Archimedean Solid2stdyust

7/29/2019 Archimedean Solid2stdyust

1/13

Archimedean Solid

The 13 Archimedean solids are theconvex polyhedrathat have a similar arrangement of nonintersecting

regularconvex polygonsof two or more different types arranged in the same way about each vertexwith

all sides the same length (Cromwell 1997, pp. 91-92).

The Archimedean solids are distinguished by having very high symmetry, thus excluding solids belonging

to adihedral groupof symmetries (e.g., the two infinite families of regular prisms and antiprisms), as well

as theelongated square gyrobicupola(because that surface's symmetry-breaking twist allows vertices

"near the equator" and those "in the polar regions" to be distinguished; Cromwell 1997, p. 92). The

Archimedean solids are sometimes also referred to as thesemiregular polyhedra.
http://mathworld.wolfram.com/ConvexPolyhedron.htmlhttp://mathworld.wolfram.com/ConvexPolyhedron.htmlhttp://mathworld.wolfram.com/ConvexPolyhedron.htmlhttp://mathworld.wolfram.com/RegularPolygon.htmlhttp://mathworld.wolfram.com/ConvexPolygon.htmlhttp://mathworld.wolfram.com/ConvexPolygon.htmlhttp://mathworld.wolfram.com/ConvexPolygon.htmlhttp://mathworld.wolfram.com/PolyhedronVertex.htmlhttp://mathworld.wolfram.com/PolyhedronVertex.htmlhttp://mathworld.wolfram.com/PolyhedronVertex.htmlhttp://mathworld.wolfram.com/DihedralGroup.htmlhttp://mathworld.wolfram.com/DihedralGroup.htmlhttp://mathworld.wolfram.com/DihedralGroup.htmlhttp://mathworld.wolfram.com/ElongatedSquareGyrobicupola.htmlhttp://mathworld.wolfram.com/ElongatedSquareGyrobicupola.htmlhttp://mathworld.wolfram.com/ElongatedSquareGyrobicupola.htmlhttp://mathworld.wolfram.com/SemiregularPolyhedron.htmlhttp://mathworld.wolfram.com/SemiregularPolyhedron.htmlhttp://mathworld.wolfram.com/SemiregularPolyhedron.htmlhttp://mathworld.wolfram.com/notebooks/Polyhedra/ArchimedeanSolid.nbhttp://mathworld.wolfram.com/notebooks/Polyhedra/ArchimedeanSolid.nbhttp://mathworld.wolfram.com/notebooks/Polyhedra/ArchimedeanSolid.nbhttp://mathworld.wolfram.com/notebooks/Polyhedra/ArchimedeanSolid.nbhttp://mathworld.wolfram.com/SemiregularPolyhedron.htmlhttp://mathworld.wolfram.com/ElongatedSquareGyrobicupola.htmlhttp://mathworld.wolfram.com/DihedralGroup.htmlhttp://mathworld.wolfram.com/PolyhedronVertex.htmlhttp://mathworld.wolfram.com/ConvexPolygon.htmlhttp://mathworld.wolfram.com/RegularPolygon.htmlhttp://mathworld.wolfram.com/ConvexPolyhedron.html


2/13

The Archimedean solids are illustrated above.


3/13

Nets of the Archimedean solids are illustrated above.

The following table lists the uniform, Schlfli, Wythoff, and Cundy and Rollett symbols for the

Archimedean solids (Wenninger 1989, p. 9).

soliduniform

polyhedron

Schlfli

symbol

Wythoff

symbol

Cundy and Rollett

symbol

1 cuboctahedron

2great

rhombicosidodecahedron t4.6.10

3 great rhombicuboctahedron t

4.6.8

4 icosidodecahedron

5small

rhombicosidodecahedron r3.4.5.4

6 small rhombicuboctahedron r
http://mathworld.wolfram.com/UniformPolyhedron.htmlhttp://mathworld.wolfram.com/UniformPolyhedron.htmlhttp://mathworld.wolfram.com/UniformPolyhedron.htmlhttp://mathworld.wolfram.com/SchlaefliSymbol.htmlhttp://mathworld.wolfram.com/SchlaefliSymbol.htmlhttp://mathworld.wolfram.com/SchlaefliSymbol.htmlhttp://mathworld.wolfram.com/WythoffSymbol.htmlhttp://mathworld.wolfram.com/WythoffSymbol.htmlhttp://mathworld.wolfram.com/WythoffSymbol.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/WythoffSymbol.htmlhttp://mathworld.wolfram.com/WythoffSymbol.htmlhttp://mathworld.wolfram.com/SchlaefliSymbol.htmlhttp://mathworld.wolfram.com/SchlaefliSymbol.htmlhttp://mathworld.wolfram.com/UniformPolyhedron.htmlhttp://mathworld.wolfram.com/UniformPolyhedron.html


