn); for n=1: no symmetry (trivial group)Achiral:

• Cnh

(n*) of order 2n - prismatic symmetry (abstract group Dn × C2); for n=1 this is denoted by Cs

(1*) andcalled reflection symmetry, also bilateral symmetry.

• Cnv

(*nn) of order 2n - pyramidal symmetry (abstract group Dn); in biology C2v is called biradial symmetry.For n=1 we have again Cs (1*).

• S2n

(n×) of order 2n (not to be confused with symmetric groups, for which the same notation is used; abstractgroup C2n); for n=1 we have S2 (1×), also denoted by C

i; this is inversion symmetry

They are the finite symmetry groups on a cone. For n = they correspond to four frieze groups. Schönfliesnotation is used, and, in parentheses, orbifold notation. The terms horizontal (h) and vertical (v) are used with respectto a vertical axis of rotation.Cnh (n*) has reflection symmetry with respect to a plane perpendicular to the n-fold rotation axis.Cnv (*nn) has vertical mirror planes. This is the symmetry group for a regular n-sided pyramid.S2n (n×) has a 2n-fold rotoreflection axis, also called 2n-fold improper rotation axis, i.e., the symmetry groupcontains a combination of a reflection in the horizontal plane and a rotation by an angle 180°/n. Thus, like Dnd, itcontains a number of improper rotations without containing the corresponding rotations.C2h (2*) and C2v (*22) of order 4 are two of the three 3D symmetry group types with the Klein four-group asabstract group. C2v applies e.g. for a rectangular tile with its top side different from its bottom side.

Examples

S2/C

i (1x): C

4v (*44): C

5v (*55):

Parallelepiped

Square pyramid

Elongated square pyramid Pentagonal pyramidhi,sophia

Reflection symmetry 28

Reflection symmetry

Figures with the axes of symmetry drawn in.

Reflection symmetry, reflectional symmetry, linesymmetry, mirror symmetry, mirror-imagesymmetry, or bilateral symmetry is symmetry withrespect to reflection.

In 2D there is an axis of symmetry, in 3D a plane ofsymmetry. An object or figure which isindistinguishable from its transformed image is calledmirror symmetric (see mirror image). Also see pattern..

The axis of symmetry or line of symmetry of atwo-dimensional figure is a line such that, for eachperpendicular constructed, if the perpendicularintersects the figure at a distance 'd' from the axis alongthe perpendicular, then there exists another intersectionof the figure and the perpendicular, at the same distance'd' from the axis, in the opposite direction along theperpendicular. Another way to think about it is that ifthe shape were to be folded in half over the axis, thetwo halves would be identical: the two halves are each other's mirror image. Thus a square has four axes ofsymmetry, because there are four different ways to fold it and have the edges all match. A circle has infinitely manyaxes of symmetry, for the same reason.

If the letter T is reflected along a vertical axis, it appears the same. Note that this is sometimes called horizontalsymmetry, and sometimes vertical symmetry. One can better use an unambiguous formulation, e.g. "T has a verticalsymmetry axis."(this may also be called a line of symmetry)The triangles with this symmetry are isosceles. The quadrilaterals with this symmetry are the kites and the isoscelestrapezoids.For each line or plane of reflection, the symmetry group is isomorphic with Cs (see point groups in threedimensions), one of the three types of order two (involutions), hence algebraically C2. The fundamental domain is ahalf-plane or half-space.In certain contexts there is rotational symmetry anyway. Then mirror-image symmetry is equivalent with inversionsymmetry; in such contexts in modern physics the term P-symmetry is used for both (P stands for parity).For more general types of reflection there are corresponding more general types of reflection symmetry. Examples:• with respect to a non-isometric affine involution (an oblique reflection in a line, plane, etc).• with respect to circle inversion.Mirrored symmetry is also found in the design of ancient structures, including Stonehenge.[1]

Reflection symmetry 29

See also• Rotational symmetry• Translational symmetry• Holstein–Herring method

References[1] Johnson, Anthony, Solving Stonehenge: The New Key to an Ancient Enigma. (Thames & Hudson, 2008) ISBN 978-0-500-05155-9

• Weyl, Hermann (1982). Symmetry. Princeton: Princeton University Press. ISBN 0-691-02374-3.

External links• Mapping with symmetry - source in Delphi (http:/ / republika. pl/ fraktal/ mapping. html)• Reflection Symmetry Examples (http:/ / www. mathsisfun. com/ geometry/ symmetry-reflection. html) from Math

Is Fun

Inverse (mathematics)In many contexts in mathematics the term inverse indicates the opposite of something. This word and its derivativesare used greatly in mathematics, as illustrated below.• Inverse element of an element x with respect to a binary operation * with identity element e is an element y such

that x * y = y * x = e. In particular,• the additive inverse of x is –x;• the multiplicative inverse of x is x–1.• Inverse function — inverse element with respect to function composition: a function that "reverses" the action

of a given function: f–1(f(x)) = x.• Inversion in a point — a geometric transform.• Circle inversion — another particular geometric transformation of a plane that maps the outside of a circle to the

inside and vice-versa.• Inverse limit — a notion in abstract algebra.• Inverse (logic) — ~p → ~q is the inverse of p → q.• Inverse matrix — inverse element with respect to matrix multiplication.• Pseudoinverse, a generalization of the inverse matrix.• Inverse proportion, also inversely proportional — a relationship between two variables x and y characterized by

the equation • Inverse problem — the task of identifying model parameters from observed data; see for example

• inverse scattering problem• inverse kinematics• inverse dynamics.

• Inverse perspective — the further the objects, the larger they are drawn.• Inversive ring geometry — classical projective geometry extended by ring theory• Inverse semigroup• Inverse of an element in a semigroup• Inverse-square law — the magnitude of a force is proportional to the inverse square of the distance.• Inverse transform sampling — generate some random numbers according to a given probability distribution.• Inverse chain rule method — related to integration and differentiation.

Inverse (mathematics) 30

• Inversion of elements, a pair of adjacent out-of-order elements of a permutation (viewed as a list).• Inverse relation

Point groups in three dimensionsIn geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the originfixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group ofall isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is asubgroup of the Euclidean group E(3) of all isometries.Symmetry groups of objects are isometry groups. Accordingly, analysis of isometry groups is analysis of possiblesymmetries. All isometries of a bounded 3D object have one or more common fixed points. We choose the origin asone of them.The symmetry group of an object is sometimes also called full symmetry group, as opposed to its rotation groupor proper symmetry group, the intersection of its full symmetry group and the rotation group SO(3) of the 3Dspace itself. The rotation group of an object is equal to its full symmetry group if and only if the object is chiral.The point groups in three dimensions are heavily used in chemistry, especially to describe the symmetries of amolecule and of molecular orbitals forming covalent bonds, and in this context they are also called molecular pointgroups.

Group structureSO(3) is a subgroup of E+(3), which consists of direct isometries, i.e., isometries preserving orientation; it containsthose that leave the origin fixed.O(3) is the direct product of SO(3) and the group generated by inversion (denoted by its matrix −I):

O(3) = SO(3) × { I , −I }Thus there is a 1-to-1 correspondence between all direct isometries and all indirect isometries, through inversion.Also there is a 1-to-1 correspondence between all groups of direct isometries H and all groups K of isometries thatcontain inversion:

K = H × { I , −I }H = K ∩ SO(3)

If a group of direct isometries H has a subgroup L of index 2, then, apart from the corresponding group containinginversion there is also a corresponding group that contains indirect isometries but no inversion:

M = L ∪ ( (H \ L) × { − I } )where isometry ( A , I ) is identified with A.Thus M is obtained from H by inverting the isometries in H \ L. This group M is as abstract group isomorphic withH. Conversely, for all isometry groups that contain indirect isometries but no inversion we can obtain a rotationgroup by inverting the indirect isometries. This is clarifying when categorizing isometry groups, see below.In 2D the cyclic group of k-fold rotations Ck is for every positive integer k a normal subgroup of O(2,R) andSO(2,R). Accordingly, in 3D, for every axis the cyclic group of k-fold rotations about that axis is a normal subgroupof the group of all rotations about that axis, and also of the group obtained by adding reflections in planes throughthe axis.

Point groups in three dimensions 31

3D isometries that leave origin fixedThe isometries of R3 that leave the origin fixed, forming the group O(3,R), can be categorized as follows:• SO(3,R):

• identity• rotation about an axis through the origin by an angle not equal to 180°• rotation about an axis through the origin by an angle of 180°

• the same with inversion (x is mapped to −x), i.e. respectively:• inversion• rotation about an axis by an angle not equal to 180°, combined with reflection in the plane through the origin

perpendicular to the axis• reflection in a plane through the origin

The 4th and 5th in particular, and in a wider sense the 6th also, are called improper rotations.See also the similar overview including translations.

ConjugacyWhen comparing the symmetry type of two objects, the origin is chosen for each separately, i.e. they need not havethe same center. Moreover, two objects are considered to be of the same symmetry type if their symmetry groups areconjugate subgroups of O(3) (two subgroups H1, H2 of a group G are conjugate, if there exists g ∈ G such that H1 =g−1H2g ).Thus two 3D objects have the same symmetry type:• if both have mirror symmetry, but with respect to a different mirror plane• if both have 3-fold rotational symmetry, but with respect to a different axis.In the case of multiple mirror planes and/or axes of rotation, two symmetry groups are of the same symmetry type ifand only if there is a single rotation mapping this whole structure of the first symmetry group to that of the second.The conjugacy definition would also allow a mirror image of the structure, but this is not needed, the structure itselfis achiral. For example, if a symmetry group contains a 3-fold axis of rotation, it contains rotations in two oppositedirections. (The structure is chiral for 11 pairs of space groups with a screw axis.)

Infinite isometry groupsWe restrict ourselves to isometry groups that are closed as topological subgroups of O(3). This excludes for examplethe group of rotations by an irrational number of turns about an axis.The whole O(3) is the symmetry group of spherical symmetry; SO(3) is the corresponding rotation group. The otherinfinite isometry groups consist of all rotations about an axis through the origin, and those with additionallyreflection in the planes through the axis, and/or reflection in the plane through the origin, perpendicular to the axis.Those with reflection in the planes through the axis, with or without reflection in the plane through the origin,perpendicular to the axis, are the symmetry groups for the two types of cylindrical symmetry.See also rotational symmetry with respect to any angle.

Point groups in three dimensions 32

Finite isometry groupsFor point groups, being finite corresponds to being discrete; infinite discrete groups as in the case of translationalsymmetry and glide reflectional symmetry do not apply.Symmetries in 3D that leave the origin fixed are fully characterized by symmetries on a sphere centered at the origin.For finite 3D point groups, see also spherical symmetry groups.Up to conjugacy the set of finite 3D point groups consists of:• 7 infinite series with at most one more-than-2-fold rotation axis; they are the finite symmetry groups on an infinite

cylinder, or equivalently, those on a finite cylinder.• 7 point groups with multiple 3-or-more-fold rotation axes; they can also be characterized as point groups with

multiple 3-fold rotation axes, because all 7 include these axes; with regard to 3-or-more-fold rotation axes thepossible combinations are:• 4×3• 4×3 and 3×4• 10×3 and 6×5

A selection of point groups is compatible with discrete translational symmetry: 27 from the 7 infinite series, and 5 ofthe 7 others, the 32 so-called crystallographic point groups. See also the crystallographic restriction theorem.

The seven infinite seriesThe infinite series have an index n, which can be any integer; in each series, the nth symmetry group contains n-foldrotational symmetry about an axis, i.e. symmetry with respect to a rotation by an angle 360°/n. n=1 covers the casesof no rotational symmetry at all. There are four series with no other axes of rotational symmetry, see cyclicsymmetries, and three with additional axes of 2-fold symmetry, see dihedral symmetry.For n = ∞ they correspond to the frieze groups. Schönflies notation is used, and, in parentheses, orbifold notation;the latter is not only conveniently related to its properties, but also to the order of the group, see below; it is a unifiednotation, also applicable for wallpaper groups and frieze groups.The 7 infinite series are:• Cn (nn ) of order n - n-fold rotational symmetry (abstract group Z

n ); for n = 1: no symmetry (trivial group)

• Cnh (n* ) of order 2n (for odd n abstract group Z2n = Zn × Z2 , for even n abstract group Zn × Z2 )• Cnv (*nn ) of order 2n - pyramidal symmetry (abstract group Dihn ); in biology C2v is called biradial symmetry.• Dn (22n ) of order 2n - dihedral symmetry (abstract group Dihn )• S2n (nx ) of order 2n (not to be confused with symmetric groups, for which the same notation is used; abstract

group Z2n )• Dnh (*22n ) of order 4n - prismatic symmetry (for odd n abstract group Dih2n = Dihn × Z2 ; for even n abstract

group Dihn × Z2 )• Dnd (or Dnv ) (2*n ) - antiprismatic symmetry of order 4n (abstract group Dih2n )The terms horizontal (h) and vertical (v), and the corresponding subscripts, refer to the additional mirror plane, thatcan be parallel to the rotation axis (vertical) or perpendicular to the rotation axis (horizontal).Involutional symmetry (abstract group Z2 ):• Ci - inversion symmetry• C2 - 2-fold rotational symmetry• Cs - reflection symmetry, also called bilateral symmetry.

Point groups in three dimensions 33

Patterns on a cylindrical bandillustrating the case n = 6 for each of the7 infinite families of point groups. Thesymmetry group of each pattern is the

indicated group.

The second of these is the first of the uniaxial groups (cyclic groups) Cn oforder n (also applicable in 2D), which are generated by a single rotation ofangle 360°/n. In addition to this, one may add a mirror plane perpendicular tothe axis, giving the group Cnh of order 2n, or a set of n mirror planescontaining the axis, giving the group Cnv, also of order 2n. The latter is thesymmetry group for a regular n-sided pyramid. A typical object withsymmetry group Cn or Dn is a propellor.

If both horizontal and vertical reflection planes are added, their intersectionsgive n axes of rotation through 180°, so the group is no longer uniaxial. Thisnew group of order 4n is called Dnh. Its subgroup of rotations is the dihedralgroup Dn of order 2n, which still has the 2-fold rotation axes perpendicular tothe primary rotation axis, but no mirror planes. Note that in 2D Dn includesreflections, which can also be viewed as flipping over flat objects withoutdistinction of front- and backside, but in 3D the two operations aredistinguished: the group contains "flipping over", not reflections.

There is one more group in this family, called Dnd (or Dnv), which has verticalmirror planes containing the main rotation axis, but instead of having ahorizontal mirror plane, it has an isometry that combines a reflection in thehorizontal plane and a rotation by an angle 180°/n. Dnh is the symmetry groupfor a regular n-sided prisms and also for a regular n-sided bipyramid. Dnd isthe symmetry group for a regular n-sided antiprism, and also for a regularn-sided trapezohedron. Dn is the symmetry group of a partially rotated prism.

Sn is generated by the combination of a reflection in the horizontal plane and arotation by an angle 360°/n. For n odd this is equal to the group generated bythe two separately, Cnh of order 2n, and therefore the notation Sn is not needed;however, for n even it is distinct, and of order n. Like Dnd it contains a numberof improper rotations without containing the corresponding rotations.

All symmetry groups in the 7 infinite series are different, except for the following four pairs of mutually equal ones:• C1h and C1v: group of order 2 with a single reflection (Cs )• D1 and C2: group of order 2 with a single 180° rotation• D1h and C2v: group of order 4 with a reflection in a plane and a 180° rotation through a line in that plane• D1d and C2h: group of order 4 with a reflection in a plane and a 180° rotation through a line perpendicular to that

planeS2 is the group of order 2 with a single inversion (Ci )"Equal" is meant here as the same up to conjugacy in space. This is stronger than "up to algebraic isomorphism". Forexample, there are three different groups of order two in the first sense, but there is only one in the second sense.Similarly, e.g. S2n is algebraically isomorphic with Z2n.

Point groups in three dimensions 34

The seven remaining point groupsThe remaining point groups are said to be of very high or polyhedral symmetry because they have more than onerotation axis of order greater than 2. Using Cn to denote an axis of rotation through 360°/n and Sn to denote an axis ofimproper rotation through the same, the groups are:• T (332) of order 12 - chiral tetrahedral symmetry. There are four C3 axes, each through two vertices of a cube

(body diagonals) or one of a regular tetrahedron, and three C2 axes, through the centers of the cube's faces, or themidpoints of the tetrahedron's edges. This group is isomorphic to A4, the alternating group on 4 elements, and isthe rotation group for a regular tetrahedron.

• Td

(*332) of order 24 - full tetrahedral symmetry. This group has the same rotation axes as T, but with sixmirror planes, each containing two edges of the cube or one edge of the tetrahedron, a single C2 axis and two C3axes. The C2 axes are now actually S4 axes. This group is the symmetry group for a regular tetrahedron. Td isisomorphic to S4, the symmetric group on 4 letters. See also the isometries of the regular tetrahedron.

• Th

(3*2) of order 24 - pyritohedral symmetry.

The structure of a volleyball has Th symmetry.

This group has the same rotation axes as T, with mirror planes parallelto the cube faces. The C3 axes become S6 axes, and there is inversionsymmetry. Th is isomorphic to A4 × C2. It is the symmetry of a cubewith on each face a line segment dividing the face into two equalrectangles, such that the line segments of adjacent faces do not meet atthe edge. The symmetries correspond to the even permutations of thebody diagonals and the same combined with inversion. It is also thesymmetry of a pyritohedron, which is similar to the cube described,with each rectangle replaced by a pentagon with one symmetry axisand 4 equal sides and 1 different side (the one corresponding to the linesegment dividing the cube's face); i.e., the cube's faces bulge out at thedividing line and become narrower there. It is a subgroup of the fullicosahedral symmetry group (as isometry group, not just as abstractgroup), with 4 of the 10 3-fold axes.

• O (432) of order 24 - chiral octahedral symmetry. This group is like T, but the C2 axes are now C4 axes, andadditionally there are 6 C2 axes, through the midpoints of the edges of the cube. This group is also isomorphic toS4, and is the rotation group of the cube and octahedron.

• Oh

(*432) of order 48 - full octahedral symmetry. This group has the same rotation axes as O, but with mirrorplanes, comprising both the mirror planes of Td and Th. This group is isomorphic to S4 × C2, and is the symmetrygroup of the cube and octahedron. See also the isometries of the cube.

• I (532) of order 60 - chiral icosahedral symmetry; the rotation group of the icosahedron and the dodecahedron.It is a normal subgroup of index 2 in the full group of symmetries I

h. The group I is isomorphic to A5, the

alternating group on 5 letters. The group contains 10 versions of D3 and 6 versions of D5 (rotational symmetrieslike prisms and antiprisms).

• Ih

(*532) of order 120 - full icosahedral symmetry; the symmetry group of the icosahedron and thedodecahedron. The group I

h is isomorphic to A5 × C2. The group contains 10 versions of D3d and 6 versions of

D5d (symmetries like antiprisms).