4/13

7 snub cubes

8 snub dodecahedron s

9 truncated cube t

10 truncated dodecahedron t

11 truncated icosahedron t

12 truncated octahedron t

13 truncated tetrahedron t

The following table gives the number of vertices , edges , and faces , together with the number of -

gonal faces for the Archimedean solids. The sorted numbers of edges are 18, 24, 36, 36, 48, 60, 60,

72, 90, 90, 120, 150, 180 (Sloane'sA092536), numbers of faces are 8, 14, 14, 14, 26, 26, 32, 32, 32, 38,

62, 62, 92 (Sloane'sA092537), and numbers of vertices are 12, 12, 24, 24, 24, 24, 30, 48, 60, 60, 60, 60,

120 (Sloane'sA092538).

solid

1 cuboctahedron 12 24 14 8 6

2 great rhombicosidodecahedron 120 180 62 30 20 12

3 great rhombicuboctahedron 48 72 26 12 8 6

4 icosidodecahedron 30 60 32 20 12

5 small rhombicosidodecahedron 60 120 62 20 30 12

6 small rhombicuboctahedron 24 48 26 8 18

7 snub cube 24 60 38 32 6

8 snub dodecahedron 60 150 92 80 12

9 truncated cube 24 36 14 8 6

10 truncated dodecahedron 60 90 32 20 12
http://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://www.research.att.com/~njas/sequences/A092536http://www.research.att.com/~njas/sequences/A092536http://www.research.att.com/~njas/sequences/A092536http://www.research.att.com/~njas/sequences/A092537http://www.research.att.com/~njas/sequences/A092537http://www.research.att.com/~njas/sequences/A092537http://www.research.att.com/~njas/sequences/A092538http://www.research.att.com/~njas/sequences/A092538http://www.research.att.com/~njas/sequences/A092538http://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://www.research.att.com/~njas/sequences/A092538http://www.research.att.com/~njas/sequences/A092537http://www.research.att.com/~njas/sequences/A092536http://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubCube.html


5/13

11 truncated icosahedron 60 90 32 12 20

12 truncated octahedron 24 36 14 6 8

13 truncated tetrahedron 12 18 8 4 4

Seven of the 13 Archimedean solids (thecuboctahedron,icosidodecahedron,truncated cube,truncated

dodecahedron,truncated octahedron,truncated icosahedron, andtruncated tetrahedron) can be obtained

bytruncationof aPlatonic solid. The three truncation series producing these seven Archimedean solids

are illustrated above.

Two additional solids (thesmall rhombicosidodecahedronandsmall rhombicuboctahedron) can be

obtained byexpansionof aPlatonic solid, and two further solids (thegreat rhombicosidodecahedronand

great rhombicuboctahedron) can be obtained byexpansionof one of the previous 9 Archimedean solids

(Stott 1910; Ball and Coxeter 1987, pp. 139-140). It is sometimes stated (e.g., Wells 1991, p. 8) that

these four solids can be obtained by truncation of other solids. The confusion originated with Kepler

himself, who used the terms "truncated icosidodecahedron" and "truncated cuboctahedron" for thegreat

rhombicosidodecahedronandgreat rhombicuboctahedron, respectively. However, truncation alone is not

capable of producing these solids, but must be combined with distorting to turn the resulting rectangles

into squares (Ball and Coxeter 1987, pp. 137-138; Cromwell 1997, p. 81).
http://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/Truncation.htmlhttp://mathworld.wolfram.com/Truncation.htmlhttp://mathworld.wolfram.com/Truncation.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/Expansion.htmlhttp://mathworld.wolfram.com/Expansion.htmlhttp://mathworld.wolfram.com/Expansion.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/Expansion.htmlhttp://mathworld.wolfram.com/Expansion.htmlhttp://mathworld.wolfram.com/Expansion.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/Expansion.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/Expansion.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/Truncation.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.html


6/13

The remaining two solids, thesnub cubeandsnub dodecahedron, can be obtained by moving the faces

of acubeanddodecahedronoutward while giving each face a twist. The resulting spaces are then filled

with ribbons ofequilateral triangles(Wells 1991, p. 8).

Pugh (1976, p. 25) points out the Archimedean solids are all capable of being circumscribed by a regular

tetrahedronso that four of their faces lie on the faces of thattetrahedron.

The Archimedean solids satisfy

(1)

where is the sum of face-angles at a vertex and is the number of vertices (Steinitz and Rademacher

1934, Ball and Coxeter 1987).

Let the cyclic sequence represent the degrees of the faces surrounding a vertex (i.e.,

is a list of the number of sides of all polygons surrounding any vertex). Then the definition of an

Archimedean solid requires that the sequence must be the same for each vertex to withinrotationand

reflection. Walsh (1972) demonstrates that represents the degrees of the faces surrounding each vertex

of a semiregular convex polyhedron ortessellationof the planeiff

1. and every member of is at least 3,

2. , with equality in the case of a planetessellation, and

3. for everyodd number , contains a subsequence ( , , ).

Condition (1) simply says that the figure consists of two or more polygons, each having at least three

sides. Condition (2) requires that the sum of interior angles at a vertex must be equal to a full rotation for

the figure to lie in the plane, and less than a full rotation for a solid figure to be convex.

The usual way of enumerating the semiregular polyhedra is to eliminate solutions of conditions (1) and (2)

using several classes of arguments and then prove that the solutions left are, in fact, semiregular (Kepler

1864, pp. 116-126; Catalan 1865, pp. 25-32; Coxeter 1940, p. 394; Coxeteret al. 1954; Lines 1965,

pp. 202-203; Walsh 1972). The following table gives all possible regular and semiregular polyhedra and

tessellations. In the table, 'P' denotesPlatonic solid, 'M' denotes aprismorantiprism, 'A' denotes an

Archimedean solid, and 'T' a plane tessellation.
http://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/Cube.htmlhttp://mathworld.wolfram.com/Cube.htmlhttp://mathworld.wolfram.com/Cube.htmlhttp://mathworld.wolfram.com/Dodecahedron.htmlhttp://mathworld.wolfram.com/Dodecahedron.htmlhttp://mathworld.wolfram.com/Dodecahedron.htmlhttp://mathworld.wolfram.com/EquilateralTriangle.htmlhttp://mathworld.wolfram.com/EquilateralTriangle.htmlhttp://mathworld.wolfram.com/EquilateralTriangle.htmlhttp://mathworld.wolfram.com/Tetrahedron.htmlhttp://mathworld.wolfram.com/Tetrahedron.htmlhttp://mathworld.wolfram.com/Tetrahedron.htmlhttp://mathworld.wolfram.com/Tetrahedron.htmlhttp://mathworld.wolfram.com/Tetrahedron.htmlhttp://mathworld.wolfram.com/Rotation.htmlhttp://mathworld.wolfram.com/Rotation.htmlhttp://mathworld.wolfram.com/Rotation.htmlhttp://mathworld.wolfram.com/Reflection.htmlhttp://mathworld.wolfram.com/Reflection.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Iff.htmlhttp://mathworld.wolfram.com/Iff.htmlhttp://mathworld.wolfram.com/Iff.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/OddNumber.htmlhttp://mathworld.wolfram.com/OddNumber.htmlhttp://mathworld.wolfram.com/OddNumber.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/Prism.htmlhttp://mathworld.wolfram.com/Prism.htmlhttp://mathworld.wolfram.com/Prism.htmlhttp://mathworld.wolfram.com/Antiprism.htmlhttp://mathworld.wolfram.com/Antiprism.htmlhttp://mathworld.wolfram.com/Antiprism.htmlhttp://mathworld.wolfram.com/Antiprism.htmlhttp://mathworld.wolfram.com/Prism.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/OddNumber.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Iff.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Reflection.htmlhttp://mathworld.wolfram.com/Rotation.htmlhttp://mathworld.wolfram.com/Tetrahedron.htmlhttp://mathworld.wolfram.com/Tetrahedron.htmlhttp://mathworld.wolfram.com/EquilateralTriangle.htmlhttp://mathworld.wolfram.com/Dodecahedron.htmlhttp://mathworld.wolfram.com/Cube.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubCube.html