Point groups in three dimensions 35

Relation between orbifold notation and orderThe order of each group is 2 divided by the orbifold Euler characteristic; the latter is 2 minus the sum of the featurevalues, assigned as follows:• n without or before * counts as (n−1)/n• n after * counts as (n−1)/(2n)• * and x count as 1This can also be applied for wallpaper groups and frieze groups: for them, the sum of the feature values is 2, givingan infinite order; see orbifold Euler characteristic for wallpaper groups

Rotation groupsThe rotation groups, i.e. the finite subgroups of SO(3), are: the cyclic groups Cn (the rotation group of a regularpyramid), the dihedral groups Dn (the rotation group of a regular prism, or regular bipyramid), and the rotationgroups T, O and I of a regular tetrahedron, octahedron/cube and icosahedron/dodecahedron.In particular, the dihedral groups D3, D4 etc. are the rotation groups of plane regular polygons embedded inthree-dimensional space, and such a figure may be considered as a degenerate regular prism. Therefore it is alsocalled a dihedron (Greek: solid with two faces), which explains the name dihedral group.• An object with symmetry group Cn, Cnh, Cnv or S2n has rotation group Cn.• An object with symmetry group Dn, Dnh, or Dnd has rotation group Dn.• An object with one of the other seven symmetry groups has as rotation group the corresponding one without

subscript: T, O or I.The rotation group of an object is equal to its full symmetry group if and only if the object is chiral. In other words,the chiral objects are those with their symmetry group in the list of rotation groups.

Correspondence between rotation groups and other groupsThe following groups contain inversion:• Cnh and Dnh for even n• S2n and Dnd for odd n (S2 = Ci is the group generated by inversion; D1d = C2h)• Th, Oh, and IhAs explained above, there is a 1-to-1 correspondence between these groups and all rotation groups:• Cnh for even n and S2n for odd n correspond to Cn• Dnh for even n and Dnd for odd n correspond to Dn• Th, Oh, and Ih correspond to T, O, and I, respectively.The other groups contain indirect isometries, but not inversion:• Cnv• Cnh and Dnh for odd n• S2n and Dnd for even n• TdThey all correspond to a rotation group H and a subgroup L of index 2 in the sense that they are obtained from H byinverting the isometries in H \ L, as explained above:• Cn is subgroup of Dn of index 2, giving Cnv• Cn is subgroup of C2n of index 2, giving Cnh for odd n and S2n for even n• Dn is subgroup of D2n of index 2, giving Dnh for odd n and Dnd for even n• T is subgroup of O of index 2, giving Td

Point groups in three dimensions 36

Maximal symmetriesThere are two discrete point groups with the property that no discrete point group has it as proper subgroup: Oh andIh. Their largest common subgroup is Th. The two groups are obtained from it by changing 2-fold rotationalsymmetry to 4-fold, and adding 5-fold symmetry, respectively. Alternatively the two groups are generated by addingfor each a reflection plane to Th.There are two crystallographic point groups with the property that no crystallographic point group has it as propersubgroup: Oh and D6h. Their maximal common subgroups, depending on orientation, are D3d and D2h.

The groups arranged by abstract group typeBelow the groups explained above are arranged by abstract group type.The smallest abstract groups that are not any symmetry group in 3D, are the quaternion group (of order 8), thedicyclic group Dic3 (of order 12), and 10 of the 14 groups of order 16.The column "# of order 2 elements" in the following tables shows the total number of isometry subgroups of typesC2 , Ci , Cs. This total number is one of the characteristics helping to distinguish the various abstract group types,while their isometry type helps to distinguish the various isometry groups of the same abstract group.Within the possibilities of isometry groups in 3D, there are infinitely many abstract group types with 0, 1 and 3elements of order 2, there are two with 2n + 1 elements of order 2, and there are three with 2n + 3 elements of order2 (for each n ≥ 2 ). There is never a positive even number of elements of order 2.

Symmetry groups in 3D that are cyclic as abstract groupThe symmetry group for n-fold rotational symmetry is Cn; its abstract group type is cyclic group Zn , which is alsodenoted by Cn. However, there are two more infinite series of symmetry groups with this abstract group type:• For even order 2n there is the group S2n (Schoenflies notation) generated by a rotation by an angle 180°/n about

an axis, combined with a reflection in the plane perpendicular to the axis. For S2 the notation Ci is used; it isgenerated by inversion.

• For any order 2n where n is odd, we have Cnh; it has an n-fold rotation axis, and a perpendicular plane ofreflection. It is generated by a rotation by an angle 360°/n about the axis, combined with the reflection. For C1hthe notation Cs is used; it is generated by reflection in a plane.

Thus we have, with bolding of the 10 cyclic crystallographic point groups, for which the crystallographic restrictionapplies:

Order Isometry groups Abstract group # of order 2 elements

1 C1

Z1 0

2 C2

, Ci , C

sZ2 1

3 C3

Z3 0

4 C4

, S4

Z4 1

5 C5 Z5 0

6 C6

, S6

, C3h

Z6 = Z3 × Z2 1

7 C7 Z7 0

8 C8 , S8 Z8 1

9 C9 Z9 0

10 C10 , S10 , C5h Z10 = Z5 × Z2 1

Point groups in three dimensions 37

etc.

Symmetry groups in 3D that are dihedral as abstract groupIn 2D dihedral group Dn includes reflections, which can also be viewed as flipping over flat objects withoutdistinction of front- and backside.However, in 3D the two operations are distinguished: the symmetry group denoted by Dn contains n 2-fold axesperpendicular to the n-fold axis, not reflections. Dn is the rotation group of the n-sided prism with regular base, andn-sided bipyramid with regular base, and also of a regular, n-sided antiprism and of a regular, n-sided trapezohedron.The group is also the full symmetry group of such objects after making them chiral by e.g. an identical chiralmarking on every face, or some modification in the shape.The abstract group type is dihedral group Dihn, which is also denoted by Dn. However, there are three more infiniteseries of symmetry groups with this abstract group type:• Cnv of order 2n, the symmetry group of a regular n-sided pyramid• Dnd of order 4n, the symmetry group of a regular n-sided antiprism• Dnh of order 4n for odd n. For n = 1 we get D2, already covered above, so n ≥ 3.Note the following property:

Dih4n+2 Dih2n+1 × Z2Thus we have, with bolding of the 12 crystallographic point groups, and writing D1d as the equivalent C2h:

Order Isometry groups Abstract group # of order 2 elements

4 D2

, C2v

, C2h

Dih2 = Z2 × Z2 3

6 D3

, C3v

Dih3 3

8 D4

, C4v

, D2d

Dih4 5

10 D5 , C5v Dih5 5

12 D6

, C6v

, D3d

, D3h

Dih6 = Dih3 × Z2 7

14 D7 , C7v Dih7 7

16 D8 , C8v , D4d Dih8 9

18 D9 , C9v Dih9 9

etc.

OtherC2n,h of order 4n is of abstract group type Z2n × Z2. For n = 1 we get Dih2 , already covered above, so n ≥ 2.Thus we have, with bolding of the 2 cyclic crystallographic point groups:

Point groups in three dimensions 38

Order Isometry group Abstract group # of order 2 elements Cycle diagram

8 C4h

Z4 × Z2 3

12 C6h

Z6 × Z2 = Z3 × Z2 × Z2 = Z3 × Dih2 3

16 C8h Z8 × Z2 3

20 C10h Z10 × Z2 = Z5 × Z2 × Z2 3

etc.Dnh of order 4n is of abstract group type Dihn × Z2. For odd n this is already covered above, so we have here D2nh oforder 8n, which is of abstract group type Dih2n × Z2 (n≥1).Thus we have, with bolding of the 3 dihedral crystallographic point groups:

Order Isometry group Abstract group # of order 2 elements Cycle diagram

8 D2h

Dih2 × Z2 7

16 D4h

Dih4 × Z2 11

24 D6h

Dih6 × Z2 15

32 D8h Dih8 × Z2 19

etc.The remaining seven are, with bolding of the 5 crystallographic point groups (see also above):• order 12: of type A4 (alternating group): T• order 24:

• of type S4 (symmetric group, not to be confused with the symmetry group with this notation): Td, O

• of type A4 × Z2: Th

.• order 48, of type S4 × Z2: O

h• order 60, of type A5: I• order 120, of type A5 × Z2: IhSee also icosahedral symmetry.

Point groups in three dimensions 39

Impossible discrete symmetriesSince the overview is exhaustive, it also shows implicitly what is not possible as discrete symmetry group. Forexample:• a C6 axis in one direction and a C3 in another• a C5 axis in one direction and a C4 in another• a C3 axis in one direction and another C3 axis in a perpendicular directionetc.

Fundamental domainThe fundamental domain of a point group is a conic solid. An object with a given symmetry in a given orientation ischaracterized by the fundamental domain. If the object is a surface it is characterized by a surface in the fundamentaldomain continuing to its radial bordal faces or surface. If the copies of the surface do not fit, radial faces or surfacescan be added. They fit anyway if the fundamental domain is bounded by reflection planes.For a polyhedron this surface in the fundamental domain can be part of an arbitrary plane. For example, in thedisdyakis triacontahedron one full face is a fundamental domain. Adjusting the orientation of the plane gives variouspossibilities of combining two or more adjacent faces to one, giving various other polyhedra with the samesymmetry. The polyhedron is convex if the surface fits to its copies and the radial line perpendicular to the plane isin the fundamental domain.Also the surface in the fundamental domain may be composed of multiple faces.

Binary polyhedral groupsThe map Spin(3) → SO(3) is the double cover of the rotation group by the spin group in 3 dimensions. (This is theonly connected cover of SO(3), since Spin(3) is simply connected.) By the lattice theorem, there is a Galoisconnection between subgroups of Spin(3) and subgroups of SO(3) (rotational point groups): the image of a subgroupof Spin(3) is a rotational point group, and the preimage of a point group is a subgroup of Spin(3).The preimage of a finite point group is called a binary polyhedral group, and is called by the same name as itspoint group, with the prefix binary. For instance, the preimage of the icosahedral group is the binary icosahedralgroup.The binary polyhedral groups are:• : binary cyclic group of an (n + 1)-gon• : binary dihedral group of an n-gon• : binary tetrahedral group• : binary octahedral group• : binary icosahedral groupThese are classified by the ADE classification, and the quotient of C2 by the action of a binary polyhedral group is aDu Val singularity.[1]

For point groups that reverse orientation, the situation is more complicated, as there are two pin groups, so there aretwo possible binary groups corresponding to a given point group.Note that this is a covering of groups, not a covering of spaces – the sphere is simply connected, and thus has nocovering spaces. There is thus no notion of a "binary polyhedron" that covers a 3-dimensional polyhedron. Binarypolyhedral groups are discrete subgroups of a Spin group, and under a representation of the spin group act on avector space, and may stabilize a polyhedron in this representation – under the map Spin(3) → SO(3) they act on thesame polyhedron that the underlying (non-binary) group acts on, while under spin representations or otherrepresentations they may stabilize other polyhedra.

Point groups in three dimensions 40

This is in contrast to projective polyhedra – the sphere does cover projective space (and also lens spaces), and thus atessellation of projective space or lens space yields a distinct notion of polyhedron.

Footnotes[1] Du Val Singularities, by Igor Burban (http:/ / enriques. mathematik. uni-mainz. de/ burban/ singul. pdf)

References• Coxeter, H. S. M. (1974), "7 The Binary Polyhedral Groups", Regular Complex Polytopes, Cambridge University

Press, pp.  73–82 (http:/ / books. google. com/ books?id=9BY9AAAAIAAJ& pg=PA73).• Conway, John Horton; Huson, Daniel H. (2002), "The Orbifold Notation for Two-Dimensional Groups",

Structural Chemistry (Springer Netherlands) 13 (3): 247–257, doi:10.1023/A:1015851621002

External links• Graphic overview of the 32 crystallographic point groups (http:/ / newton. ex. ac. uk/ research/ qsystems/ people/

goss/ symmetry/ Solids. html) - form the first parts (apart from skipping n=5) of the 7 infinite series and 5 of the 7separate 3D point groups

• Overview of properties of point groups (http:/ / newton. ex. ac. uk/ research/ qsystems/ people/ goss/ symmetry/CC_All. html)

• Simplest Canonical Polyhedra of Each Symmetry Type (http:/ / homepage. mac. com/ dmccooey/ polyhedra/Simplest. html) (uses Java)

Cyclic groupIn group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group hasan element g (called a "generator" of the group) such that, when written multiplicatively, every element of the groupis a power of g (a multiple of g when the notation is additive).

Definition

The six 6th complex roots of unity form acyclic group under multiplication. z is aprimitive element, but z2 is not, because

the odd powers of z are not a power of z2.

A group G is called cyclic if there exists an element g in G such that G = <g>= { gn | n is an integer }. Since any group generated by an element in a groupis a subgroup of that group, showing that the only subgroup of a group G thatcontains g is G itself suffices to show that G is cyclic.

For example, if G = { g0, g1, g2, g3, g4, g5 } is a group, then g6 = g0, and G iscyclic. In fact, G is essentially the same as (that is, isomorphic to) the set { 0,1, 2, 3, 4, 5 } with addition modulo 6. For example, 1 + 2 = 3 (mod 6)corresponds to g1·g2 = g3, and 2 + 5 = 1 (mod 6) corresponds to g2·g5 = g7 =g1, and so on. One can use the isomorphism φ defined by φ(gi) = i.

For every positive integer n there is exactly one cyclic group (up toisomorphism) whose order is n, and there is exactly one infinite cyclic group(the integers under addition). Hence, the cyclic groups are the simplest groupsand they are completely classified.

The name "cyclic" may be misleading: it is possible to generate infinitely many elements and not form any literal cycles; that is, every is distinct. (It can be said that it has one infinitely long cycle.) A group generated in this

Cyclic group 41

way is called an infinite cyclic group, and is isomorphic to the additive group of integers Z.Furthermore, the circle group (whose elements are uncountable) is not a cyclic group—a cyclic group always hascountable elements.Since the cyclic groups are abelian, they are often written additively and denoted Zn. However, this notation can beproblematic for number theorists because it conflicts with the usual notation for p-adic number rings or localizationat a prime ideal. The quotient notations Z/nZ, Z/n, and Z/(n) are standard alternatives. We adopt the first of thesehere to avoid the collision of notation. See also the section Subgroups and notation below.One may write the group multiplicatively, and denote it by Cn, where n is the order (which can be ∞). For example,g3g4 = g2 in C5, whereas 3 + 4 = 2 in Z/5Z.

PropertiesThe fundamental theorem of cyclic groups states that if G is a cyclic group of order n then every subgroup of G iscyclic. Moreover, the order of any subgroup of G is a divisor of n and for each positive divisor k of n the group G hasexactly one subgroup of order k. This property characterizes finite cyclic groups: a group of order n is cyclic if andonly if for every divisor d of n the group has at most one subgroup of order d. Sometimes the equivalent statement isused: a group of order n is cyclic if and only if for every divisor d of n the group has exactly one subgroup of order d.Every finite cyclic group is isomorphic to the group { [0], [1], [2], ..., [n − 1] } of integers modulo n under addition,and any infinite cyclic group is isomorphic to Z (the set of all integers) under addition. Thus, one only needs to lookat such groups to understand the properties of cyclic groups in general. Hence, cyclic groups are one of the simplestgroups to study and a number of nice properties are known.Given a cyclic group G of order n (n may be infinity) and for every g in G,• G is abelian; that is, their group operation is commutative: gh = hg (for all h in G). This is so since g + h mod n =

h + g mod n.• If n is finite, then gn = g0 is the identity element of the group, since kn mod n = 0 for any integer k.• If n = ∞, then there are exactly two elements that generate the group on their own: namely 1 and −1 for Z• If n is finite, then there are exactly φ(n) elements that generate the group on their own, where φ is the Euler

totient function• Every subgroup of G is cyclic. Indeed, each finite subgroup of G is a group of { 0, 1, 2, 3, ... m − 1} with addition

modulo m. And each infinite subgroup of G is mZ for some m, which is bijective to (so isomorphic to) Z.• Gn is isomorphic to Z/nZ (factor group of Z over nZ) since Z/nZ = {0 + nZ, 1 + nZ, 2 + nZ, 3 + nZ, 4 + nZ, ..., n

− 1 + nZ} { 0, 1, 2, 3, 4, ..., n − 1} under addition modulo n.More generally, if d is a divisor of n, then the number of elements in Z/n which have order d is φ(d). The order of theresidue class of m is n / gcd(n,m).If p is a prime number, then the only group (up to isomorphism) with p elements is the cyclic group Cp or Z/pZ.There are more numbers with the same property, see cyclic number.The direct product of two cyclic groups Z/nZ and Z/mZ is cyclic if and only if n and m are coprime. Thus e.g. Z/12Zis the direct product of Z/3Z and Z/4Z, but not the direct product of Z/6Z and Z/2Z.The definition immediately implies that cyclic groups have very simple group presentation C∞ = < x | > and Cn = < x| xn > for finite n.A primary cyclic group is a group of the form Z/pk where p is a prime number. The fundamental theorem of abeliangroups states that every finitely generated abelian group is the direct product of finitely many finite primary cyclicand infinite cyclic groups.Z/nZ and Z are also commutative rings. If p is a prime, then Z/pZ is a finite field, also denoted by Fp or GF(p).Every field with p elements is isomorphic to this one.

Cyclic group 42

The units of the ring Z/nZ are the numbers coprime to n. They form a group under multiplication modulo n withφ(n) elements (see above). It is written as (Z/nZ)×. For example, when n = 6, we get (Z/nZ)× = {1,5}. When n = 8,we get (Z/nZ)× = {1,3,5,7}.In fact, it is known that (Z/nZ)× is cyclic if and only if n is 1 or 2 or 4 or pk or 2 pk for an odd prime number p and k≥ 1, in which case every generator of (Z/nZ)× is called a primitive root modulo n. Thus, (Z/nZ)× is cyclic for n = 6,but not for n = 8, where it is instead isomorphic to the Klein four-group.The group (Z/pZ)× is cyclic with p − 1 elements for every prime p, and is also written (Z/pZ)* because it consists ofthe non-zero elements. More generally, every finite subgroup of the multiplicative group of any field is cyclic.

ExamplesIn 2D and 3D the symmetry group for n-fold rotational symmetry is Cn, of abstract group type Zn. In 3D there arealso other symmetry groups which are algebraically the same, see Symmetry groups in 3D that are cyclic as abstractgroup.Note that the group S1 of all rotations of a circle (the circle group) is not cyclic, since it is not even countable.The nth roots of unity form a cyclic group of order n under multiplication. e.g.,

where and a group of undermultiplication is cyclic.The Galois group of every finite field extension of a finite field is finite and cyclic; conversely, given a finite field Fand a finite cyclic group G, there is a finite field extension of F whose Galois group is G.

RepresentationThe cycle graphs of finite cyclic groups are all n-sided polygons with the elements at the vertices. The dark vertex inthe cycle graphs below stand for the identity element, and the other vertices are the other elements of the group. Acycle consists of successive powers of either of the elements connected to the identity element.

C1 C2 C3 C4 C5 C6 C7 C8

The representation theory of the cyclic group is a critical base case for the representation theory of more generalfinite groups. In the complex case, a representation of a cyclic group decomposes into a direct sum of linearcharacters, making the connection between character theory and representation theory transparent. In the positivecharacteristic case, the indecomposable representations of the cyclic group form a model and inductive basis for therepresentation theory of groups with cyclic Sylow subgroups and more generally the representation theory of blocksof cyclic defect.