7/13

fg. solid Schlfli symbol

(3, 3, 3) P tetrahedron

(3, 4, 4) M triangular prism t

(3, 6, 6) A truncated tetrahedron t

(3, 8, 8) A truncated cube t

(3, 10, 10) A truncated dodecahedron t

(3, 12, 12) T tessellation t

(4, 4, ) M -gonalprism t

(4, 4, 4) P cube

(4, 6, 6) A truncated octahedron t

(4, 6, 8) A great rhombicuboctahedron t

(4, 6, 10) A great rhombicosidodecahedron t



(5, 5, 5) P dodecahedron

(5, 6, 6) A truncated icosahedron t

(6, 6, 6) T tessellation

(3, 3, 3, ) M -gonalantiprisms

(3, 3, 3, 3) P octahedron

(3, 4, 3, 4) A cuboctahedron

(3, 5, 3, 5) A icosidodecahedron
http://mathworld.wolfram.com/SchlaefliSymbol.htmlhttp://mathworld.wolfram.com/SchlaefliSymbol.htmlhttp://mathworld.wolfram.com/Tetrahedron.htmlhttp://mathworld.wolfram.com/Tetrahedron.htmlhttp://mathworld.wolfram.com/TriangularPrism.htmlhttp://mathworld.wolfram.com/TriangularPrism.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Prism.htmlhttp://mathworld.wolfram.com/Prism.htmlhttp://mathworld.wolfram.com/Prism.htmlhttp://mathworld.wolfram.com/Cube.htmlhttp://mathworld.wolfram.com/Cube.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Dodecahedron.htmlhttp://mathworld.wolfram.com/Dodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Antiprism.htmlhttp://mathworld.wolfram.com/Antiprism.htmlhttp://mathworld.wolfram.com/Antiprism.htmlhttp://mathworld.wolfram.com/Octahedron.htmlhttp://mathworld.wolfram.com/Octahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/Octahedron.htmlhttp://mathworld.wolfram.com/Antiprism.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/Dodecahedron.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/Cube.htmlhttp://mathworld.wolfram.com/Prism.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TriangularPrism.htmlhttp://mathworld.wolfram.com/Tetrahedron.htmlhttp://mathworld.wolfram.com/SchlaefliSymbol.html


8/13

(3, 6, 3, 6) T tessellation

(3, 4, 4, 4) A small rhombicuboctahedronr

(3, 4, 5, 4) A small rhombicosidodecahedron r

(3, 4, 6, 4) T tessellation r

(4, 4, 4, 4) T tessellation

(3, 3, 3, 3, 3) P icosahedron

(3, 3, 3, 3, 4) A snub cube

s

(3, 3, 3, 3, 5) A snub dodecahedron s

(3, 3, 3, 3, 6) T tessellation s

(3, 3, 3, 4, 4) T tessellation --

(3, 3, 4, 3, 4) T tessellation s

(3, 3, 3, 3, 3, 3) T tessellation

As shown in the above table, there are exactly 13 Archimedean solids (Walsh 1972, Ball and Coxeter

1987). They are called thecuboctahedron,great rhombicosidodecahedron,great rhombicuboctahedron,

icosidodecahedron,small rhombicosidodecahedron,small rhombicuboctahedron,snub cube,snub

dodecahedron,truncated cube,truncated dodecahedron,truncated icosahedron(soccer ball),truncated

octahedron, andtruncated tetrahedron.

Let be theinradiusof the dual polyhedron (corresponding to theinsphere, which touches the faces of

the dual solid), be themidradiusof both the polyhedron and its dual (corresponding to the

midsphere, which touches the edges of both the polyhedron and its duals), thecircumradius

(corresponding to thecircumsphereof the solid which touches the vertices of the solid) of the

Archimedean solid, and the edge length of the solid Since thecircumsphereandinsphereare dual to

each other, they obey the relationship
http://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Icosahedron.htmlhttp://mathworld.wolfram.com/Icosahedron.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/Inradius.htmlhttp://mathworld.wolfram.com/Inradius.htmlhttp://mathworld.wolfram.com/Inradius.htmlhttp://mathworld.wolfram.com/Insphere.htmlhttp://mathworld.wolfram.com/Insphere.htmlhttp://mathworld.wolfram.com/Insphere.htmlhttp://mathworld.wolfram.com/Midradius.htmlhttp://mathworld.wolfram.com/Midradius.htmlhttp://mathworld.wolfram.com/Midradius.htmlhttp://mathworld.wolfram.com/Midsphere.htmlhttp://mathworld.wolfram.com/Midsphere.htmlhttp://mathworld.wolfram.com/Circumradius.htmlhttp://mathworld.wolfram.com/Circumradius.htmlhttp://mathworld.wolfram.com/Circumradius.htmlhttp://mathworld.wolfram.com/Circumsphere.htmlhttp://mathworld.wolfram.com/Circumsphere.htmlhttp://mathworld.wolfram.com/Circumsphere.htmlhttp://mathworld.wolfram.com/Circumsphere.htmlhttp://mathworld.wolfram.com/Circumsphere.htmlhttp://mathworld.wolfram.com/Circumsphere.htmlhttp://mathworld.wolfram.com/Insphere.htmlhttp://mathworld.wolfram.com/Insphere.htmlhttp://mathworld.wolfram.com/Insphere.htmlhttp://mathworld.wolfram.com/Insphere.htmlhttp://mathworld.wolfram.com/Circumsphere.htmlhttp://mathworld.wolfram.com/Circumsphere.htmlhttp://mathworld.wolfram.com/Circumradius.htmlhttp://mathworld.wolfram.com/Midsphere.htmlhttp://mathworld.wolfram.com/Midradius.htmlhttp://mathworld.wolfram.com/Insphere.htmlhttp://mathworld.wolfram.com/Inradius.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/Icosahedron.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/Tessellation.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/Tessellation.html