Subgroups and notationAll subgroups and quotient groups of cyclic groups are cyclic. Specifically, all subgroups of Z are of the form mZ, with m an integer ≥0. All of these subgroups are different, and apart from the trivial group (for m=0) all are isomorphic to Z. The lattice of subgroups of Z is isomorphic to the dual of the lattice of natural numbers ordered by divisibility. All factor groups of Z are finite, except for the trivial exception Z/{0} = Z/0Z. For every positive divisor d of n, the quotient group Z/nZ has precisely one subgroup of order d, the one generated by the residue class of n/d. There are no other subgroups. The lattice of subgroups is thus isomorphic to the set of divisors of n, ordered by

Cyclic group 43

divisibility. In particular, a cyclic group is simple if and only if its order (the number of its elements) is prime.[1]

Using the quotient group formalism, Z/nZ is a standard notation for the additive cyclic group with n elements. Inring terminology, the subgroup nZ is also the ideal (n), so the quotient can also be written Z/(n) or Z/n without abuseof notation. These alternatives do not conflict with the notation for the p-adic integers. The last form is very commonin informal calculations; it has the additional advantage that it reads the same way that the group or ring is oftendescribed verbally, "Zee mod en".As a practical problem, one may be given a finite subgroup C of order n, generated by an element g, and asked tofind the size m of the subgroup generated by gk for some integer k. Here m will be the smallest integer > 0 such thatmk is divisible by n. It is therefore n/m where m = (k, n) is the greatest common divisor of k and n. Put another way,the index of the subgroup generated by gk is m. This reasoning is known as the index calculus algorithm, in numbertheory.

EndomorphismsThe endomorphism ring of the abelian group Z/nZ is isomorphic to Z/nZ itself as a ring. Under this isomorphism,the number r corresponds to the endomorphism of Z/nZ that maps each element to the sum of r copies of it. This is abijection if and only if r is coprime with n, so the automorphism group of Z/nZ is isomorphic to the unit group(Z/nZ)× (see above).Similarly, the endomorphism ring of the additive group Z is isomorphic to the ring Z. Its automorphism group isisomorphic to the group of units of the ring Z, i.e. to {−1, +1} C2.

Virtually cyclic groupsA group is called virtually cyclic if it contains a cyclic subgroup of finite index (the number of cosets that thesubgroup has). In other words, any element in a virtually cyclic group can be arrived at by applying a member of thecyclic subgroup to a member in a certain finite set. Every cyclic group is virtually cyclic, as is every finite group. Itis known that a finitely generated discrete group with exactly two ends is virtually cyclic (for instance the product ofZ/n and Z). Every abelian subgroup of a Gromov hyperbolic group is virtually cyclic.

External links• An introduction to cyclic groups [2]

Notes[1] Gannon (2006), p. 18 (http:/ / books. google. com/ books?id=ehrUt21SnsoC& pg=PA18& dq="Zn+ is+ simple+ iff+ n+ is+ prime")[2] http:/ / members. tripod. com/ ~dogschool/ cyclic. html

References• Gallian, Joseph (1998) (in English), Contemporary abstract algebra (4th ed.), Boston: Houghton Mifflin,

ISBN 978-0-669-86179-2, especially chapter 4.• Herstein, I. N. (1996), Abstract algebra (3rd ed.), Prentice Hall, MR1375019, ISBN 978-0-13-374562-7,

especially pages 53–60.• Gannon, Terry (2006). Moonshine beyond the monster: the bridge connecting algebra, modular forms and

physics. Cambridge monographs on mathematical physics. Cambridge University Press. ISBN 9780521835312.

Dihedral group 44

Dihedral group

This snowflake has the dihedral symmetry of aregular hexagon.

In mathematics, a dihedral group is the group of symmetries of aregular polygon, including both rotations and reflections.[1] Dihedralgroups are among the simplest examples of finite groups, and they playan important role in group theory, geometry, and chemistry.

See also: Dihedral symmetry in three dimensions.

Notation

There are two competing notations for the dihedral group associated toa polygon with n sides. In geometry the group is denoted Dn, while inalgebra the same group is denoted by D2n to indicate the number ofelements.

In this article, Dn (and sometimes Dihn) refers to the symmetries of aregular polygon with n sides.

Definition

Elements

The six reflection symmetries of a regularhexagon

A regular polygon with n sides has 2n different symmetries: nrotational symmetries and n reflection symmetries. The associatedrotations and reflections make up the dihedral group Dn. If n is oddeach axis of symmetry connects the mid-point of one side to theopposite vertex. If n is even there are n/2 axes of symmetry connectingthe mid-points of opposite sides and n/2 axes of symmetry connectingopposite vertices. In either case, there are n axes of symmetryaltogether and 2n elements in the symmetry group. Reflecting in oneaxis of symmetry followed by reflecting in another axis of symmetryproduces a rotation through twice the angle between the axes. Thefollowing picture shows the effect of the sixteen elements of D8 on astop sign:

The first row shows the effect of the eight rotations, and the second row shows the effect of the eight reflections.

Dihedral group 45

Group structureAs with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry. Thisoperation gives the symmetries of a polygon the algebraic structure of a finite group.

The composition of these two reflections is arotation.

The following Cayley table shows the effect of composition in thegroup D3 (the symmetries of an equilateral triangle). R0 denotes theidentity; R1 and R2 denote counterclockwise rotations by 120 and 240degrees; and S0, S1, and S2 denote reflections across the three linesshown in the picture to the right.

R0 R1 R2 S0 S1 S2R0 R0 R1 R2 S0 S1 S2R1 R1 R2 R0 S1 S2 S0R2 R2 R0 R1 S2 S0 S1S0 S0 S2 S1 R0 R2 R1S1 S1 S0 S2 R1 R0 R2S2 S2 S1 S0 R2 R1 R0

For example, S2S1 = R1 because the reflection S1 followed by the reflection S2 results in a 120-degree rotation. (Thisis the normal backwards order for composition.) Note that the composition operation is not commutative.In general, the group Dn has elements R0,...,Rn−1 and S0,...,Sn−1, with composition given by the following formulae:

In all cases, addition and subtraction of subscripts should be performed using modular arithmetic with modulus n.

Dihedral group 46

Matrix representation

The symmetries of this pentagon are lineartransformations.

If we center the regular polygon at the origin, then elements of thedihedral group act as linear transformations of the plane. This lets usrepresent elements of Dn as matrices, with composition being matrixmultiplication. This is an example of a (2-dimensional) grouprepresentation.

For example, the elements of the group D4 can be represented by thefollowing eight matrices:

In general, the matrices for elements of Dn have the following form:

Rk is a rotation matrix, expressing a counterclockwise rotation through an angle of 2πk ⁄ n. Sk is a reflection across aline that makes an angle of πk ⁄ n with the x-axis.

Small dihedral groupsFor n = 1 we have Dih1. This notation is rarely used except in the framework of the series, because it is equal to Z2.For n = 2 we have Dih2, the Klein four-group. Both are exceptional within the series:• They are abelian; for all other values of n the group Dihn is not abelian.• They are not subgroups of the symmetric group Sn, corresponding to the fact that 2n > n ! for these n.The cycle graphs of dihedral groups consist of an n-element cycle and n 2-element cycles. The dark vertex in thecycle graphs below of various dihedral groups stand for the identity element, and the other vertices are the otherelements of the group. A cycle consists of successive powers of either of the elements connected to the identityelement.

Dih1

Dih2

Dih3

Dih4

Dih5

Dih6

Dih7

Dihedral group 47

The dihedral group as symmetry group in 2D and rotation group in 3DAn example of abstract group Dihn, and a common way to visualize it, is the group Dn of Euclidean plane isometrieswhich keep the origin fixed. These groups form one of the two series of discrete point groups in two dimensions. Dnconsists of n rotations of multiples of 360°/n about the origin, and reflections across n lines through the origin,making angles of multiples of 180°/n with each other. This is the symmetry group of a regular polygon with n sides(for n ≥3, and also for the degenerate case n = 2, where we have a line segment in the plane).Dihedral group Dn is generated by a rotation r of order n and a reflection s of order 2 such that

(in geometric terms: in the mirror a rotation looks like an inverse rotation).In matrix form, an anti-clockwise rotation and a reflection in the x-axis are given by

(in terms of complex numbers: multiplication by and complex conjugation).By setting

and defining and for we can write the product rules for  Dn as

(Compare coordinate rotations and reflections.)The dihedral group D2 is generated by the rotation r of 180 degrees, and the reflection s across the x-axis. Theelements of D2 can then be represented as {e, r, s, rs}, where e is the identity or null transformation and rs is thereflection across the y-axis.

The four elements of D2 (x-axis is vertical here)

D2 is isomorphic to the Kleinfour-group.

If the order of Dn is greater than 4, theoperations of rotation and reflection ingeneral do not commute and Dn is notabelian; for example, in D4, a rotationof 90 degrees followed by a reflectionyields a different result from a reflection followed by a rotation of 90 degrees:

D4 is nonabelian (x-axis is vertical here).

Thus, beyond their obvious applicationto problems of symmetry in the plane,these groups are among the simplestexamples of non-abelian groups, andas such arise frequently as easycounterexamples to theorems whichare restricted to abelian groups.

Dihedral group 48

The 2n elements of Dn can be written as e, r, r2, ..., rn−1, s, r s, r2 s, ..., rn−1 s. The first n listed elements are rotationsand the remaining n elements are axis-reflections (all of which have order 2). The product of two rotations or tworeflections is a rotation; the product of a rotation and a reflection is a reflection.So far, we have considered Dn to be a subgroup of O(2), i.e. the group of rotations (about the origin) and reflections(across axes through the origin) of the plane. However, notation Dn is also used for a subgroup of SO(3) which isalso of abstract group type Dihn: the proper symmetry group of a regular polygon embedded in three-dimensionalspace (if n ≥ 3). Such a figure may be considered as a degenerate regular solid with its face counted twice. Thereforeit is also called a dihedron (Greek: solid with two faces), which explains the name dihedral group (in analogy totetrahedral, octahedral and icosahedral group, referring to the proper symmetry groups of a regular tetrahedron,octahedron, and icosahedron respectively).

Examples of 2D dihedral symmetry

2D D6 symmetry – The Red Star of David 2D D24 symmetry – Ashoka Chakra, asdepicted on the National flag of the Republic

of India.

Equivalent definitionsFurther equivalent definitions of Dihn are:• The automorphism group of the graph consisting only of a cycle with n vertices (if n ≥ 3).• The group with presentation

or

(The only finite groups that can be generated by two elements of order 2 are the dihedral groups and the cyclicgroups. If the two elements of order 2 are distinct, then the group generated is dihedral.)From the second presentation follows that Dihn belongs to the class of coxeter groups.

• The semidirect product of cyclic groups Zn and Z2, with Z2 acting on Zn by inversion (thus, Dihn always has anormal subgroup isomorphic to the group Zn

is isomorphic to Dihn if φ(0) is the identity and φ(1) is inversion.

Dihedral group 49

PropertiesIf we consider Dihn (n ≥ 3) as the symmetry group of a regular n-gon and number the polygon's vertices, we see thatDihn is a subgroup of the symmetric group Sn via this permutation representation.The properties of the dihedral groups Dihn with n ≥ 3 depend on whether n is even or odd. For example, the center ofDihn consists only of the identity if n is odd, but if n is even the center has two elements, namely the identity and theelement rn / 2 (with Dn as a subgroup of O(2), this is inversion; since it is scalar multiplication by −1, it is clear that itcommutes with any linear transformation).For odd n, abstract group Dih2n is isomorphic with the direct product of Dihn and Z2.In the case of 2D isometries, this corresponds to adding inversion, giving rotations and mirrors in between theexisting ones.If m divides n, then Dihn has n / m subgroups of type Dihm, and one subgroup Zm. Therefore the total number ofsubgroups of Dihn (n ≥ 1), is equal to d(n) + σ(n), where d(n) is the number of positive divisors of n and σ(n) is thesum of the positive divisors of n. See list of small groups for the cases n ≤ 8.

Conjugacy classes of reflectionsAll the reflections are conjugate to each other in case n is odd, but they fall into two conjugacy classes if n is even. Ifwe think of the isometries of a regular n-gon: for odd n there are rotations in the group between every pair of mirrors,while for even n only half of the mirrors can be reached from one by these rotations. Geometrically, in an oddpolygon every axis of symmetry passes through a vertex and a side, while in an even polygon half the axes passthrough two vertices, and half pass through two sides.Algebraically, this is an instance of the conjugate Sylow theorem (for n odd): for n odd, each reflection, togetherwith the identity, form a subgroup of order 2, which is a Sylow 2-subgroup ( is the maximum power of 2dividing ), while for n even, these order 2 subgroups are not Sylow subgroups because 4 (ahigher power of 2) divides the order of the group.For n even there is instead an outer automorphism interchanging the two types of reflections (properly, a class ofouter automorphisms, which are all conjugate by an inner automorphism).

Automorphism groupThe automorphism group of Dihn is isomorphic to the affine group Aff(Z/nZ) and hasorder where is Euler's totient function, the number of k in coprime to n.It can be understood in terms of the generators of a reflection and an elementary rotation (rotation by , fork coprime to n); which automorphisms are inner and outer depends on the parity of n.• For n odd, the dihedral group is centerless, so any element defines a non-trivial inner automorphism; for n even,

the rotation by 180° (reflection through the origin) is the non-trivial element of the center.• Thus for n odd, the inner automorphism group has order 2n, and for n even the inner automorphism group has

order n.• For n odd, all reflections are conjugate; for n even, they fall into two classes (those through two vertices and those

through two faces), related by an outer automorphism, which can be represented by rotation by (half theminimal rotation).

• The rotations are a normal subgroup; conjugation by a reflection changes the sign (direction) of the rotation, butotherwise leaves them unchanged. Thus automorphisms that multiply angles by k (coprime to n) are outer unless

Dihedral group 50

Examples of automorphism groupsDih9 has 18 inner automorphisms. As 2D isometry group D9, the group has mirrors at 20° intervals. The 18 innerautomorphisms provide rotation of the mirrors by multiples of 20°, and reflections. As isometry group these are allautomorphisms. As abstract group there are in addition to these, 36 outer automorphisms, e.g. multiplying angles ofrotation by 2.Dih10 has 10 inner automorphisms. As 2D isometry group D10, the group has mirrors at 18° intervals. The 10 innerautomorphisms provide rotation of the mirrors by multiples of 36°, and reflections. As isometry group there are 10more automorphisms; they are conjugates by isometries outside the group, rotating the mirrors 18° with respect tothe inner automorphisms. As abstract group there are in addition to these 10 inner and 10 outer automorphisms, 20more outer automorphisms, e.g. multiplying rotations by 3.Compare the values 6 and 4 for Euler's totient function, the multiplicative group of integers modulo n for n = 9 and10, respectively. This triples and doubles the number of automorphisms compared with the two automorphisms asisometries (keeping the order of the rotations the same or reversing the order).

GeneralizationsThere are several important generalizations of the dihedral groups:• The infinite dihedral group is an infinite group with algebraic structure similar to the finite dihedral groups. It can

be viewed as the group of symmetries of the integers.• The orthogonal group O(2), i.e. the symmetry group of the circle, also has similar properties to the dihedral

groups.• The family of generalized dihedral groups includes both of the examples above, as well as many other groups.• The quasidihedral groups are family of finite groups with similar properties to the dihedral groups.

References[1] Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.

External links• Dihedral Group n of Order 2n (http:/ / demonstrations. wolfram. com/ DihedralGroupNOfOrder2n/ ) by Shawn

Dudzik, Wolfram Demonstrations Project.

Lattice (group) 51

Lattice (group)

A lattice in the Euclidean plane.

In mathematics, especially in geometry and grouptheory, a lattice in Rn is a discrete subgroup of Rn

which spans the real vector space Rn. Every lattice inRn can be generated from a basis for the vector spaceby forming all linear combinations with integercoefficients. A lattice may be viewed as a regular tilingof a space by a primitive cell.

Lattices have many significant applications in puremathematics, particularly in connection to Lie algebras,number theory and group theory. They also arise inapplied mathematics in connection with coding theory,in cryptography because of conjectured computationalhardness of several lattice problems, and are used invarious ways in the physical sciences. For instance, in materials science and solid-state physics, a lattice is asynonym for the "frame work" of a crystalline structure, a 3-dimensional array of regularly spaced points coincidingwith the atom or molecule positions in a crystal. More generally, lattice models are studied in physics, often by thetechniques of computational physics.

Symmetry considerations and examplesA lattice is the symmetry group of discrete translational symmetry in n directions. A pattern with this lattice oftranslational symmetry cannot have more, but may have less symmetry than the lattice itself.A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with e.g. the atom or moleculepositions in a crystal, or more generally, the orbit of a group action under translational symmetry, is a translate of thetranslation lattice: a coset, which need not contain the origin, and therefore need not be a lattice in the previoussense.A simple example of a lattice in Rn is the subgroup Zn. A more complicated example is the Leech lattice, which is alattice in R24. The period lattice in R2 is central to the study of elliptic functions, developed in nineteenth centurymathematics; it generalises to higher dimensions in the theory of abelian functions.

Dividing space according to a latticeA typical lattice Λ in Rn thus has the form

where {v1, ..., vn} is a basis for Rn. Different bases can generate the same lattice, but the absolute value of thedeterminant of the vectors vi is uniquely determined by Λ, and is denoted by d(Λ). If one thinks of a lattice asdividing the whole of Rn into equal polyhedra (copies of an n-dimensional parallelepiped, known as the fundamentalregion of the lattice), then d(Λ) is equal to the n-dimensional volume of this polyhedron. This is why d(Λ) issometimes called the covolume of the lattice.

Lattice (group) 52

Lattice points in convex setsMinkowski's theorem relates the number d(Λ) and the volume of a symmetric convex set S to the number of latticepoints contained in S. The number of lattice points contained in a polytope all of whose vertices are elements of thelattice is described by the polytope's Ehrhart polynomial. Formulas for some of the coefficients of this polynomialinvolve d(Λ) as well.Theorem: let P be the polytope: fundamental region of a basis which is a weighted square self-blocking clutter S.then covolume(P) = k and P contains k - 1 integer interior points, where k is the wheight of the edges of S.

See also: Integer points in polyhedra

Computing with latticesLattice basis reduction is the problem of finding a short and nearly orthogonal lattice basis. TheLenstra-Lenstra-Lovász lattice basis reduction algorithm (LLL) approximates such a lattice basis in polynomial time;it has found numerous applications, particularly in public-key cryptography.

Lattices in two dimensions: detailed discussionThere are five 2D lattice types as given by the crystallographic restriction theorem. Below, the wallpaper group ofthe lattice is given in parentheses; note that a pattern with this lattice of translational symmetry cannot have more,but may have less symmetry than the lattice itself. If the symmetry group of a pattern contains an n-fold rotation thenthe lattice has n-fold symmetry for even n and 2n-fold for odd n.• a rhombic lattice, also called centered rectangular lattice or isosceles triangular lattice (cmm), with evenly

spaced rows of evenly spaced points, with the rows alternatingly shifted one half spacing (symmetricallystaggered rows):

• a hexagonal lattice or equilateral triangular lattice (p6m)• a square lattice (p4m):• a rectangular lattice, also called primitive rectangular lattice (pmm):• more generally, a parallelogrammic lattice, also called oblique lattice (p2)(with asymmetrically staggered

rows):For the classification of a given lattice, start with one point and take a nearest second point. For the third point, noton the same line, consider its distances to both points. Among the points for which the smaller of these two distancesis least, choose a point for which the larger of the two is least. (Not logically equivalent but in the case of latticesgiving the same result is just "Choose a point for which the larger of the two is least".)The five cases correspond to the triangle being equilateral, right isosceles, right, isosceles, and scalene. In a rhombiclattice, the shortest distance may either be a diagonal or a side of the rhombus, i.e., the line segment connecting thefirst two points may or may not be one of the equal sides of the isosceles triangle. This depends on the smaller angleof the rhombus being less than 60° or between 60° and 90°.The general case is known as a period lattice. If the vectors p and q generate the lattice, instead of p and q we can also take p and p-q, etc. In general in 2D, we can take a p + b q and c p + d q for integers a,b, c and d such that

Lattice (group) 53

ad-bc is 1 or -1. This ensures that p and q themselves are integer linear combinations of the other two vectors. Eachpair p, q defines a parallelogram, all with the same area, the magnitude of the cross product. One parallelogram fullydefines the whole object. Without further symmetry, this parallelogram is a fundamental parallelogram.

The fundamental domain of the period lattice.

The vectors p and q can be represented by complex numbers. Up to size and orientation, a pair can be represented bytheir quotient. Expressed geometrically: if two lattice points are 0 and 1, we consider the position of a third latticepoint. Equivalence in the sense of generating the same lattice is represented by the modular group: represents choosing a different third point in the same grid, represents choosing a different side ofthe triangle as reference side 0-1, which in general implies changing the scaling of the lattice, and rotating it. Each"curved triangle" in the image contains for each 2D lattice shape one complex number, the grey area is a canonicalrepresentation, corresponding to the classification above, with 0 and 1 two lattice points that are closest to eachother; duplication is avoided by including only half of the boundary. The rhombic lattices are represented by thepoints on its boundary, with the hexagonal lattice as vertex, and i for the square lattice. The rectangular lattices are atthe imaginary axis, and the remaining area represents the parallelogrammetic lattices, with the mirror image of aparallelogram represented by the mirror image in the imaginary axis.

Lattices in three dimensionsThe 14 lattice types in 3D are called Bravais lattices. They are characterized by their space group. 3D patterns withtranslational symmetry of a particular type cannot have more, but may have less symmetry than the lattice itself.

Lattices in complex spaceA lattice in Cn is a discrete subgroup of Cn which spans the 2n-dimensional real vector space Cn. For example, theGaussian integers form a lattice in C.Every lattice in Rn is a free abelian group of rank n; every lattice in Cn is a free abelian group of rank 2n.

In Lie groupsMore generally, a lattice Γ in a Lie group G is a discrete subgroup, such that the quotient G/Γ is of finite measure,for the measure on it inherited from Haar measure on G (left-invariant, or right-invariant—the definition isindependent of that choice). That will certainly be the case when G/Γ is compact, but that sufficient condition is notnecessary, as is shown by the case of the modular group in SL2(R), which is a lattice but where the quotient isn'tcompact (it has cusps). There are general results stating the existence of lattices in Lie groups.A lattice is said to be uniform or cocompact if G/Γ is compact; otherwise the lattice is called non-uniform.

Lattice (group) 54

Lattices over general vector-spacesWhilst we normally consider lattices in this concept can be generalised to any finite dimensional vectorspace over any field. This can be done as follows:

Let be a field, let be an -dimensional -vector space, let be a -basis for and let be a ring contained within . Then the lattice in generated by is given by:

Different bases will in general generate different lattices. However, if the transition matrix between the basesis in - the general linear group of R (in simple terms this means that all the entries of are in and allthe entries of are in - which is equivalent to saying that the determinant of is in - the unit group ofelements in with multiplicative inverses) then the lattices generated by these bases will be isomorphic since induces an isomorphism between the two lattices.Important cases of such lattices occur in number theory with K a p-adic field and R the p-adic integers.For a vector space which is also an inner product space, the dual lattice can be concretely described by the set:

or equivalently as,

ReferencesBirkhoff, Garrett (1967). Lattice Theory (3 ed.). American Mathematical Society. ISBN 0821810251.

Wallpaper group

Example of an Egyptian design with wallpaper group p4m

A wallpaper group (or plane symmetry group orplane crystallographic group) is a mathematicalclassification of a two-dimensional repetitive pattern,based on the symmetries in the pattern. Such patternsoccur frequently in architecture and decorative art.There are 17 possible distinct groups.

Wallpaper groups are two-dimensional symmetrygroups, intermediate in complexity between the simplerfrieze groups and the three-dimensionalcrystallographic groups (also called space groups).

Introduction

Wallpaper groups categorize patterns by theirsymmetries. Subtle differences may place similarpatterns in different groups, while patterns that are verydifferent in style, color, scale or orientation may belongto the same group.Consider the following examples:

Wallpaper group 55

Example A: Cloth, Tahiti Example B: Ornamental painting, Nineveh,Assyria

Example C: Painted porcelain, China

Examples A and B have the same wallpaper group; it is called p4m. Example C has a different wallpaper group,called p4g. The fact that A and B have the same wallpaper group means that they have the same symmetries,regardless of details of the designs, whereas C has a different set of symmetries despite any superficial similarities.A complete list of all seventeen possible wallpaper groups can be found below.

Symmetries of patternsA symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that the pattern looks exactly thesame after the transformation. For example, translational symmetry is present when the pattern can be translated(shifted) some finite distance and appear unchanged. Think of shifting a set of vertical stripes horizontally by onestripe. The pattern is unchanged. Strictly speaking, a true symmetry only exists in patterns that repeat exactly andcontinue indefinitely. A set of only, say, five stripes does not have translational symmetry — when shifted, the stripeon one end "disappears" and a new stripe is "added" at the other end. In practice, however, classification is applied tofinite patterns, and small imperfections may be ignored.Sometimes two categorizations are meaningful, one based on shapes alone and one also including colors. Whencolors are ignored there may be more symmetry. In black and white there are also 17 wallpaper groups; e.g., acolored tiling is equivalent with one in black and white with the colors coded radially in a circularly symmetric "barcode" in the centre of mass of each tile.The types of transformations that are relevant here are called Euclidean plane isometries. For example:• If we shift example B one unit to the right, so that each square covers the square that was originally adjacent to it,

then the resulting pattern is exactly the same as the pattern we started with. This type of symmetry is called atranslation. Examples A and C are similar, except that the smallest possible shifts are in diagonal directions.

• If we turn example B clockwise by 90°, around the centre of one of the squares, again we obtain exactly the samepattern. This is called a rotation. Examples A and C also have 90° rotations, although it requires a little moreingenuity to find the correct centre of rotation for C.

• We can also flip example B across a horizontal axis that runs across the middle of the image. This is called areflection. Example B also has reflections across a vertical axis, and across two diagonal axes. The same can besaid for A.

However, example C is different. It only has reflections in horizontal and vertical directions, not across diagonalaxes. If we flip across a diagonal line, we do not get the same pattern back; what we do get is the original patternshifted across by a certain distance. This is part of the reason that the wallpaper group of A and B is different fromthe wallpaper group of C.

Wallpaper group 56

HistoryAll 17 groups were used by Egyptian craftsmen, and used extensively in the Muslim world. A proof that there wereonly 17 possible patterns was first carried out by Evgraf Fedorov in 1891[1] and then derived independently byGeorge Pólya in 1924.[2] [3]

Formal definition and discussionMathematically, a wallpaper group or plane crystallographic group is a type of topologically discrete group ofisometries of the Euclidean plane that contains two linearly independent translations.Two such isometry groups are of the same type (of the same wallpaper group) if they are the same up to an affinetransformation of the plane. Thus e.g. a translation of the plane (hence a translation of the mirrors and centres ofrotation) does not affect the wallpaper group. The same applies for a change of angle between translation vectors,provided that it does not add or remove any symmetry (this is only the case if there are no mirrors and no glidereflections, and rotational symmetry is at most of order 2).Unlike in the three-dimensional case, we can equivalently restrict the affine transformations to those that preserveorientation.It follows from the Bieberbach theorem that all wallpaper groups are different even as abstract groups (as opposed toe.g. Frieze groups, of which two are isomorphic with Z).2D patterns with double translational symmetry can be categorized according to their symmetry group type.

Isometries of the Euclidean planeIsometries of the Euclidean plane fall into four categories (see the article Euclidean plane isometry for moreinformation).• Translations, denoted by Tv, where v is a vector in R2. This has the effect of shifting the plane applying

displacement vector v.• Rotations, denoted by Rc,θ, where c is a point in the plane (the centre of rotation), and θ is the angle of rotation.• Reflections, or mirror isometries, denoted by FL, where L is a line in R2. (F is for "flip"). This has the effect of

reflecting the plane in the line L, called the reflection axis or the associated mirror.• Glide reflections, denoted by GL,d, where L is a line in R2 and d is a distance. This is a combination of a

reflection in the line L and a translation along L by a distance d.

The independent translations conditionThe condition on linearly independent translations means that there exist linearly independent vectors v and w (inR2) such that the group contains both Tv and Tw.The purpose of this condition is to distinguish wallpaper groups from frieze groups, which possess a translation butnot two linearly independent ones, and from two-dimensional discrete point groups, which have no translations at all.In other words, wallpaper groups represent patterns that repeat themselves in two distinct directions, in contrast tofrieze groups, which only repeat along a single axis.(It is possible to generalise this situation. We could for example study discrete groups of isometries of Rn with mlinearly independent translations, where m is any integer in the range 0 ≤ m ≤ n.)

Wallpaper group 57

The discreteness conditionThe discreteness condition means that there is some positive real number ε, such that for every translation Tv in thegroup, the vector v has length at least ε (except of course in the case that v is the zero vector).The purpose of this condition is to ensure that the group has a compact fundamental domain, or in other words, a"cell" of nonzero, finite area, which is repeated through the plane. Without this condition, we might have forexample a group containing the translation Tx for every rational number x, which would not correspond to anyreasonable wallpaper pattern.One important and nontrivial consequence of the discreteness condition in combination with the independenttranslations condition is that the group can only contain rotations of order 2, 3, 4, or 6; that is, every rotation in thegroup must be a rotation by 180°, 120°, 90°, or 60°. This fact is known as the crystallographic restriction theorem,and can be generalised to higher-dimensional cases.

Notations for wallpaper groups

Crystallographic notation

Crystallography has 230 space groups to distinguish, far more than the 17 wallpaper groups, but many of thesymmetries in the groups are the same. Thus we can use a similar notation for both kinds of groups, that of CarlHermann and Charles-Victor Mauguin. An example of a full wallpaper name in Hermann-Mauguin style is p31m,with four letters or digits; more usual is a shortened name like cmm or pg.For wallpaper groups the full notation begins with either p or c, for a primitive cell or a face-centred cell; these areexplained below. This is followed by a digit, n, indicating the highest order of rotational symmetry: 1-fold (none),2-fold, 3-fold, 4-fold, or 6-fold. The next two symbols indicate symmetries relative to one translation axis of thepattern, referred to as the "main" one; if there is a mirror perpendicular to a translation axis we choose that axis asthe main one (or if there are two, one of them). The symbols are either m, g, or 1, for mirror, glide reflection, ornone. The axis of the mirror or glide reflection is perpendicular to the main axis for the first letter, and either parallelor tilted 180°/n (when n > 2) for the second letter. Many groups include other symmetries implied by the given ones.The short notation drops digits or an m that can be deduced, so long as that leaves no confusion with another group.A primitive cell is a minimal region repeated by lattice translations. All but two wallpaper symmetry groups aredescribed with respect to primitive cell axes, a coordinate basis using the translation vectors of the lattice. In theremaining two cases symmetry description is with respect to centred cells that are larger than the primitive cell, andhence have internal repetition; the directions of their sides is different from those of the translation vectors spanninga primitive cell. Hermann-Mauguin notation for crystal space groups uses additional cell types.Examples

• p2 (p211): Primitive cell, 2-fold rotation symmetry, no mirrors or glide reflections.• p4g (p4gm): Primitive cell, 4-fold rotation, glide reflection perpendicular to main axis, mirror axis at 45°.• cmm (c2mm): Centred cell, 2-fold rotation, mirror axes both perpendicular and parallel to main axis.• p31m (p31m): Primitive cell, 3-fold rotation, mirror axis at 60°.Here are all the names that differ in short and full notation.

Wallpaper group 58

Crystallographic short and full names

Short p2 pm pg cm pmm pmg pgg cmm p4m p4g p6m

Full p211 p1m1 p1g1 c1m1 p2mm p2mg p2gg c2mm p4mm p4gm p6mm

The remaining names are p1, p3, p3m1, p31m, p4, and p6.

Orbifold notation

Orbifold notation for wallpaper groups, introduced by John Horton Conway (Conway, 1992), is based not oncrystallography, but on topology. We fold the infinite periodic tiling of the plane into its essence, an orbifold, thendescribe that with a few symbols.• A digit, n, indicates a centre of n-fold rotation corresponding to a cone point on the orbifold. By the

crystallographic restriction theorem, n must be 2, 3, 4, or 6.• An asterisk, *, indicates a mirror symmetry corresponding to a boundary of the orbifold. It interacts with the

digits as follows:1. Digits before * denote centres of pure rotation (cyclic).2. Digits after * denote centres of rotation with mirrors through them, corresponding to "corners" on the

boundary of the orbifold (dihedral).• A cross, x, occurs when a glide reflection is present and indicates a crosscap on the orbifold. Pure mirrors

combine with lattice translation to produce glides, but those are already accounted for so we do not notate them.• The "no symmetry" symbol, o, stands alone, and indicates we have only lattice translations with no other

symmetry. The orbifold with this symbol is a torus; in general the symbol o denotes a handle on the orbifold.Consider the group denoted in crystallographic notation by cmm; in Conway's notation, this will be 2*22. The 2before the * says we have a 2-fold rotation centre with no mirror through it. The * itself says we have a mirror. Thefirst 2 after the * says we have a 2-fold rotation centre on a mirror. The final 2 says we have an independent second2-fold rotation centre on a mirror, one that is not a duplicate of the first one under symmetries.The group denoted by pgg will be 22x. We have two pure 2-fold rotation centres, and a glide reflection axis.Contrast this with pmg, Conway 22*, where crystallographic notation mentions a glide, but one that is implicit in theother symmetries of the orbifold.

Conway and crystallographic correspondence

Conway o xx *x ** 632 *632

Crystal. p1 pg cm pm p6 p6m

Conway 333 *333 3*3 442 *442 4*2

Crystal. p3 p3m1 p31m p4 p4m p4g

Conway 2222 22x 22* *2222 2*22

Crystal. p2 pgg pmg pmm cmm

Why there are exactly seventeen groupsAn orbifold can be viewed as a polygon with face, edges, and vertices, which can be unfolded to form a possibly infinite set of polygons which tile either the sphere, the plane or the hyperbolic plane. When it tiles the plane it will give a wallpaper group and when it tiles the sphere or hyperbolic plane it gives either a spherical symmetry groups or Hyperbolic symmetry group. The type of space the polygons tile can be found by calculating the Euler characteristic, χ = V − E + F, where V is the number of corners (vertices), E is the number of edges and F is the number of faces. If

Wallpaper group 59

the Euler characteristic is positive then the orbifold has a elliptic (spherical) structure; if it is zero then it has aparabolic structure, i.e. a wallpaper group; and if it is negative it will have a hyperbolic structure. When the full setof possible orbifolds is enumerated it is found that only 17 have Euler characteristic 0.When an orbifold replicates by symmetry to fill the plane, its features create a structure of vertices, edges, andpolygon faces, which must be consistent with the Euler characteristic. Reversing the process, we can assign numbersto the features of the orbifold, but fractions, rather than whole numbers. Because the orbifold itself is a quotient ofthe full surface by the symmetry group, the orbifold Euler characteristic is a quotient of the surface Eulercharacteristic by the order of the symmetry group.The orbifold Euler characteristic is 2 minus the sum of the feature values, assigned as follows:• A digit n before a * counts as (n−1)/n.• A digit n after a * counts as (n−1)/2n.• Both * and x count as 1.• The "no symmetry" o counts as 2.For a wallpaper group, the sum for the characteristic must be zero; thus the feature sum must be 2.Examples

• 632: 5/6 + 2/3 + 1/2 = 2• 3*3: 2/3 + 1 + 1/3 = 2• 4*2: 3/4 + 1 + 1/4 = 2• 22x: 1/2 + 1/2 + 1 = 2Now enumeration of all wallpaper groups becomes a matter of arithmetic, of listing all feature strings with valuessumming to 2.Incidentally, feature strings with other sums are not nonsense; they imply non-planar tilings, not discussed here.(When the orbifold Euler characteristic is negative, the tiling is hyperbolic; when positive, spherical or bad).

Guide to recognising wallpaper groupsTo work out which wallpaper group corresponds to a given design, one may use the following table.

Size ofsmallestrotation

Has reflection?

Yes No

360° / 6 p6m p6

360° / 4 Has mirrors at 45°? p4

Yes: p4m No: p4g

360° / 3 Has rot. centre off mirrors? p3

Yes: p31m No: p3m1

360° / 2 Has perpendicular reflections? Has glide reflection?

Yes No

Has rot. centre off mirrors? pmg Yes: pgg No: p2

Yes: cmm No: pmm

none Has glide axis off mirrors? Has glide reflection?

Yes: cm No: pm Yes: pg No: p1

See also this overview with diagrams.

Wallpaper group 60

The seventeen groupsEach of the groups in this section has two cell structure diagrams, which are to be interpreted as follows:

a centre of rotation of order two (180°).

a centre of rotation of order three (120°̊).

a centre of rotation of order four (90°̊).

a centre of rotation of order six (60°).

an axis of reflection.

an axis of glide reflection.

On the right-hand side diagrams, different equivalence classes of symmetry elements are colored (and rotated)differently.The brown or yellow area indicates a fundamental domain, i.e. the smallest part of the pattern that is repeated.The diagrams on the right show the cell of the lattice corresponding to the smallest translations; those on the leftsometimes show a larger area.

Group p1

Example and diagram for p1

Cell structure for p1

Cell structure for p1

• Orbifold notation: o.• The group p1 contains only translations; there are no rotations,

reflections, or glide reflections.

Examples of group p1

Wallpaper group 61

Computer generated Mediæval wall diapering

The two translations (cell sides) can each have different lengths, and can form any angle.

Group p2

Example and diagram for p2

Cell structure for p2

Cell structure for p2

• Orbifold notation: 2222.• The group p2 contains four rotation centres of order two (180°), but

no reflections or glide reflections.

Examples of group p2

Wallpaper group 62

Computer generated Cloth, HawaiianIslandsSandwich Islands

(Hawaii)

Mat on which AncientEgyptEgyptian king stood

Egyptian mat (detail)

Ceiling of AncientEgyptEgyptian tomb

Wire fence, U.S.

Group pm

Example and diagram for pm

Cell structure for pm

• Orbifold notation: **.• The group pm has no rotations. It has reflection axes, they are all

parallel.

Examples of group pm

(The first three have a vertical symmetry axis, and the last two eachhave a different diagonal one.)

Wallpaper group 63

Cell structure for pm

Computer generated Dress of a figure in a tomb atBiban el Moluk, Ancient

EgyptEgypt

Ancient EgyptEgyptian tomb,Thebes (Egypt)Thebes

Ceiling of a tomb at Gourna,Ancient EgyptEgypt. Reflection

axis is diagonal.

Indian metalwork at the GreatExhibition in 1851. This isalmost pm (ignoring short

diagonal lines between ovalsmotifs, which make it #Group

p1p1).

Group pg

Example and diagram for pg

• Orbifold notation: xx.• The group pg contains glide reflections only, and their axes are all

parallel. There are no rotations or reflections.

Examples of group pg

Wallpaper group 64

Cell structure for pg

Cell structure for pg

Computer generated Mat with herringbonepattern on which AncientEgyptEgyptian king stood

Egyptian mat (detail) Pavement with herringbonepattern in Salzburg. Glide

reflection axis runsnortheast-southwest.

Wallpaper group 65

One of the colorings of the snubsquare tiling; the glide reflectionlines are in the direction upper

left / lower right; ignoring colorsthere is much more symmetrythan just pg, then it is p4g (see

there for this image with equallycolored triangles)It helps toconsider the squares as the

background, then we see a simplepatterns of rows of rhombuses.