9/13

(2)

(Cundy and Rollett 1989, Table II following p. 144). In addition,

(3)

(4)

(5)

(6)

(7)

(8)

The following tables give the analytic and numerical values of , , and for the Archimedean solids with

polyhedron edgesof unit length (Coxeteret al. 1954; Cundy and Rollett 1989, Table II following p. 144).

Hume (1986) gives approximate expressions for thedihedral anglesof the Archimedean solid (and exact

expressions for their duals).

solid

1 cuboctahedron 1

2

great

rhombicosidodecahedro

n

3great

rhombicuboctahedron

4 icosidodecahedron

5

small

rhombicosidodecahedro

n
http://mathworld.wolfram.com/PolyhedronEdge.htmlhttp://mathworld.wolfram.com/PolyhedronEdge.htmlhttp://mathworld.wolfram.com/DihedralAngle.htmlhttp://mathworld.wolfram.com/DihedralAngle.htmlhttp://mathworld.wolfram.com/DihedralAngle.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/DihedralAngle.htmlhttp://mathworld.wolfram.com/PolyhedronEdge.html


10/13

6small

rhombicuboctahedron

7 snub cube * * *

8 snub dodecahedron * * *

9 truncated cube

1

0

truncated

dodecahedron

1

1truncated icosahedron

1

2truncated octahedron

1

3truncated tetrahedron

*The complicated analytic expressions for thecircumradiiof these solids are given in the entries for the

snub cubeandsnub dodecahedron.

solid

1 cuboctahedron 0.75 0.86603 1

2 great rhombicosidodecahedron 3.73665 3.76938 3.80239

3 great rhombicuboctahedron 2.20974 2.26303 2.31761

4 icosidodecahedron 1.46353 1.53884 1.61803

5 small rhombicosidodecahedron 2.12099 2.17625 2.23295

6 small rhombicuboctahedron 1.22026 1.30656 1.39897

7 snub cube 1.15763 1.24719 1.34371

8 snub dodecahedron 2.03969 2.09688 2.15583
http://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/Circumradius.htmlhttp://mathworld.wolfram.com/Circumradius.htmlhttp://mathworld.wolfram.com/Circumradius.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/Icosidodecahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/GreatRhombicosidodecahedron.htmlhttp://mathworld.wolfram.com/Cuboctahedron.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/Circumradius.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/SnubDodecahedron.htmlhttp://mathworld.wolfram.com/SnubCube.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.htmlhttp://mathworld.wolfram.com/SmallRhombicuboctahedron.html


11/13

9 truncated cube 1.63828 1.70711 1.77882

10 truncated dodecahedron 2.88526 2.92705 2.96945

11 truncated icosahedron 2.37713 2.42705 2.47802

12 truncated octahedron 1.42302 1.5 1.58114

13 truncated tetrahedron 0.95940 1.06066 1.17260

The Archimedean solids and theirdualsare allcanonical polyhedra. Since the Archimedean solids are

convex, theconvex hullof each Archimedean solid is the solid itself.

SEE ALSO:Archimedean Solid Stellations,Catalan Solid,Deltahedron,Isohedron,Johnson Solid,Kepler-

Poinsot Solid,Platonic Solid,Quasiregular Polyhedron,Semiregular Polyhedron,Uniform Polyhedron,

Uniform Tessellation

REFERENCES:

Ball, W. W. R. and Coxeter, H. S. M.Mathematical Recreations and Essays, 13th ed.New York: Dover, p. 136, 1987.

Behnke, H.; Bachman, F.; Fladt, K.; and Kunle, H. (Eds.).Fundamentals of Mathematics, Vol. 2: Geometry.Cambridge, MA:

MIT Press, pp. 269-286, 1974.