Without the details inside the zigzag bands the mat is pmg; with the details but without the distinction betweenbrown and black it is pgg.Ignoring the wavy borders of the tiles, the pavement is pgg.

Group cm

Cell structure for cm

Cell structure for cm

• Orbifold notation: *x.• The group cm contains no rotations. It has reflection axes, all

parallel. There is at least one glide reflection whose axis is not areflection axis; it is halfway between two adjacent parallel reflectionaxes.

This group applies for symmetrically staggered rows (i.e. there is ashift per row of half the translation distance inside the rows) ofidentical objects, which have a symmetry axis perpendicular to therows.Examples of group cm

Wallpaper group 66

Computer generated Dress of Amun, from AbuSimbel, Ancient EgyptEgypt

Dado (architecture)Dado fromBiban el Moluk, Ancient

EgyptEgypt

Bronze vessel in Nimroud,Assyria

Spandrils of arches,the Alhambra,

Spain

Soffitt of arch, the Alhambra,Spain

Persian empirePersiantapestry

Indian metalwork at the GreatExhibition in 1851

Dress of a figure in a tomb atBiban el Moluk, Ancient

EgyptEgypt

Group pmm

Example and diagram for pmm

• Orbifold notation: *2222.• The group pmm has reflections in two perpendicular directions, and

four rotation centres of order two (180°) located at the intersectionsof the reflection axes.

Examples of group pmm

Wallpaper group 67

Cell structure for pmm

Cell structure for pmm

Computer generated 2D image of lattice fence, U.S.(in 3D there is additional

symmetry)

Mummy case stored in TheLouvre

Ceiling of AncientEgyptEgyptian tomb. Ignoringminor asymmetries, this would

be cmm.

Mummy case stored in TheLouvre. Would be type p4mexcept for the mismatched

coloring.

Compact packing of two sizes ofcircle.

Another compact packingof two sizes of circle.

Another compact packingof two sizes of circle.

Wallpaper group 68

Group pmg

Example and diagram for pmg

Cell structure for pmg

Cell structure for pmg

• Orbifold notation: 22*.• The group pmg has two rotation centres of order two (180°), and

reflections in only one direction. It has glide reflections whose axesare perpendicular to the reflection axes. The centres of rotation alllie on glide reflection axes.

Examples of group pmg

Computer generated Cloth, HawaiianIslandsSandwich Islands

(Hawaii)

Ceiling of AncientEgyptEgyptian tomb

Floor tiling in Prague, the CzechRepublic

Wallpaper group 69

Bowl from Kingdom ofKermaKerma

Pentagon packing

Group pgg

Example and diagram for pgg

Cell structure for pgg

Cell structure for pgg

• Orbifold notation: 22x.• The group pgg contains two rotation centres of order two (180°),

and glide reflections in two perpendicular directions. The centres ofrotation are not located on the glide reflection axes. There are noreflections.

Examples of group pgg

Wallpaper group 70

Computer generated Bronze vessel in Nimroud, Assyria Pavement (roads)Pavement in Budapest,Hungary. Glide reflection axes are diagonal.

Group cmm

Cell structure for cmm

Cell structure for cmm

• Orbifold notation: 2*22.• The group cmm has reflections in two perpendicular directions, and

a rotation of order two (180°) whose centre is not on a reflectionaxis. It also has two rotations whose centres are on a reflection axis.

• This group is frequently seen in everyday life, since the mostcommon arrangement of bricks in a brick building utilises thisgroup (see example below).

The rotational symmetry of order 2 with centres of rotation at thecentres of the sides of the rhombus is a consequence of the otherproperties.The pattern corresponds to each of the following:• symmetrically staggered rows of identical doubly symmetric objects• a checkerboard pattern of two alternating rectangular tiles, of which

each, by itself, is doubly symmetric• a checkerboard pattern of alternatingly a 2-fold rotationally

symmetric rectangular tile and its mirror imageExamples of group cmm

Wallpaper group 71

Computer generated one of the 8 Tilings of regularpolygons#Archimedean,uniform or semiregular

tilingssemi-regulartessellations; ignoring color this

is this group cmm, otherwisegroup p1

Suburban brick wall, U.S. Ceiling of AncientEgyptEgyptian tomb. Ignoringcolors, this would be #Group

p4gp4g.

Ancient EgyptEgyptian PersianempirePersian

tapestry

Ancient EgyptEgyptian tomb Turkic peoplesTurkish dish

Group p4

Example and diagram for p4

• Orbifold notation: 442.• The group p4 has two rotation centres of order four (90°̊), and one

rotation centre of order two (180°̊). It has no reflections or glidereflections.

Examples of group p4

A p4 pattern can be looked upon as a repetition in rows and columns ofequal square tiles with 4-fold rotational symmetry. Also it can be looked upon as a checkerboard pattern of two suchtiles, a factor smaller and rotated 45°.

Wallpaper group 72

Cell structure for p4

Cell structure for p4

Computer generated Ceiling of AncientEgyptEgyptian tomb;

ignoring colors this is p4,otherwise #Group p2p2

Ceiling of AncientEgyptEgyptian tomb

Frieze, the Alhambra, Spain.Requires close inspection to see

why there are no reflections.

Viennese cane Renaissance earthenware

Wallpaper group 73

Group p4m

Example and diagram for p4m

Cell structure for p4m

Cell structure for p4m

• Orbifold notation: *442.• The group p4m has two rotation centres of order four (90°), and

reflections in four distinct directions (horizontal, vertical, anddiagonals). It has additional glide reflections whose axes are notreflection axes; rotations of order two (180°) are centred at theintersection of the glide reflection axes. All rotation centres lie onreflection axes.

This corresponds to a straightforward grid of rows and columns ofequal squares with the four reflection axes. Also it corresponds to acheckerboard pattern of two of such squares.Examples of group p4m

Examples displayed with the smallest translations horizontal andvertical (like in the diagram):

Computer generated one of the 3 Tilings of regularpolygons#Regular tilingsregular

tessellations (in thischeckerboard coloring, smallest

translations are diagonal)

Tetrakis squaretilingDemiregular tiling with

triangles; ignoring colors, this isp4m, otherwise #Group

cmmcmm

one of the 8 Tilings of regularpolygons#Archimedean, uniform

or semiregulartilingssemi-regular tessellations

(ignoring color also, with smallertranslations)

Wallpaper group 74

Ornamental painting,Nineveh, Assyria

Storm drain, U.S. Ancient EgyptEgyptianmummy case

Persian EmpirePersian glaze(painting technique)glazed

tile

Compact packing of two sizes ofcircle.

Examples displayed with the smallest translations diagonal (like on a checkerboard):

Cloth, Otaheite(Tahiti)

AncientEgyptEgyptian tomb

Cathedral of Bourges Dish from Turkey,Ottoman

EmpireOttomanperiod

Group p4g

Example and diagram for p4g

• Orbifold notation: 4*2.• The group p4g has two centres of rotation of order four (90°), which

are each other's mirror image, but it has reflections in only twodirections, which are perpendicular. There are rotations of order two(180°) whose centres are located at the intersections of reflectionaxes. It has glide reflections axes parallel to the reflection axes, inbetween them, and also at an angle of 45° with these.

A p4g pattern can be looked upon as a checkerboard pattern of copies of a square tile with 4-fold rotationalsymmetry, and its mirror image.

Wallpaper group 75

Cell structure for p4g

Cell structure for p4g

Alternatively it can be looked upon (by shifting half a tile) as acheckerboard pattern of copies of a horizontally and verticallysymmetric tile and its 90° rotated version. Note that neither applies fora plain checkerboard pattern of black and white tiles, this is group p4m(with diagonal translation cells).

Note that the diagram on the left represents in area twice the smallestsquare that is repeated by translation.Examples of group p4g

Computer generated Bathroom linoleum, U.S. Painted porcelain, China Fly screen, U.S.

Painting, China one of the colorings of the snubsquare tiling (see also at pg)

Wallpaper group 76

Group p3

Cell structure for p3 (the rotation centres at thecentres of the triangles are not shown)

Cell structure for p3

• Orbifold notation: 333.• The group p3 has three different rotation centres of order three

(120°), but no reflections or glide reflections.

Imagine a tessellation of the plane with equilateral triangles of equalsize, with the sides corresponding to the smallest translations. Thenhalf of the triangles are in one orientation, and the other half upsidedown. This wallpaper group corresponds to the case that all triangles ofthe same orientation are equal, while both types have rotationalsymmetry of order three, but the two are not equal, not each other'smirror image, and not both symmetric (if the two are equal we have p6,if they are each other's mirror image we have p31m, if they are bothsymmetric we have p3m1; if two of the three apply then the third also,and we have p6m). For a given image, three of these tessellations arepossible, each with rotation centres as vertices, i.e. for any tessellationtwo shifts are possible. In terms of the image: the vertices can be thered, the blue or the green triangles.

Equivalently, imagine a tessellation of the plane with regular hexagons,with sides equal to the smallest translation distance divided by √3.Then this wallpaper group corresponds to the case that all hexagons are

equal (and in the same orientation) and have rotational symmetry of order three, while they have no mirror imagesymmetry (if they have rotational symmetry of order six we have p6, if they are symmetric with respect to the maindiagonals we have p31m, if they are symmetric with respect to lines perpendicular to the sides we have p3m1; if twoof the three apply then the third also, and we have p6m). For a given image, three of these tessellations are possible,each with one third of the rotation centres as centres of the hexagons. In terms of the image: the centres of thehexagons can be the red, the blue or the green triangles.

Examples of group p3

Computer generated one of the 8 Tilings ofregular

polygons#Archimedean,uniform or semiregular

tilingssemi-regulartessellations (ignoring thecolors: p6); the translation

vectors are rotated a little tothe right compared with thedirections in the underlying

hexagonal lattice of theimage

Street pavement inZakopane, Poland

Wall tiling in the Alhambra,Spain (and the whole wall);ignoring all colors this is p3

(ignoring only star colors it is#Group p1p1)

Wallpaper group 77

Group p3m1

Example and diagram for p3m1

Cell structure for p3m1

Cell structure for p3m1

• Orbifold notation: *333.• The group p3m1 has three different rotation centres of order three

(120°). It has reflections in the three sides of an equilateral triangle.The centre of every rotation lies on a reflection axis. There areadditional glide reflections in three distinct directions, whose axesare located halfway between adjacent parallel reflection axes.

Like for p3, imagine a tessellation of the plane with equilateraltriangles of equal size, with the sides corresponding to the smallesttranslations. Then half of the triangles are in one orientation, and theother half upside down. This wallpaper group corresponds to the casethat all triangles of the same orientation are equal, while both typeshave rotational symmetry of order three, and both are symmetric, butthe two are not equal, and not each other's mirror image. For a givenimage, three of these tessellations are possible, each with rotationcentres as vertices. In terms of the image: the vertices can be the red,the dark blue or the green triangles.

Examples of group p3m1

one of the 3 Tilings of regularpolygons#Regular tilingsregular

tessellations (ignoring colors:p6m)

another regular tessellation(ignoring colors: p6m)

one of the 8 Tilings of regularpolygons#Archimedean, uniform

or semiregulartilingssemi-regular tessellations

(ignoring colors: p6m)

Persian EmpirePersianglaze (painting

technique)glazed tile(ignoring colors: p6m)

Wallpaper group 78

Persian EmpirePersian ornament Painting, China (see detailedimage)

Computer generated Compact packing of twosizes of circle.

Group p31m

Example and diagram for p31m

Cell structure for p31m

Cell structure for p31m

• Orbifold notation: 3*3.• The group p31m has three different rotation centres of order three

(120°), of which two are each other's mirror image. It has reflectionsin three distinct directions. It has at least one rotation whose centredoes not lie on a reflection axis. There are additional glidereflections in three distinct directions, whose axes are locatedhalfway between adjacent parallel reflection axes.

Like for p3 and p3m1, imagine a tessellation of the plane withequilateral triangles of equal size, with the sides corresponding to thesmallest translations. Then half of the triangles are in one orientation,and the other half upside down. This wallpaper group corresponds tothe case that all triangles of the same orientation are equal, while bothtypes have rotational symmetry of order three and are each other'smirror image, but not symmetric themselves, and not equal. For agiven image, only one such tessellation is possible. In terms of theimage: the vertices can not be dark blue triangles.

Examples of group p31m

Wallpaper group 79

Persian empirePersian glaze(painting technique)glazed

tile

Painted porcelain, China Painting, China Computer generated

Group p6

Example and diagram for p6

Cell structure for p6

Cell structure for p6

• Orbifold notation: 632.• The group p6 has one rotation centre of order six (60°); it has also

two rotation centres of order three, which only differ by a rotationof 60° (or, equivalently, 180°), and three of order two, which onlydiffer by a rotation of 60°. It has no reflections or glide reflections.

A pattern with this symmetry can be looked upon as a tessellation ofthe plane with equal triangular tiles with C3 symmetry, or equivalently,a tessellation of the plane with equal hexagonal tiles with C6 symmetry(with the edges of the tiles not necessarily part of the pattern).

Examples of group p6

Wallpaper group 80

Computer generated Wall panelling, the Alhambra, Spain Persian EmpirePersianornament

Group p6m

Example and diagram for p6m

Cell structure for p6m

Cell structure for p6m

• Orbifold notation: *632.• The group p6m has one rotation centre of order six (60°); it has also

two rotation centres of order three, which only differ by a rotationof 60° (or, equivalently, 180°), and three of order two, which onlydiffer by a rotation of 60°. It has also reflections in six distinctdirections. There are additional glide reflections in six distinctdirections, whose axes are located halfway between adjacentparallel reflection axes.

A pattern with this symmetry can be looked upon as a tessellation ofthe plane with equal triangular tiles with D3 symmetry, or equivalently,a tessellation of the plane with equal hexagonal tiles with D6 symmetry(with the edges of the tiles not necessarily part of the pattern). Thus thesimplest examples are a triangular lattice with or without connectinglines, and a hexagonal tiling with one color for outlining the hexagonsand one for the background.

Examples of group p6m

Wallpaper group 81

Computer generated one of the 8 Tilings of regularpolygons#Archimedean,uniform or semiregular

tilingssemi-regulartessellations

another semi-regulartessellation

another semi-regulartessellation

Persian empirePersian glaze(painting technique)glazed tile

King's dress,Khorsabad, Assyria;this is almost p6m

(ignoring inner partsof flowers, whichmake it #Group

cmmcmm)

Bronze vessel inNimroud, Assyria

ByzantineartByzantine marble

pavement, Rome

Painted porcelain, China Painted porcelain, China Compact packing of two sizes ofcircle.

Another compact packing of twosizes of circle.

Lattice typesThere are five lattice types, corresponding to the five possible wallpaper groups of the lattice itself. The wallpapergroup of a pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than thelattice itself.• In the 5 cases of rotational symmetry of order 3 or 6, the cell consists of two equilateral triangles (hexagonal

lattice, itself p6m).• In the 3 cases of rotational symmetry of order 4, the cell is a square (square lattice, itself p4m).• In the 5 cases of reflection or glide reflection, but not both, the cell is a rectangle (rectangular lattice, itself pmm),

therefore the diagrams show a rectangle, but a special case is that it actually is a square.

Wallpaper group 82

• In the 2 cases of reflection combined with glide reflection, the cell is a rhombus (rhombic lattice, itself cmm); aspecial case is that it actually is a square.

• In the case of only rotational symmetry of order 2, and the case of no other symmetry than translational, the cell isin general a parallelogram (parallelogrammatic lattice, itself p2), therefore the diagrams show a parallelogram,but special cases are that it actually is a rectangle, rhombus, or square.

Symmetry groupsThe actual symmetry group should be distinguished from the wallpaper group. Wallpaper groups are collections ofsymmetry groups. There are 17 of these collections, but for each collection there are infinitely many symmetrygroups, in the sense of actual groups of isometries. These depend, apart from the wallpaper group, on a number ofparameters for the translation vectors, the orientation and position of the reflection axes and rotation centers.The numbers of degrees of freedom are:• 6 for p2• 5 for pmm, pmg, pgg, and cmm• 4 for the rest.However, within each wallpaper group, all symmetry groups are algebraically isomorphic.Some symmetry group isomorphisms:• p1: Z2

• pm: Z × D∞• pmm: D∞ × D∞.

Dependence of wallpaper groups on transformations• The wallpaper group of a pattern is invariant under isometries and uniform scaling (similarity transformations).• Translational symmetry is preserved under arbitrary bijective affine transformations.• Rotational symmetry of order two ditto; this means also that 4- and 6-fold rotation centres at least keep 2-fold

rotational symmetry.• Reflection in a line and glide reflection are preserved on expansion/contraction along, or perpendicular to, the axis

of reflection and glide reflection. It changes p6m, p4g, and p3m1 into cmm, p3m1 into cm, and p4m, dependingon direction of expansion/contraction, into pmm or cmm. A pattern of symmetrically staggered rows of points isspecial in that it can convert by expansion/contraction from p6m to p4m.

Note that when a transformation decreases symmetry, a transformation of the same kind (the inverse) obviously forsome patterns increases the symmetry. Such a special property of a pattern (e.g. expansion in one direction producesa pattern with 4-fold symmetry) is not counted as a form of extra symmetry.Change of colors does not affect the wallpaper group if any two points that have the same color before the change,also have the same color after the change, and any two points that have different colors before the change, also havedifferent colors after the change.If the former applies, but not the latter, such as when converting a color image to one in black and white, thensymmetries are preserved, but they may increase, so that the wallpaper group can change.

Wallpaper group 83

Web demo and softwareThere exist several software graphic tools that will let you create 2D patterns using wallpaper symmetry groups.Usually, you can edit the original tile and its copies in the entire pattern are updated automatically.• MadPattern [5], a free set of Adobe Illustrator templates that support the 17 wallpaper groups• Tess [6], a nagware tessellation program for multiple platforms, supports all wallpaper, frieze, and rosette groups,

as well as Heesch tilings.• Kali [7], online graphical symmetry editor applet.• Kali [8], free downloadable Kali for Windows and Mac Classic.• Inkscape, a free vector graphics editor, supports all 17 groups plus arbitrary scales, shifts, rotates, and color

changes per row or per column, optionally randomized to a given degree. (See [9])• SymmetryWorks [10] is a commercial plugin for Adobe Illustrator, supports all 17 groups.• Arabeske [11] is a free standalone tool, supports a subset of wallpaper groups.

Notes[1] E. Fedorov (1891) "Simmetrija na ploskosti" [Symmetry in the plane], Zapiski Imperatorskogo Sant-Petersburgskogo Mineralogicheskogo

Obshchestva [Proceedings of the Imperial St. Petersburg Mineralogical Society], series 2, vol. 28, pages 345-291 (in Russian).[2] George Pólya (1924) "Über die Analogie der Kristallsymmetrie in der Ebene," Zeitschrift für Kristallographie, vol. 60, pages 278–282.[3] Weyl, Hermann (1952), Symmetry, Princeton University Press, ISBN 0-691-02374-3[4] It helps to consider the squares as the background, then we see a simple patterns of rows of rhombuses.[5] http:/ / www. madpattern. com/[6] http:/ / www. peda. com/ tess/[7] http:/ / www. scienceu. com/ geometry/ handson/ kali/[8] http:/ / www. geometrygames. org/ Kali/ index. html[9] http:/ / tavmjong. free. fr/ INKSCAPE/ MANUAL/ html/ Tiles-Symmetries. html[10] http:/ / www. artlandia. com/ products/ SymmetryWorks/[11] http:/ / www. wozzeck. net/ arabeske/ index. html

References• The Grammar of Ornament (http:/ / www. animationarchive. org/ 2006/ 04/

media-grammar-of-ornament-part-one. html) (1856), by Owen Jones. Many of the images in this article are fromthis book; it contains many more.

• J. H. Conway (1992). "The Orbifold Notation for Surface Groups". In: M. W. Liebeck and J. Saxl (eds.), Groups,Combinatorics and Geometry, Proceedings of the L.M.S. Durham Symposium, July 5–15, Durham, UK, 1990;London Math. Soc. Lecture Notes Series 165. Cambridge University Press, Cambridge. pp. 438–447

• Grünbaum, Branko; Shephard, G. C. (1987): Tilings and Patterns. New York: Freeman. ISBN 0-7167-1193-1.• Pattern Design, Lewis F. Day

Wallpaper group 84

External links• The 17 plane symmetry groups (http:/ / www. clarku. edu/ ~djoyce/ wallpaper/ seventeen. html) by David E.

Joyce• Introduction to wallpaper patterns (http:/ / www. geom. uiuc. edu/ education/ math5337/ Wallpaper/ introduction.

html) by Chaim Goodman-Strauss and Heidi Burgiel• Description (http:/ / www. geom. uiuc. edu/ docs/ reference/ CRC-formulas/ node12. html) by Silvio Levy• Example tiling for each group, with dynamic demos of properties (http:/ / clowder. net/ hop/ 17walppr/ 17walppr.

html)• Overview with example tiling for each group (http:/ / www. math. toronto. edu/ ~drorbn/ Gallery/ Symmetry/

Tilings/ Sanderson/ index. html)• Tiling plane and fancy (http:/ / www. spsu. edu/ math/ tile/ ) by Steve Edwards• Escher Web Sketch, a java applet with interactive tools for drawing in all 17 plane symmetry groups (http:/ /

escher. epfl. ch/ escher/ )• Burak, a Java applet for drawing symmetry groups. (http:/ / www-viz. tamu. edu/ faculty/ ergun/ research/

artisticdepiction/ symmetric/ program/ index. html)• Beobachtungen zum geometrischen Motiv der Pelta (http:/ / www. peltenwirbel. de/ )• Seventeen Kinds of Wallpaper Patterns (http:/ / mathmuse. sci. ibaraki. ac. jp/ pattrn/ PatternE. html) the 17

symmetries found in traditional Japanese patterns.• Math on the Walls- Natalie Wolchover (http:/ / www. factodiem. com/ 2010/ 09/ wallpaper-math. html)

Article Sources and Contributors 85

Article Sources and ContributorsList of spherical symmetry groups  Source: http://en.wikipedia.org/w/index.php?oldid=338505211  Contributors: CambridgeBayWeather, Charles Matthews, F1f2f3f4, Giants27, Giftlite, JitseNiesen, KSmrq, Linas, Michael Hardy, Patrick, Rich Farmbrough, Tamfang, Tomruen, Woohookitty, 3 anonymous edits

List of planar symmetry groups  Source: http://en.wikipedia.org/w/index.php?oldid=365554506  Contributors: 345Kai, Charles Matthews, Doctormatt, KSmrq, Patrick, Tomruen, Weston.pace,Winhunter, ZeroOne

Dihedral symmetry in three dimensions  Source: http://en.wikipedia.org/w/index.php?oldid=409235355  Contributors: Chuunen Baka, FyShu^, Jim.belk, KSmrq, Maksim-e, Michael Hardy,Oleg Alexandrov, Patrick, Quibik, SMasters, Tomruen, Wayne Miller, 3 anonymous edits

Tetrahedral symmetry  Source: http://en.wikipedia.org/w/index.php?oldid=369615802  Contributors: AndrewKepert, Euclidthegreek, Experiment123, Fropuff, Giftlite, JackSchmidt,JamesBWatson, Maksim-e, MatthewMain, Nbarth, Noe, Pak21, Patrick, Professor Fiendish, Raven4x4x, Schutz, Tohd8BohaithuGh1, Tomruen, 7 anonymous edits

Octahedral symmetry  Source: http://en.wikipedia.org/w/index.php?oldid=365533508  Contributors: 4C, Baccyak4H, Experiment123, Fropuff, Giftlite, JackSchmidt, Jwanders, Katzmik,Maksim-e, Mets501, Mysid, Pak21, Patrick, Paul D. Anderson, Pjvpjv, Saxbryn, Stannered, Tomruen, 4 anonymous edits

Icosahedral symmetry  Source: http://en.wikipedia.org/w/index.php?oldid=405953029  Contributors: BD2412, Baccyak4H, Charles Matthews, ChrisRuvolo, Colonies Chris, Dendrophilos,DrBob, Euclidthegreek, Experiment123, Fropuff, Giftlite, Jim.belk, Koavf, Maksim-e, Michael Hardy, Nbarth, Oleg Alexandrov, Ospalh, Patrick, Paul D. Anderson, Sango123, Stephen Bain,Tintazul, Tomruen, Woscafrench, 12 anonymous edits

Cyclic symmetries  Source: http://en.wikipedia.org/w/index.php?oldid=345482886  Contributors: Oleg Alexandrov, Patrick, TheLimbicOne, Tomruen, 1 anonymous edits

Reflection symmetry  Source: http://en.wikipedia.org/w/index.php?oldid=409083350  Contributors: Abu-Fool Danyal ibn Amir al-Makhiri, Adam majewski, Allmightyduck, AmosWolfe,AndrewHowse, Anomaly1, Byeee, CanadianLinuxUser, Ceyockey, Charles Matthews, Davwillev, Dbiel, Eaefremov, Enormousdude, Favonian, Fifelfoo, Gadfium, GainLine, Gandalf61, Giftlite,Husond, Incnis Mrsi, JStor, JWilk, Japanese Searobin, Jedibob5, Johnuniq, Jtir, Kayau, Kitkatkool, Linas, Loggie, MathsIsFun, Mxn, Nsaa, Oleg Alexandrov, OverlordQ, Pampas Cat, Pasajero,Patrick, Paul August, Paul D. Anderson, Pfalstad, Pontificake, RJASE1, Ronhjones, RoyBoy, Siddhanteocker, Sitehut, Skater, Smack, The Thing That Should Not Be, TheLimbicOne, Tide rolls,Ummit, Useight, WeLeb14, Woohookitty, Yaragn, Zaslav, 71 anonymous edits

Inverse (mathematics)  Source: http://en.wikipedia.org/w/index.php?oldid=318111010  Contributors: CBM, Ciphers, Funandtrvl, Krishnachandranvn, Lambiam, Oleg Alexandrov, Patrick,PhotoBox, Rgdboer, So9q, The Great Redirector, 9 anonymous edits

Point groups in three dimensions  Source: http://en.wikipedia.org/w/index.php?oldid=397756821  Contributors: AndrewKepert, Baccyak4H, Bduke, Charles Matthews, Colonies Chris, Crystalwhacker, DMacks, Dalf, DrBob, Eg-T2g, Eric Kvaalen, Fratrep, Giftlite, Itub, Jim.belk, KSmrq, Mets501, Michael Hardy, Nbarth, Noe, Oakwood, Oleg Alexandrov, Oysteinp, Pak21, Patrick,R.e.b., Tamfang, TheLimbicOne, Tobias Bergemann, Tomruen, Woohookitty, Zundark, 15 anonymous edits

Cyclic group  Source: http://en.wikipedia.org/w/index.php?oldid=408480709  Contributors: A2r4e1, Arcfrk, Arthur Rubin, Arved, AxelBoldt, Bentong Isles, Charles Matthews, Colonies Chris,DYLAN LENNON, David Eppstein, David.kaplan, DavidHouse, Dbenbenn, Dcoetzee, Deville, DniQ, Drschawrz, Dysprosia, Elroch, Eric Kvaalen, Fadereu, Fibonacci, Giftlite, Greg Kuperberg,Grubber, Hao2lian, Harrisonmetz, Helder.wiki, HellFire, Herbee, Ht686rg90, Iridescent, JackSchmidt, Jakob.scholbach, Jim.belk, Jjalexand, Joth, Juan Marquez, Kilva, LOL, Lambiam,LarryLACa, Linas, Lowellian, MathMartin, Michael Hardy, Michael Slone, Mike409, Minesweeper, Nahkh, Nubiatech, Obradovic Goran, PAR, Pako, Patrick, Paul D. Anderson, Pbroks13,Peruvianllama, Phys, PierreAbbat, Pocketfox, Raven in Orbit, Revolver, Rvollmert, SMP, Salix alba, Schildt.a, Selfworm, Seqsea, Siroxo, Sleske, Spindled, TakuyaMurata, Tarquin, The Stickler,Tosha, Vipul, Weregerbil, Zundark, 46 anonymous edits

Dihedral group  Source: http://en.wikipedia.org/w/index.php?oldid=407842612  Contributors: 4C, Albmont, AugPi, AxelBoldt, Babomb, Baccyak4H, Bender2k14, Bkell, Bobo192, CharlesMatthews, Chas zzz brown, CommonsDelinker, Dalf, David Radcliffe, Denelson83, Derek farn, Doubtingapostle, Dougofborg, DrBob, Dysprosia, Eliadtsai, Enchanter, Experiment123, Fropuff,Gabbe, Gauge, Giftlite, Hammer Raccoon, Happy-melon, Heath.gerhardt, Hetar, Idiazabal, Incnis Mrsi, JackSchmidt, Jim.belk, Jleedev, JoshuaZ, Kevin Lamoreau, LOL, MSGJ, Michael Hardy,Miyagawa, MrRedwood, Nbarth, Niteowlneils, PAR, Patrick, Paul August, PierreAbbat, Pleasantville, Prumpf, Rghthndsd, Rjwilmsi, Shahab, Tomruen, Tosha, Turgidson, Zundark, 33anonymous edits

Lattice (group)  Source: http://en.wikipedia.org/w/index.php?oldid=406367782  Contributors: Anonymous anonymous, Arcfrk, AxelBoldt, Bandwidthjunkie, Charles Matthews, David Eppstein,Flammifer, Flandre, Gene Ward Smith, Giftlite, Gregbard, Gvozdet, GyRo567, Hrushikesh Tilak, Ilmari Karonen, Jim.belk, Jitse Niesen, Js coron, Justin W Smith, Linas, Lopkiol, Matusz,McKay, Merewyn, Michael Hardy, Mikespedia, Msh210, Natalya, Natox, Patrick, R.e.b., Simetrical, Stebulus, Sverigekillen, Tobias Bergemann, Twri, Underdog, Zhw, Zundark, 33 anonymousedits

Wallpaper group  Source: http://en.wikipedia.org/w/index.php?oldid=406317983  Contributors: 99of9, Ahoerstemeier, Anomalocaris, AnomalousArtemis, Asmeurer, BigrTex, C S,Calculuslover, CesarB, Charles Matthews, Cimon Avaro, CommonsDelinker, Cwkmail, David Eppstein, Dmharvey, Dogears, Fibonacci, Gaius Cornelius, Giftlite, Greg Kuperberg, Hajor,HenkvD, JaGa, JackSchmidt, Jeff3000, Jim.belk, John Baez, Johntinker, Joseph Myers, KSmrq, Karlscherer3, Keenan Pepper, Kjoonlee, Lasunncty, Lowellian, Madmarigold, Magicmonster,Maproom, Martin von Gagern, Mayooranathan, Mhym, Michael Hardy, Mordomo, Mpatel, Nevit, Noe, Oleg Alexandrov, P0lyglut, Pak21, Paolo.dL, Patrick, Paul D. Anderson, Pengo, Phys,Prodego, RDBury, Rich Farmbrough, Salix alba, Schneelocke, Sculleyjp, Seberle, SilentC, Tedder, TimBentley, Titian1962, Tkircher, Tomruen, Tosha, Trapolator, Wikidsoup, Woohookitty, 73anonymous edits

Image Sources, Licenses and Contributors 86

Image Sources, Licenses and ContributorsImage:Sphere_symmetry_group_d2.png  Source: http://en.wikipedia.org/w/index.php?title=File:Sphere_symmetry_group_d2.png  License: Public Domain  Contributors: User:TomruenImage:Sphere_symmetry_group_d3.png  Source: http://en.wikipedia.org/w/index.php?title=File:Sphere_symmetry_group_d3.png  License: Public Domain  Contributors: Tom RuenImage:Sphere_symmetry_group_d2h.png  Source: http://en.wikipedia.org/w/index.php?title=File:Sphere_symmetry_group_d2h.png  License: Public Domain  Contributors: Maksim,NonenmacImage:Sphere_symmetry_group_d3h.png  Source: http://en.wikipedia.org/w/index.php?title=File:Sphere_symmetry_group_d3h.png  License: Public Domain  Contributors: Maksim,NonenmacImage:Sphere symmetry group d2d.png  Source: http://en.wikipedia.org/w/index.php?title=File:Sphere_symmetry_group_d2d.png  License: Public Domain  Contributors:Image:Sphere symmetry group d3d.png  Source: http://en.wikipedia.org/w/index.php?title=File:Sphere_symmetry_group_d3d.png  License: Public Domain  Contributors:Image:Sphere symmetry group t.png  Source: http://en.wikipedia.org/w/index.php?title=File:Sphere_symmetry_group_t.png  License: Public Domain  Contributors: MaksimImage:Sphere symmetry group td.png  Source: http://en.wikipedia.org/w/index.php?title=File:Sphere_symmetry_group_td.png  License: Public Domain  Contributors: TomruenImage:Sphere symmetry group th.png  Source: http://en.wikipedia.org/w/index.php?title=File:Sphere_symmetry_group_th.png  License: Public Domain  Contributors: MaksimImage:Sphere symmetry group o.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Sphere_symmetry_group_o.svg  License: Public Domain  Contributors: User:TomruenImage:Sphere symmetry group oh.png  Source: http://en.wikipedia.org/w/index.php?title=File:Sphere_symmetry_group_oh.png  License: Public Domain  Contributors: MaksimImage:Sphere symmetry group i.png  Source: http://en.wikipedia.org/w/index.php?title=File:Sphere_symmetry_group_i.png  License: Public Domain  Contributors: MaksimImage:Sphere symmetry group ih.png  Source: http://en.wikipedia.org/w/index.php?title=File:Sphere_symmetry_group_ih.png  License: Public Domain  Contributors: MaksimImage:Sphere symmetry group c1.png  Source: http://en.wikipedia.org/w/index.php?title=File:Sphere_symmetry_group_c1.png  License: Public Domain  Contributors: MaksimImage:Sphere symmetry group_c2.png  Source: http://en.wikipedia.org/w/index.php?title=File:Sphere_symmetry_group_c2.png  License: Public Domain  Contributors: TomruenImage:Sphere_symmetry_group_cs.png  Source: http://en.wikipedia.org/w/index.php?title=File:Sphere_symmetry_group_cs.png  License: Public Domain  Contributors: MaksimImage:Sphere_symmetry_group_c2v.png  Source: http://en.wikipedia.org/w/index.php?title=File:Sphere_symmetry_group_c2v.png  License: Public Domain  Contributors: Maksim, VerneEquinoxImage:Sphere_symmetry_group_c3v.png  Source: http://en.wikipedia.org/w/index.php?title=File:Sphere_symmetry_group_c3v.png  License: Public Domain  Contributors: Maksim, VerneEquinoxImage:Sphere symmetry group c2h.png  Source: http://en.wikipedia.org/w/index.php?title=File:Sphere_symmetry_group_c2h.png  License: Public Domain  Contributors: TomruenImage:Sphere symmetry group ci.png  Source: http://en.wikipedia.org/w/index.php?title=File:Sphere_symmetry_group_ci.png  License: Public Domain  Contributors: TomruenFile:GeometricKite.svg  Source: http://en.wikipedia.org/w/index.php?title=File:GeometricKite.svg  License: Public Domain  Contributors: User:DriniImage:Flag of Hong Kong.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Flag_of_Hong_Kong.svg  License: Public Domain  Contributors: Designed byImage:Snowflake8.png  Source: http://en.wikipedia.org/w/index.php?title=File:Snowflake8.png  License: Public Domain  Contributors: Howcheng, Nauticashades, Saperaud, WillowWImage:frieze2b.png  Source: http://en.wikipedia.org/w/index.php?title=File:Frieze2b.png  License: GNU Free Documentation License  Contributors: HenkvD, Ronaldino, 1 anonymous editsImage:Wallpaper group diagram p2.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_p2.png  License: Public Domain  Contributors: User:Martin vonGagernImage:Wallpaper group diagram p4.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_p4.png  License: Public Domain  Contributors: User:Martin vonGagernImage:Wallpaper group diagram p3.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_p3.png  License: Public Domain  Contributors: User:Martin vonGagernImage:Wallpaper group diagram p6.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_p6.png  License: Public Domain  Contributors: User:Martin vonGagernImage:Wallpaper group diagram pmm.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_pmm.png  License: Public Domain  Contributors: User:Martinvon GagernImage:Wallpaper group diagram cmm.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_cmm.png  License: Public Domain  Contributors: User:Martinvon GagernImage:Wallpaper group diagram p4m.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_p4m.png  License: Public Domain  Contributors: Denniss,Patrick, Zscout370Image:Wallpaper group diagram p6mm.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_p6mm.png  License: Public Domain  Contributors:User:Martin von GagernImage:Wallpaper group diagram p3m1.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_p3m1.png  License: Public Domain  Contributors: User:Martinvon GagernImage:Wallpaper group diagram p31m.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_p31m.png  License: Public Domain  Contributors: User:Martinvon GagernImage:Wallpaper group diagram p4g.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_p4g.png  License: Public Domain  Contributors: Denniss, Patrick,Zscout370Image:Wallpaper group diagram pm.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_pm.png  License: Public Domain  Contributors: User:Martin vonGagernImage:Wallpaper group diagram p1.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_p1.png  License: Public Domain  Contributors: User:Martin vonGagernImage:Wallpaper group diagram pg.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_pg.png  License: Public Domain  Contributors: User:Martin vonGagernImage:Wallpaper group diagram pmg.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_pmg.png  License: Public Domain  Contributors: User:Martinvon GagernImage:Wallpaper group diagram cm.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_cm.png  License: Public Domain  Contributors: User:Martin vonGagernImage:Wallpaper group diagram pgg.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_pgg.png  License: Public Domain  Contributors: User:Martin vonGagernImage:Geometricprisms.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Geometricprisms.gif  License: Public Domain  Contributors: MaksimImage:Pentagrammic prism.png  Source: http://en.wikipedia.org/w/index.php?title=File:Pentagrammic_prism.png  License: unknown  Contributors: TomruenImage:Pentagrammic antiprism.png  Source: http://en.wikipedia.org/w/index.php?title=File:Pentagrammic_antiprism.png  License: unknown  Contributors: TomruenImage:Snub square antiprism.png  Source: http://en.wikipedia.org/w/index.php?title=File:Snub_square_antiprism.png  License: GNU Free Documentation License  Contributors: Jidari,SharkDImage:antiprism5.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Antiprism5.jpg  License: GNU Free Documentation License  Contributors: Hellisp, SharkDImage:Pentagrammic crossed antiprism.png  Source: http://en.wikipedia.org/w/index.php?title=File:Pentagrammic_crossed_antiprism.png  License: unknown  Contributors: TomruenImage:Trapezohedron5.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Trapezohedron5.jpg  License: GNU Free Documentation License  Contributors: Hellisp, Joseolgon, SnailyImage:antiprism17.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Antiprism17.jpg  License: GNU Free Documentation License  Contributors: Geniac, Hellisp, Nonenmac, PieterKuiper, Snaily