Catalan, E. "Mmoire sur la Thorie des Polydres." J. l'cole Polytechnique (Paris)41, 1-71, 1865.

Coxeter, H. S. M. "The Pure Archimedean Polytopes in Six and Seven Dimensions." Proc. Cambridge Phil. Soc.24, 1-9,

1928.

Coxeter, H. S. M. "Regular and Semi-Regular Polytopes I." Math. Z.46, 380-407, 1940.

Coxeter, H. S. M. 2.9 inRegular Polytopes, 3rd ed."Historical Remarks." New York: Dover, pp. 30-32, 1973.

Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. London Ser. A

246, 401-450, 1954.

Critchlow, K.Order in Space: A Design Source Book.New York: Viking Press, 1970.
http://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedCube.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/DualPolyhedron.htmlhttp://mathworld.wolfram.com/DualPolyhedron.htmlhttp://mathworld.wolfram.com/DualPolyhedron.htmlhttp://mathworld.wolfram.com/CanonicalPolyhedron.htmlhttp://mathworld.wolfram.com/CanonicalPolyhedron.htmlhttp://mathworld.wolfram.com/CanonicalPolyhedron.htmlhttp://mathworld.wolfram.com/ConvexHull.htmlhttp://mathworld.wolfram.com/ConvexHull.htmlhttp://mathworld.wolfram.com/ConvexHull.htmlhttp://mathworld.wolfram.com/ArchimedeanSolidStellations.htmlhttp://mathworld.wolfram.com/ArchimedeanSolidStellations.htmlhttp://mathworld.wolfram.com/ArchimedeanSolidStellations.htmlhttp://mathworld.wolfram.com/CatalanSolid.htmlhttp://mathworld.wolfram.com/CatalanSolid.htmlhttp://mathworld.wolfram.com/CatalanSolid.htmlhttp://mathworld.wolfram.com/Deltahedron.htmlhttp://mathworld.wolfram.com/Deltahedron.htmlhttp://mathworld.wolfram.com/Deltahedron.htmlhttp://mathworld.wolfram.com/Isohedron.htmlhttp://mathworld.wolfram.com/Isohedron.htmlhttp://mathworld.wolfram.com/Isohedron.htmlhttp://mathworld.wolfram.com/JohnsonSolid.htmlhttp://mathworld.wolfram.com/JohnsonSolid.htmlhttp://mathworld.wolfram.com/JohnsonSolid.htmlhttp://mathworld.wolfram.com/Kepler-PoinsotSolid.htmlhttp://mathworld.wolfram.com/Kepler-PoinsotSolid.htmlhttp://mathworld.wolfram.com/Kepler-PoinsotSolid.htmlhttp://mathworld.wolfram.com/Kepler-PoinsotSolid.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/QuasiregularPolyhedron.htmlhttp://mathworld.wolfram.com/QuasiregularPolyhedron.htmlhttp://mathworld.wolfram.com/QuasiregularPolyhedron.htmlhttp://mathworld.wolfram.com/SemiregularPolyhedron.htmlhttp://mathworld.wolfram.com/SemiregularPolyhedron.htmlhttp://mathworld.wolfram.com/SemiregularPolyhedron.htmlhttp://mathworld.wolfram.com/UniformPolyhedron.htmlhttp://mathworld.wolfram.com/UniformPolyhedron.htmlhttp://mathworld.wolfram.com/UniformPolyhedron.htmlhttp://mathworld.wolfram.com/UniformTessellation.htmlhttp://mathworld.wolfram.com/UniformTessellation.htmlhttp://www.amazon.com/exec/obidos/ASIN/0486253570/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486253570/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486253570/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0262020696/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0262020696/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0262020696/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486614808/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486614808/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486614808/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0500340331/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0500340331/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0500340331/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0500340331/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486614808/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0262020696/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486253570/ref=nosim/weisstein-20http://mathworld.wolfram.com/UniformTessellation.htmlhttp://mathworld.wolfram.com/UniformPolyhedron.htmlhttp://mathworld.wolfram.com/SemiregularPolyhedron.htmlhttp://mathworld.wolfram.com/QuasiregularPolyhedron.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/Kepler-PoinsotSolid.htmlhttp://mathworld.wolfram.com/Kepler-PoinsotSolid.htmlhttp://mathworld.wolfram.com/JohnsonSolid.htmlhttp://mathworld.wolfram.com/Isohedron.htmlhttp://mathworld.wolfram.com/Deltahedron.htmlhttp://mathworld.wolfram.com/CatalanSolid.htmlhttp://mathworld.wolfram.com/ArchimedeanSolidStellations.htmlhttp://mathworld.wolfram.com/ConvexHull.htmlhttp://mathworld.wolfram.com/CanonicalPolyhedron.htmlhttp://mathworld.wolfram.com/DualPolyhedron.htmlhttp://mathworld.wolfram.com/TruncatedTetrahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedIcosahedron.htmlhttp://mathworld.wolfram.com/TruncatedDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedCube.html


12/13

Cromwell, P. R.Polyhedra.New York: Cambridge University Press, pp. 79-86, 1997.