Image Sources, Licenses and Contributors 87

image:tetrahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Tetrahedron.jpg  License: GNU Free Documentation License  Contributors: Dbenbenn, Kjell André, Matthias M.,Quasipalm, SharkD, WardenImage:Tetrahedral group 2.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Tetrahedral_group_2.svg  License: GNU Free Documentation License  Contributors: User Debivort onen.wikipediaImage:Tetrakishexahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Tetrakishexahedron.jpg  License: GNU Free Documentation License  Contributors: Maxim Razin,Oodzunadaira, Paddy, Quasipalm, SharkDimage:Gaelic football ball.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Gaelic_football_ball.jpg  License: GNU Free Documentation License  Contributors: Kanchelskis,TFCforeverImage:Snub tetrahedron.png  Source: http://en.wikipedia.org/w/index.php?title=File:Snub_tetrahedron.png  License: unknown  Contributors: Frankee 67, GeorgHH, Kilom691, MrBogus,RosarioVanTulpe, Sanbecimage:truncatedtetrahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Truncatedtetrahedron.jpg  License: GNU Free Documentation License  Contributors: BLueFiSH.as,Farmer Jan, Hellisp, Paddy, Quadell, SharkD, Tropyliumimage:triakistetrahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Triakistetrahedron.jpg  License: GNU Free Documentation License  Contributors: Maxim Razin,Oodzunadaira, Paddy, Quasipalm, SharkDImage:Tetrahemihexahedron.png  Source: http://en.wikipedia.org/w/index.php?title=File:Tetrahemihexahedron.png  License: unknown  Contributors: JidariImage:Octahemioctahedron 3-color.png  Source: http://en.wikipedia.org/w/index.php?title=File:Octahemioctahedron_3-color.png  License: Public Domain  Contributors: User:Tomruenimage:hexahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Hexahedron.jpg  License: GNU Free Documentation License  Contributors: Dbenbenn, Kjell André, Matthias M.,Quasipalm, SharkD, Str4nd, WikipediaMaster, 1 anonymous editsImage:Dual Cube-Octahedron.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Dual_Cube-Octahedron.svg  License: Creative Commons Attribution-Sharealike 2.5  Contributors:4CImage:Disdyakisdodecahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Disdyakisdodecahedron.jpg  License: GNU Free Documentation License  Contributors: Conscious,Maxim Razin, Oodzunadaira, Paddy, Quasipalm, SharkDimage:snubhexahedronccw.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Snubhexahedronccw.jpg  License: GNU Free Documentation License  Contributors: BLueFiSH.as,Hellisp, Paddy, SharkD, Tropyliumimage:pentagonalicositetrahedronccw.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Pentagonalicositetrahedronccw.jpg  License: GNU Free Documentation License Contributors: Maxim Razin, Oodzunadaira, Paddy, Quasipalm, SharkDimage:snubhexahedroncw.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Snubhexahedroncw.jpg  License: GNU Free Documentation License  Contributors: BLueFiSH.as, Hellisp,Paddy, SharkD, Tropyliumimage:pentagonalicositetrahedroncw.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Pentagonalicositetrahedroncw.jpg  License: GNU Free Documentation License  Contributors:Maxim Razin, Oodzunadaira, Paddy, Quasipalm, SharkDimage:octahedron.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Octahedron.svg  License: GNU Free Documentation License  Contributors: User:Stanneredimage:cuboctahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Cuboctahedron.jpg  License: GNU Free Documentation License  Contributors: Aknorals, BLueFiSH.as,Cbrown1023, Erina, Hellisp, Himasaram, Paddy, Quadell, SharkD, Tropyliumimage:truncatedhexahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Truncatedhexahedron.jpg  License: GNU Free Documentation License  Contributors: BLueFiSH.as,Cgoe, Hellisp, Paddy, Quadell, SharkD, Tropyliumimage:truncatedoctahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Truncatedoctahedron.jpg  License: GNU Free Documentation License  Contributors: BLueFiSH.as,Hellisp, Paddy, Quadell, SharkD, Tropyliumimage:rhombicuboctahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Rhombicuboctahedron.jpg  License: GNU Free Documentation License  Contributors: BLueFiSH.as,Hellisp, Lipedia, Paddy, SharkDimage:truncatedcuboctahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Truncatedcuboctahedron.jpg  License: GNU Free Documentation License  Contributors:BLueFiSH.as, Hellisp, Lipedia, Paddy, Quadell, SharkDimage:rhombicdodecahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Rhombicdodecahedron.jpg  License: GNU Free Documentation License  Contributors: Amalthea,Cyberpunk, EugeneZelenko, Lipedia, SharkD, Tintazulimage:triakisoctahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Triakisoctahedron.jpg  License: GNU Free Documentation License  Contributors: Angr, Maxim Razin,Oodzunadaira, Paddy, Quasipalm, SharkDimage:tetrakishexahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Tetrakishexahedron.jpg  License: GNU Free Documentation License  Contributors: Maxim Razin,Oodzunadaira, Paddy, Quasipalm, SharkDimage:deltoidalicositetrahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Deltoidalicositetrahedron.jpg  License: GNU Free Documentation License  Contributors:EugeneZelenko, Maxim Razin, Oodzunadaira, Paddy, Quasipalm, SharkDimage:disdyakisdodecahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Disdyakisdodecahedron.jpg  License: GNU Free Documentation License  Contributors: Conscious,Maxim Razin, Oodzunadaira, Paddy, Quasipalm, SharkDImage:Stella octangula.png  Source: http://en.wikipedia.org/w/index.php?title=File:Stella_octangula.png  License: Public Domain  Contributors: User:FropuffFile:Soccer ball.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Soccer_ball.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: User:Pumbaa80File:Sphere symmetry group i.png  Source: http://en.wikipedia.org/w/index.php?title=File:Sphere_symmetry_group_i.png  License: Public Domain  Contributors: MaksimFile:Sphere symmetry group ih.png  Source: http://en.wikipedia.org/w/index.php?title=File:Sphere_symmetry_group_ih.png  License: Public Domain  Contributors: MaksimFile:Disdyakistriacontahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Disdyakistriacontahedron.jpg  License: GNU Free Documentation License  Contributors: MaximRazin, Oodzunadaira, Paddy, Quasipalm, SharkDFile:POV-Ray-Dodecahedron.svg  Source: http://en.wikipedia.org/w/index.php?title=File:POV-Ray-Dodecahedron.svg  License: GNU Free Documentation License  Contributors: User:DTRFile:Icosahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Icosahedron.jpg  License: GNU Free Documentation License  Contributors: Bukk, Dbenbenn, Juiced lemon, KjellAndré, Matthias M., Nevit, Quasipalm, SharkDFile:truncateddodecahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Truncateddodecahedron.jpg  License: GNU Free Documentation License  Contributors: BLueFiSH.as,Hellisp, Paddy, Quadell, SharkD, TropyliumFile:truncatedicosidodecahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Truncatedicosidodecahedron.jpg  License: GNU Free Documentation License  Contributors:BLueFiSH.as, Hellisp, Lipedia, Paddy, Quadell, SharkDFile:truncatedicosahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Truncatedicosahedron.jpg  License: GNU Free Documentation License  Contributors: Hellisp, Lipedia,Paddy, Quadell, Rocket000, SharkDFile:rhombicosidodecahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Rhombicosidodecahedron.jpg  License: GNU Free Documentation License  Contributors: Amalthea,BLueFiSH.as, Hellisp, Lipedia, Paddy, SharkDFile:icosidodecahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Icosidodecahedron.jpg  License: GNU Free Documentation License  Contributors: Atropos235, BLueFiSH.as,Hellisp, Paddy, SharkD, Tropylium, 1 anonymous editsFile:triakisicosahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Triakisicosahedron.jpg  License: GNU Free Documentation License  Contributors: Maxim Razin,Oodzunadaira, Paddy, Quasipalm, SharkDFile:disdyakistriacontahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Disdyakistriacontahedron.jpg  License: GNU Free Documentation License  Contributors: MaximRazin, Oodzunadaira, Paddy, Quasipalm, SharkDFile:pentakisdodecahedron.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Pentakisdodecahedron.jpg  License: GNU Free Documentation License  Contributors: Maxim Razin,Oodzunadaira, Paddy, Quasipalm, SharkDFile:deltoidalhexecontahedron.jpg  Source: 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Image Sources, Licenses and Contributors 88

File:Rhombictriacontahedron.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Rhombictriacontahedron.svg  License: GNU Free Documentation License  Contributors: User:DTRImage:Parallelepipedon.png  Source: http://en.wikipedia.org/w/index.php?title=File:Parallelepipedon.png  License: Public Domain  Contributors: User:SverdrupImage:Square pyramid.png  Source: http://en.wikipedia.org/w/index.php?title=File:Square_pyramid.png  License: GNU Free Documentation License  Contributors: Jidari, Man vyi, SharkD,TamorlanImage:Elongated square pyramid.png  Source: http://en.wikipedia.org/w/index.php?title=File:Elongated_square_pyramid.png  License: GNU Free Documentation License  Contributors: Jidari,SharkDImage:Pentagonal pyramid.png  Source: http://en.wikipedia.org/w/index.php?title=File:Pentagonal_pyramid.png  License: GNU Free Documentation License  Contributors: GDK, Jidari,Oodzunadaira, SharkDImage:Symmetry.png  Source: http://en.wikipedia.org/w/index.php?title=File:Symmetry.png  License: unknown  Contributors: User:Dbc334Image:Uniaxial.png  Source: http://en.wikipedia.org/w/index.php?title=File:Uniaxial.png  License: Creative Commons Attribution-Sharealike 2.5  Contributors: AndrewKepert, AnonMoos,Derlay, 1 anonymous editsImage:Volleyball seams diagram.png  Source: http://en.wikipedia.org/w/index.php?title=File:Volleyball_seams_diagram.png  License: Public Domain  Contributors: TamfangImage:GroupDiagramMiniC2C4.png  Source: http://en.wikipedia.org/w/index.php?title=File:GroupDiagramMiniC2C4.png  License: Public Domain  Contributors: Helder.wiki, Irigi,Joey-das-WBF, Kilom691, NbarthImage:GroupDiagramMiniC2C6.png  Source: http://en.wikipedia.org/w/index.php?title=File:GroupDiagramMiniC2C6.png  License: Public Domain  Contributors: Irigi, 1 anonymous editsImage:GroupDiagramMiniC2C8.png  Source: http://en.wikipedia.org/w/index.php?title=File:GroupDiagramMiniC2C8.png  License: Public Domain  Contributors: Helder.wiki,Joey-das-WBF, Maksim, NbarthImage:GroupDiagramMiniC2x3.png  Source: http://en.wikipedia.org/w/index.php?title=File:GroupDiagramMiniC2x3.png  License: Public Domain  Contributors: Helder.wiki, Joey-das-WBF,Maksim, NbarthImage:GroupDiagramMiniC2D8.png  Source: http://en.wikipedia.org/w/index.php?title=File:GroupDiagramMiniC2D8.png  License: Public Domain  Contributors: Helder.wiki,Joey-das-WBF, Maksim, NbarthFile:Cyclic group.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Cyclic_group.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: User:Jakob.scholbach,User:Pbroks13File:GroupDiagramMiniC1.png  Source: http://en.wikipedia.org/w/index.php?title=File:GroupDiagramMiniC1.png  License: Public Domain  Contributors: Amit6, Joey-das-WBF, Maksim,Nbarth, 1 anonymous editsFile:GroupDiagramMiniC2.png  Source: http://en.wikipedia.org/w/index.php?title=File:GroupDiagramMiniC2.png  License: Public Domain  Contributors: Helder.wiki, Joey-das-WBF,Maksim, NbarthFile:GroupDiagramMiniC3.png  Source: http://en.wikipedia.org/w/index.php?title=File:GroupDiagramMiniC3.png  License: Public Domain  Contributors: Helder.wiki, Ilmari Karonen,Joey-das-WBF, Maksim, Nbarth, PARFile:GroupDiagramMiniC4.png  Source: http://en.wikipedia.org/w/index.php?title=File:GroupDiagramMiniC4.png  License: Public Domain  Contributors: Helder.wiki, Joey-das-WBF,Maksim, NbarthFile:GroupDiagramMiniC5.png  Source: http://en.wikipedia.org/w/index.php?title=File:GroupDiagramMiniC5.png  License: Public Domain  Contributors: Helder.wiki, Ilmari Karonen,Joey-das-WBF, Maksim, NbarthFile:GroupDiagramMiniC6.png  Source: http://en.wikipedia.org/w/index.php?title=File:GroupDiagramMiniC6.png  License: Public Domain  Contributors: Helder.wiki, Joey-das-WBF,Maksim, NbarthFile:GroupDiagramMiniC7.png  Source: http://en.wikipedia.org/w/index.php?title=File:GroupDiagramMiniC7.png  License: Public Domain  Contributors: Helder.wiki, Joey-das-WBF,Maksim, NbarthFile:GroupDiagramMiniC8.png  Source: http://en.wikipedia.org/w/index.php?title=File:GroupDiagramMiniC8.png  License: Public Domain  Contributors: Helder.wiki, Joey-das-WBF,Maksim, NbarthFile:Snowflake8.png  Source: http://en.wikipedia.org/w/index.php?title=File:Snowflake8.png  License: Public Domain  Contributors: Howcheng, Nauticashades, Saperaud, WillowWFile:Hexagon Reflections.png  Source: http://en.wikipedia.org/w/index.php?title=File:Hexagon_Reflections.png  License: Public Domain  Contributors: Grafite, NbarthFile:Dihedral8.png  Source: http://en.wikipedia.org/w/index.php?title=File:Dihedral8.png  License: Public Domain  Contributors: Jim.belkFile:Labeled Triangle Reflections.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Labeled_Triangle_Reflections.svg  License: Public Domain  Contributors: User:Jim.belkFile:Two Reflection Rotation.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Two_Reflection_Rotation.svg  License: Public Domain  Contributors: User:Jim.belkFile:Pentagon Linear.png  Source: http://en.wikipedia.org/w/index.php?title=File:Pentagon_Linear.png  License: Public Domain  Contributors: Jim.belkFile:GroupDiagramMiniD4.png  Source: http://en.wikipedia.org/w/index.php?title=File:GroupDiagramMiniD4.png  License: Public Domain  Contributors: Helder.wiki, Joey-das-WBF,Maksim, NbarthFile:GroupDiagramMiniD6.png  Source: http://en.wikipedia.org/w/index.php?title=File:GroupDiagramMiniD6.png  License: Public Domain  Contributors: Helder.wiki, Ilmari Karonen,Joey-das-WBF, Maksim, Nbarth, PARFile:GroupDiagramMiniD8.png  Source: http://en.wikipedia.org/w/index.php?title=File:GroupDiagramMiniD8.png  License: Public Domain  Contributors: Helder.wiki, Joey-das-WBF,Maksim, NbarthFile:GroupDiagramMiniD10.png  Source: http://en.wikipedia.org/w/index.php?title=File:GroupDiagramMiniD10.png  License: Public Domain  Contributors: Helder.wiki, Joey-das-WBF,Maksim, NbarthFile:GroupDiagramMiniD12.png  Source: http://en.wikipedia.org/w/index.php?title=File:GroupDiagramMiniD12.png  License: Public Domain  Contributors: Helder.wiki, Joey-das-WBF,Maksim, NbarthFile:GroupDiagramMiniD14.png  Source: http://en.wikipedia.org/w/index.php?title=File:GroupDiagramMiniD14.png  License: Public Domain  Contributors: Helder.wiki, Joey-das-WBF,Maksim, NbarthFile:Dihedral4.png  Source: http://en.wikipedia.org/w/index.php?title=File:Dihedral4.png  License: GNU Free Documentation License  Contributors: MaksimFile:d8isNonAbelian.png  Source: http://en.wikipedia.org/w/index.php?title=File:D8isNonAbelian.png  License: GNU Free Documentation License  Contributors: MaksimFile:Red Star of David.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Red_Star_of_David.svg  License: Public Domain  Contributors: User:Denelson83, user:Denelson83File:Ashoka Chakra.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Ashoka_Chakra.svg  License: Public Domain  Contributors: Abhishekjoshi, Davin7, Fred the Oyster, Fry1989,Homo lupus, Incnis Mrsi, Kwasura, Madden, Nichalp, Patrick, Rocket000, Roland zh, Xiengyod, Zscout370, 1 anonymous editsFile:Equilateral Triangle Lattice.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Equilateral_Triangle_Lattice.svg  License: Public Domain  Contributors: User:Jim.belkFile:Rhombic Lattice.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Rhombic_Lattice.svg  License: Public Domain  Contributors: User:Jim.belkFile:ModularGroup-FundamentalDomain-01.png  Source: http://en.wikipedia.org/w/index.php?title=File:ModularGroup-FundamentalDomain-01.png  License: GNU Free DocumentationLicense  Contributors: FropuffImage:Wallpaper group-p4m-5.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p4m-5.jpg  License: Public Domain  Contributors: Owen JonesImage:Wallpaper_group-p4m-2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p4m-2.jpg  License: Public Domain  Contributors: Dogears, GeorgHH, Juicedlemon, MaksimImage:Wallpaper_group-p4m-1.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p4m-1.jpg  License: Public Domain  Contributors: Dogears, GeorgHH, Juicedlemon, MaksimImage:Wallpaper_group-p4g-2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p4g-2.jpg  License: Public Domain  Contributors: Dogears, GeorgHH, Juicedlemon, MaksimImage:Wallpaper group diagram legend rotation2.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_legend_rotation2.svg  License: Public Domain Contributors: User:Martin von GagernImage:Wallpaper group diagram legend rotation3.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_legend_rotation3.svg  License: Public Domain Contributors: User:Martin von Gagern