Cundy, H. and Rollett, A. "Stellated Archimedean Polyhedra." 3.9 inMathematical Models, 3rd ed.Stradbroke, England:

Tarquin Pub., pp. 123-128 and Table II following p. 144, 1989.

Fejes Tth, L. Ch. 4 inRegular Figures.Oxford, England: Pergamon Press, 1964.

Geometry Technologies. "The 5 Platonic Solids and the 13 Archimedean Solids."

http://www.scienceu.com/geometry/facts/solids/ .

Holden, A.Shapes, Space, and Symmetry.New York: Dover, p. 54, 1991.

Hovinga, S. "Regular and Semi-Regular Convex Polytopes: A Short Historical Overview."

http://presh.com/hovinga/regularandsemiregularconvexpolytopesashorthistoricaloverview.html.

Hume, A. "Exact Descriptions of Regular and Semi-Regular Polyhedra and Their Duals." Computing Science Tech. Rep.,

No. 130. Murray Hill, NJ: AT&T Bell Laboratories, 1986.

Kepler, J. "Harmonice Mundi." Opera Omnia, Vol. 5. Frankfurt, pp. 75-334, 1864.

Kraitchik, M.Mathematical Recreations.New York: W. W. Norton, pp. 199-207, 1942.

Le, Ha. "Archimedean Solids."http://www.scg.uwaterloo.ca/~hqle/Polyhedra/archimedean.html.

Lines, L.Solid Geometry, with Chapters on Space-Lattices, Sphere-Packs, and Crystals.New York: Dover, 1965.

Maehara, H. "On the Sphericity of the Graphs of Semi-Regular Polyhedra." Discr. Math.58, 311-315, 1986.

Nooshin, H.; Disney, P. L.; and Champion, O. C. "Properties of Platonic and Archimedean Polyhedra." Table 12.1 in

"Computer-Aided Processing of Polyhedric Configurations." Ch. 12 inBeyond the Cube: The Architecture of Space Frames

and Polyhedra(Ed. J. F. Gabriel). New York: Wiley, pp. 360-361, 1997.

Pearce, P.Structure in Nature Is a Strategy for Design.Cambridge, MA: MIT Press, pp. 34-35, 1978.

Pedagoguery Software. Poly.http://www.peda.com/poly/.

Pugh, A.Polyhedra: A Visual Approach.Berkeley: University of California Press, p. 25, 1976.

Rawles, B. A. "Platonic and Archimedean Solids--Faces, Edges, Areas, Vertices, Angles, Volumes, Sphere Ratios."
http://www.amazon.com/exec/obidos/ASIN/0521664055/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0521664055/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0521664055/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0906212200/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0906212200/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0906212200/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0080100589/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0080100589/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0080100589/ref=nosim/weisstein-20http://www.scienceu.com/geometry/facts/solids/http://www.scienceu.com/geometry/facts/solids/http://www.amazon.com/exec/obidos/ASIN/0486268519/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486268519/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486268519/ref=nosim/weisstein-20http://presh.com/hovinga/regularandsemiregularconvexpolytopesashorthistoricaloverview.htmlhttp://presh.com/hovinga/regularandsemiregularconvexpolytopesashorthistoricaloverview.htmlhttp://www.amazon.com/exec/obidos/ASIN/0486201635/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486201635/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486201635/ref=nosim/weisstein-20http://www.scg.uwaterloo.ca/~hqle/Polyhedra/archimedean.htmlhttp://www.scg.uwaterloo.ca/~hqle/Polyhedra/archimedean.htmlhttp://www.scg.uwaterloo.ca/~hqle/Polyhedra/archimedean.htmlhttp://www.amazon.com/exec/obidos/ASIN/B0006BMVAU/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/B0006BMVAU/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/B0006BMVAU/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0471122610/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0471122610/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0471122610/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0471122610/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0262660458/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0262660458/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0262660458/ref=nosim/weisstein-20http://www.peda.com/poly/http://www.peda.com/poly/http://www.peda.com/poly/http://www.amazon.com/exec/obidos/ASIN/0520029267/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0520029267/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0520029267/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0520029267/ref=nosim/weisstein-20http://www.peda.com/poly/http://www.amazon.com/exec/obidos/ASIN/0262660458/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0471122610/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0471122610/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/B0006BMVAU/ref=nosim/weisstein-20http://www.scg.uwaterloo.ca/~hqle/Polyhedra/archimedean.htmlhttp://www.amazon.com/exec/obidos/ASIN/0486201635/ref=nosim/weisstein-20http://presh.com/hovinga/regularandsemiregularconvexpolytopesashorthistoricaloverview.htmlhttp://www.amazon.com/exec/obidos/ASIN/0486268519/ref=nosim/weisstein-20http://www.scienceu.com/geometry/facts/solids/http://www.amazon.com/exec/obidos/ASIN/0080100589/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0906212200/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0521664055/ref=nosim/weisstein-20


13/13

http://www.intent.com/sg/polyhedra.html.

Robertson, S. A. and Carter, S. "On the Platonic and Archimedean Solids." J. London Math. Soc.2, 125-132, 1970.

Rorres, C. "Archimedean Solids: Pappus."http://www.mcs.drexel.edu/~crorres/Archimedes/Solids/Pappus.html.

Sloane, N. J. A. SequencesA092536,A092537, andA092538in "The On-Line Encyclopedia of Integer Sequences."

Steinitz, E. and Rademacher, H. Vorlesungen ber die Theorie der Polyheder. Berlin, p. 11, 1934.

Stott, A. B. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der

Koninklijke Akad. Wetenschappen Amsterdam11, 3-24, 1910.

Vichera, M. "Archimedean Polyhedra."http://www.vicher.cz/puzzle/telesa/telesa.htm.

Walsh, T. R. S. "Characterizing the Vertex Neighbourhoods of Semi-Regular Polyhedra." Geometriae Dedicata1, 117-123,

1972.

Webb, R. "Archimedean Solids and Catalan Solids."http://www.software3d.com/Archimedean.html.

Wells, D.The Penguin Dictionary of Curious and Interesting Geometry.London: Penguin, pp. 6-8, 1991.

Wenninger, M. J. "The Thirteen Semiregular Convex Polyhedra and Their Duals." Ch. 2 inDual Models.Cambridge,

England: Cambridge University Press, pp. 14-35, 1983.

Wenninger, M. J.Polyhedron Models.New York: Cambridge University Press, 1989.
http://www.intent.com/sg/polyhedra.htmlhttp://www.intent.com/sg/polyhedra.htmlhttp://www.mcs.drexel.edu/~crorres/Archimedes/Solids/Pappus.htmlhttp://www.mcs.drexel.edu/~crorres/Archimedes/Solids/Pappus.htmlhttp://www.mcs.drexel.edu/~crorres/Archimedes/Solids/Pappus.htmlhttp://www.research.att.com/~njas/sequences/A092536http://www.research.att.com/~njas/sequences/A092536http://www.research.att.com/~njas/sequences/A092536http://www.research.att.com/~njas/sequences/A092537http://www.research.att.com/~njas/sequences/A092537http://www.research.att.com/~njas/sequences/A092537http://www.research.att.com/~njas/sequences/A092538http://www.research.att.com/~njas/sequences/A092538http://www.research.att.com/~njas/sequences/A092538http://www.vicher.cz/puzzle/telesa/telesa.htmhttp://www.vicher.cz/puzzle/telesa/telesa.htmhttp://www.vicher.cz/puzzle/telesa/telesa.htmhttp://www.software3d.com/Archimedean.htmlhttp://www.software3d.com/Archimedean.htmlhttp://www.software3d.com/Archimedean.htmlhttp://www.amazon.com/exec/obidos/ASIN/0140118136/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0140118136/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0140118136/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0521245249/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0521245249/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0521245249/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0521098599/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0521098599/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0521098599/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0521098599/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0521245249/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0140118136/ref=nosim/weisstein-20http://www.software3d.com/Archimedean.htmlhttp://www.vicher.cz/puzzle/telesa/telesa.htmhttp://www.research.att.com/~njas/sequences/A092538http://www.research.att.com/~njas/sequences/A092537http://www.research.att.com/~njas/sequences/A092536http://www.mcs.drexel.edu/~crorres/Archimedes/Solids/Pappus.htmlhttp://www.intent.com/sg/polyhedra.html

Archimedean Solid2stdyust

Documents

Transcript of Archimedean Solid2stdyust