Image Sources, Licenses and Contributors 89

Image:Wallpaper group diagram legend rotation4.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_legend_rotation4.svg  License: Public Domain Contributors: User:Martin von GagernImage:Wallpaper group diagram legend rotation6.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_legend_rotation6.svg  License: Public Domain Contributors: User:Martin von GagernImage:Wallpaper group diagram legend reflection.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_legend_reflection.svg  License: Public Domain Contributors: User:Martin von GagernImage:Wallpaper group diagram legend glide reflection.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_legend_glide_reflection.svg  License: PublicDomain  Contributors: User:Martin von GagernImage:SymBlend p1.svg  Source: http://en.wikipedia.org/w/index.php?title=File:SymBlend_p1.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Martin von GagernImage:Wallpaper_group-cell-p1.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cell-p1.png  License: GNU Free Documentation License  Contributors:ChongDae, Dongseok86Image:Wallpaper group diagram p1.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_p1.svg  License: Public Domain  Contributors: User:Martin vonGagernImage:WallpaperP1.GIF  Source: http://en.wikipedia.org/w/index.php?title=File:WallpaperP1.GIF  License: GNU Free Documentation License  Contributors: AnomalousArtemis, Quadell,Sfan00 IMG, Zscout370Image:Wallpaper_group-p1-3.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p1-3.jpg  License: GNU Free Documentation License  Contributors: Owen JonesImage:SymBlend p2.svg  Source: http://en.wikipedia.org/w/index.php?title=File:SymBlend_p2.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Martin von GagernImage:Wallpaper_group-cell-p2.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cell-p2.png  License: GNU Free Documentation License  Contributors:ChongDae, GeorgHHImage:Wallpaper group diagram p2.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_p2.svg  License: Public Domain  Contributors: User:Martin vonGagernImage:WallpaperP2.GIF  Source: http://en.wikipedia.org/w/index.php?title=File:WallpaperP2.GIF  License: GNU Free Documentation License  Contributors: AnomalousArtemis, Quadell,Sfan00 IMG, Zscout370Image:Wallpaper_group-p2-1.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p2-1.jpg  License: Public Domain  Contributors: w:Owen Jones (architect)OwenJones, uploaded to the English Wikipedia byImage:Wallpaper_group-p2-2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p2-2.jpg  License: Public Domain  Contributors: w:Owen Jones (architect)OwenJones, uploaded to the English Wikipedia byImage:Wallpaper_group-p2-2 detail 2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p2-2_detail_2.jpg  License: Public Domain  Contributors: GeorgHH, Joolz,Patrick, 1 anonymous editsImage:Wallpaper_group-p2-3.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p2-3.jpg  License: Public Domain  Contributors: w:Owen Jones (architect)OwenJones, uploaded to the English Wikipedia byImage:Wallpaper_group-p2-4.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p2-4.jpg  License: unknown  Contributors: User:Dmharvey at the English WikipediaImage:SymBlend pm.svg  Source: http://en.wikipedia.org/w/index.php?title=File:SymBlend_pm.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Martin von GagernImage:Wallpaper_group-cell-pm.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cell-pm.png  License: GNU Free Documentation License  Contributors: ToePeuImage:Wallpaper group diagram pm.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_pm.svg  License: Public Domain  Contributors: User:Martin vonGagernImage:WallpaperPM.gif  Source: http://en.wikipedia.org/w/index.php?title=File:WallpaperPM.gif  License: GNU Free Documentation License  Contributors: Original uploader wasAnomalousArtemis at en.wikipediaImage:Wallpaper_group-pm-3.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-pm-3.jpg  License: Public Domain  Contributors: Original uploader was Dmharveyat en.wikipediaImage:Wallpaper_group-pm-4.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-pm-4.jpg  License: Public Domain  Contributors: Original uploader was Dmharveyat en.wikipediaImage:Wallpaper_group-pm-1.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-pm-1.jpg  License: Public Domain  Contributors: Original uploader was Dmharveyat en.wikipediaImage:Wallpaper_group-pm-5.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-pm-5.jpg  License: Public Domain  Contributors: Original uploader was Dmharveyat en.wikipediaImage:SymBlend pg.svg  Source: http://en.wikipedia.org/w/index.php?title=File:SymBlend_pg.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Martin von GagernImage:Wallpaper_group-cell-pg.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cell-pg.png  License: GNU Free Documentation License  Contributors:ChongDae, GeorgHHImage:Wallpaper group diagram pg.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_pg.svg  License: Public Domain  Contributors: User:Martin vonGagernImage:WallpaperPG.GIF  Source: http://en.wikipedia.org/w/index.php?title=File:WallpaperPG.GIF  License: GNU Free Documentation License  Contributors: AnomalousArtemis, Quadell,Sfan00 IMG, Zscout370, 1 anonymous editsImage:Wallpaper_group-pg-1.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-pg-1.jpg  License: Public Domain  Contributors: Original uploader was Dmharveyat en.wikipediaImage:Wallpaper_group-pg-1 detail.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-pg-1_detail.jpg  License: Public Domain  Contributors: Denniss, Patrick,ToobazImage:Wallpaper_group-pg-2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-pg-2.jpg  License: Public Domain  Contributors: Dmharvey, TomruenImage:Tile 33434.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Tile_33434.svg  License: GNU Free Documentation License  Contributors: User:Fibonacci, User:FibonacciImage:Wallpaper group-cell-cm.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cell-cm.png  License: GNU Free Documentation License  Contributors:GeorgHH, Maksim, 1 anonymous editsImage:Wallpaper group diagram cm.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_cm.svg  License: Public Domain  Contributors: User:Martin vonGagernImage:WallpaperCM.GIF  Source: http://en.wikipedia.org/w/index.php?title=File:WallpaperCM.GIF  License: GNU Free Documentation License  Contributors: AnomalousArtemis, Quadell,Sfan00 IMG, Zscout370Image:Wallpaper_group-cm-1.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cm-1.jpg  License: Public Domain  Contributors: ChongDae, GeorgHH, Nevit,NonenmacImage:Wallpaper_group-cm-2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cm-2.jpg  License: Public Domain  Contributors: ChongDae, GeorgHH, Nevit,NonenmacImage:Wallpaper_group-cm-3.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cm-3.jpg  License: Public Domain  Contributors: ChongDae, GeorgHH, Nevit,NonenmacImage:Wallpaper_group-cm-4.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cm-4.jpg  License: Public Domain  Contributors: ChongDae, GeorgHH, Nevit,NonenmacImage:Wallpaper_group-cm-5.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cm-5.jpg  License: Public Domain  Contributors: ChongDae, GeorgHH, Nevit,NonenmacImage:Wallpaper_group-cm-6.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cm-6.jpg  License: Public Domain  Contributors: UnknownImage:Wallpaper_group-cm-7.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cm-7.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:Wallpaper_group-pm-2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-pm-2.jpg  License: Public Domain  Contributors: Dmharvey, Dogears

Image Sources, Licenses and Contributors 90

Image:SymBlend pmm.svg  Source: http://en.wikipedia.org/w/index.php?title=File:SymBlend_pmm.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Martin vonGagernImage:Wallpaper_group-cell-pmm.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cell-pmm.png  License: GNU Free Documentation License  Contributors:ChongDae, GeorgHHImage:Wallpaper group diagram pmm.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_pmm.svg  License: Public Domain  Contributors: User:Martinvon GagernImage:WallpaperPMM.GIF  Source: http://en.wikipedia.org/w/index.php?title=File:WallpaperPMM.GIF  License: GNU Free Documentation License  Contributors: AnomalousArtemis,Lasunncty, QuadellImage:Wallpaper_group-pmm-1.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-pmm-1.jpg  License: Public Domain  Contributors: ChongDae, GeorgHH, Wst,Zscout370Image:Wallpaper_group-pmm-2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-pmm-2.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:Wallpaper_group-pmm-3.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-pmm-3.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:Wallpaper_group-pmm-4.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-pmm-4.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:2-d dense packing r1.svg  Source: http://en.wikipedia.org/w/index.php?title=File:2-d_dense_packing_r1.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors:User:99of9Image:2-d dense packing r3.svg  Source: http://en.wikipedia.org/w/index.php?title=File:2-d_dense_packing_r3.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors:User:99of9Image:2-d dense packing r7.svg  Source: http://en.wikipedia.org/w/index.php?title=File:2-d_dense_packing_r7.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors:User:99of9Image:SymBlend pmg.svg  Source: http://en.wikipedia.org/w/index.php?title=File:SymBlend_pmg.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Martin von GagernImage:Wallpaper_group-cell-pmg.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cell-pmg.png  License: GNU Free Documentation License  Contributors:Original uploader was Dmharvey at en.wikipediaImage:Wallpaper group diagram pmg.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_pmg.svg  License: Public Domain  Contributors: User:Martin vonGagernImage:WallpaperPMG.GIF  Source: http://en.wikipedia.org/w/index.php?title=File:WallpaperPMG.GIF  License: GNU Free Documentation License  Contributors: AnomalousArtemis,Quadell, Sfan00 IMG, Zscout370Image:Wallpaper_group-pmg-1.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-pmg-1.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:Wallpaper_group-pmg-2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-pmg-2.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:Wallpaper_group-pmg-3.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-pmg-3.jpg  License: Public Domain  Contributors: ChongDae, GeorgHH,Zscout370Image:Wallpaper_group-pmg-4.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-pmg-4.jpg  License: Public Domain  Contributors: ChongDae, GeorgHH,NeithsabesImage:2-d pentagon packing.svg  Source: http://en.wikipedia.org/w/index.php?title=File:2-d_pentagon_packing.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors:User:99of9Image:SymBlend pgg.svg  Source: http://en.wikipedia.org/w/index.php?title=File:SymBlend_pgg.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Martin von GagernImage:Wallpaper_group-cell-pgg.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cell-pgg.png  License: GNU Free Documentation License  Contributors:Original uploader was Dmharvey at en.wikipediaImage:Wallpaper group diagram pgg.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_pgg.svg  License: Public Domain  Contributors: User:Martin vonGagernImage:WallpaperPGG.GIF  Source: http://en.wikipedia.org/w/index.php?title=File:WallpaperPGG.GIF  License: GNU Free Documentation License  Contributors: Original uploader wasAnomalousArtemis at en.wikipediaImage:Wallpaper_group-pgg-1.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-pgg-1.jpg  License: Public Domain  Contributors: Original uploader wasDmharvey at en.wikipediaImage:Wallpaper_group-pgg-2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-pgg-2.jpg  License: Public Domain  Contributors: Original uploader wasDmharvey at en.wikipediaImage:Wallpaper group-cell-cmm.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cell-cmm.png  License: GNU Free Documentation License  Contributors:GeorgHH, Maksim, 1 anonymous editsImage:Wallpaper group diagram cmm.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_cmm.svg  License: Public Domain  Contributors: User:Martinvon GagernImage:WallpaperCMM.GIF  Source: http://en.wikipedia.org/w/index.php?title=File:WallpaperCMM.GIF  License: GNU Free Documentation License  Contributors: AnomalousArtemis,Quadell, Sfan00 IMG, Zscout370Image:Tile 33344.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Tile_33344.svg  License: GNU Free Documentation License  Contributors: User:Fibonacci, User:FibonacciImage:Wallpaper_group-cmm-1.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cmm-1.jpg  License: Public Domain  Contributors: ChongDae, Zscout370Image:Wallpaper_group-cmm-2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cmm-2.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:Wallpaper_group-cmm-3.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cmm-3.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:Wallpaper_group-cmm-4.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cmm-4.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:Wallpaper_group-cmm-5.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cmm-5.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:Wallpaper_group-cmm-6.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cmm-6.jpg  License: Public Domain  Contributors: ChongDae, GeorgHH, NevitImage:SymBlend p4.svg  Source: http://en.wikipedia.org/w/index.php?title=File:SymBlend_p4.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Martin von GagernImage:Wallpaper_group-cell-p4.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cell-p4.png  License: GNU Free Documentation License  Contributors: Originaluploader was Dmharvey at en.wikipediaImage:Wallpaper group diagram p4.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_p4.svg  License: Public Domain  Contributors: User:Martin vonGagernImage:WallpaperP4.GIF  Source: http://en.wikipedia.org/w/index.php?title=File:WallpaperP4.GIF  License: GNU Free Documentation License  Contributors: AnomalousArtemis, Quadell,Sfan00 IMG, Zscout370Image:Wallpaper_group-p4-1.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p4-1.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:Wallpaper_group-p4-2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p4-2.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:Wallpaper_group-p4-3.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p4-3.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:Wallpaper_group-p4-4.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p4-4.jpg  License: Public Domain  Contributors: ChongDae, GeorgHH, Nevit,Zscout370Image:Wallpaper_group-p4-5.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p4-5.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:SymBlend p4m.svg  Source: http://en.wikipedia.org/w/index.php?title=File:SymBlend_p4m.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Martin von GagernImage:Wallpaper_group-cell-p4m.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cell-p4m.png  License: GNU Free Documentation License  Contributors:ChongDae, GeorgHHImage:Wallpaper group diagram p4m.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_p4m.svg  License: Public Domain  Contributors: User:Martin vonGagern

Image Sources, Licenses and Contributors 91

Image:WallpaperP4M.GIF  Source: http://en.wikipedia.org/w/index.php?title=File:WallpaperP4M.GIF  License: GNU Free Documentation License  Contributors: AnomalousArtemis, Quadell,Sfan00 IMG, Zscout370Image:Tile 4,4.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Tile_4,4.svg  License: Public Domain  Contributors: User:Fibonacci, User:FibonacciImage:Tile V488.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Tile_V488.svg  License: Public Domain  Contributors: User:Fibonacci, User:FibonacciImage:Tile 488.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Tile_488.svg  License: GNU Free Documentation License  Contributors: User:Fibonacci, User:FibonacciImage:Wallpaper_group-p4m-3.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p4m-3.jpg  License: Public Domain  Contributors: ChongDae, GeorgHH, Nevit,Zscout370Image:Wallpaper_group-p4m-5.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p4m-5.jpg  License: Public Domain  Contributors: Owen JonesImage:Wallpaper_group-p4m-6.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p4m-6.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:2-d dense packing r4.svg  Source: http://en.wikipedia.org/w/index.php?title=File:2-d_dense_packing_r4.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors:User:99of9Image:Wallpaper_group-p4m-4.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p4m-4.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:Wallpaper_group-p4m-7.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p4m-7.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:Wallpaper_group-p4m-8.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p4m-8.jpg  License: Public Domain  Contributors: ChongDae, GeorgHH, NevitImage:SymBlend p4g.svg  Source: http://en.wikipedia.org/w/index.php?title=File:SymBlend_p4g.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Martin von GagernImage:Wallpaper_group-cell-p4g.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cell-p4g.png  License: GNU Free Documentation License  Contributors:GeorgHH, MaksimImage:Wallpaper group diagram p4g.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_p4g.svg  License: Public Domain  Contributors: User:Martin vonGagernImage:WallpaperP4G.GIF  Source: http://en.wikipedia.org/w/index.php?title=File:WallpaperP4G.GIF  License: GNU Free Documentation License  Contributors: AnomalousArtemis,Lasunncty, QuadellImage:Wallpaper_group-p4g-1.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p4g-1.jpg  License: Public Domain  Contributors: ChongDae, Zscout370Image:Wallpaper_group-p4g-3.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p4g-3.jpg  License: Public Domain  Contributors: ChongDae, GeorgHH, Nevit,Zscout370Image:Wallpaper_group-p4g-4.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p4g-4.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHFile:Uniform tiling 44-h01.png  Source: http://en.wikipedia.org/w/index.php?title=File:Uniform_tiling_44-h01.png  License: Public Domain  Contributors: TomruenImage:Wallpaper group-cell-p3.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cell-p3.png  License: GNU Free Documentation License  Contributors: GeorgHH,Maksim, 1 anonymous editsImage:Wallpaper group diagram p3.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_p3.svg  License: Public Domain  Contributors: User:Martin vonGagernImage:WallpaperP3.GIF  Source: http://en.wikipedia.org/w/index.php?title=File:WallpaperP3.GIF  License: GNU Free Documentation License  Contributors: AnomalousArtemis, Quadell,Sfan00 IMG, Zscout370Image:Tile 33336.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Tile_33336.svg  License: GNU Free Documentation License  Contributors: User:Fibonacci, User:FibonacciImage:Wallpaper_group-p3-1.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p3-1.jpg  License: Public Domain  Contributors: w:en:User:DmharveyDmharveyImage:Alhambra-p3-closeup.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Alhambra-p3-closeup.jpg  License: Creative Commons Attribution-Sharealike 2.5  Contributors:User:DmharveyImage:SymBlend p3m1.svg  Source: http://en.wikipedia.org/w/index.php?title=File:SymBlend_p3m1.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Martin vonGagernImage:Wallpaper_group-cell-p3m1.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cell-p3m1.png  License: GNU Free Documentation License  Contributors:Original uploader was Dmharvey at en.wikipediaImage:Wallpaper group diagram p3m1.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_p3m1.svg  License: Public Domain  Contributors: User:Martinvon GagernImage:Tile 3,6.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Tile_3,6.svg  License: Public Domain  Contributors: User:Fibonacci, User:FibonacciImage:Tile 6,3.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Tile_6,3.svg  License: Public Domain  Contributors: User:Fibonacci, User:FibonacciImage:Tile 3bb.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Tile_3bb.svg  License: GNU Free Documentation License  Contributors: User:Fibonacci, User:FibonacciImage:Wallpaper_group-p3m1-1.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p3m1-1.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:Wallpaper_group-p3m1-3.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p3m1-3.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:Wallpaper_group-p3m1-2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p3m1-2.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:WallpaperP3M1.GIF  Source: http://en.wikipedia.org/w/index.php?title=File:WallpaperP3M1.GIF  License: GNU Free Documentation License  Contributors: AnomalousArtemis,Lasunncty, QuadellImage:2-d dense packing r2.svg  Source: http://en.wikipedia.org/w/index.php?title=File:2-d_dense_packing_r2.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors:User:99of9Image:SymBlend p31m.svg  Source: http://en.wikipedia.org/w/index.php?title=File:SymBlend_p31m.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Martin vonGagernImage:Wallpaper_group-cell-p31m.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cell-p31m.png  License: GNU Free Documentation License  Contributors:Original uploader was Dmharvey at en.wikipediaImage:Wallpaper group diagram p31m.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_p31m.svg  License: Public Domain  Contributors: User:Martinvon GagernImage:Wallpaper_group-p31m-1.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p31m-1.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:Wallpaper_group-p31m-2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p31m-2.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:Wallpaper_group-p31m-3.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p31m-3.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:WallpaperP31M.GIF  Source: http://en.wikipedia.org/w/index.php?title=File:WallpaperP31M.GIF  License: GNU Free Documentation License  Contributors: AnomalousArtemis,Lasunncty, QuadellImage:SymBlend p6.svg  Source: http://en.wikipedia.org/w/index.php?title=File:SymBlend_p6.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Martin von GagernImage:Wallpaper_group-cell-p6.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cell-p6.png  License: GNU Free Documentation License  Contributors: Originaluploader was Dmharvey at en.wikipediaImage:Wallpaper group diagram p6.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_p6.svg  License: Public Domain  Contributors: User:Martin vonGagernImage:WallpaperP6.GIF  Source: http://en.wikipedia.org/w/index.php?title=File:WallpaperP6.GIF  License: GNU Free Documentation License  Contributors: AnomalousArtemis, Quadell,Sfan00 IMG, Zscout370Image:Wallpaper_group-p6-1.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p6-1.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:Wallpaper_group-p6-2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p6-2.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:SymBlend p6m.svg  Source: http://en.wikipedia.org/w/index.php?title=File:SymBlend_p6m.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Martin von GagernImage:Wallpaper_group-cell-p6m.png  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-cell-p6m.png  License: GNU Free Documentation License  Contributors:ChongDae, GeorgHH

Image Sources, Licenses and Contributors 92

Image:Wallpaper group diagram p6m.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group_diagram_p6m.svg  License: Public Domain  Contributors: User:Martin vonGagernImage:WallpaperP6M.GIF  Source: http://en.wikipedia.org/w/index.php?title=File:WallpaperP6M.GIF  License: GNU Free Documentation License  Contributors: ChongDae, NevitImage:Tile 3636.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Tile_3636.svg  License: Public Domain  Contributors: User:Fibonacci, User:FibonacciImage:Tile 3464.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Tile_3464.svg  License: GNU Free Documentation License  Contributors: User:Fibonacci, User:FibonacciImage:Tile 46b.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Tile_46b.svg  License: GNU Free Documentation License  Contributors: User:Fibonacci, User:FibonacciImage:Wallpaper_group-p6m-1.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p6m-1.jpg  License: Public Domain  Contributors: Anarkman, Kilom691, NevitImage:Wallpaper_group-p6m-2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p6m-2.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:Wallpaper_group-p6m-3.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p6m-3.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:Wallpaper_group-p6m-4.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p6m-4.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:Wallpaper_group-p6m-5.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p6m-5.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:Wallpaper_group-p6m-6.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Wallpaper_group-p6m-6.jpg  License: Public Domain  Contributors: ChongDae, GeorgHHImage:2-d dense packing r5.svg  Source: http://en.wikipedia.org/w/index.php?title=File:2-d_dense_packing_r5.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors:User:99of9Image:2-d dense packing r6.svg  Source: http://en.wikipedia.org/w/index.php?title=File:2-d_dense_packing_r6.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors:User:99of9

License 93

LicenseCreative Commons Attribution-Share Alike 3.0 Unportedhttp:/ / creativecommons. org/ licenses/ by-sa/ 3. 0/