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Chapter 11:The Mathematics
of Symmetry
11.1 Rigid Motions
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As are many other core concepts, symmetry is rather
hard to define, and we will not even attempt a
proper definition until Section 11.6. We will start our
discussion with just an informal stab at the
mathematical (or geometric if you prefer)
interpretation of symmetry.
Symmetry
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The figure shows three triangles: (a) a scalene triangle
(all three sides are different), (b) an isosceles triangle,
and (c) an equilateral triangle. In terms of symmetry,
how do these triangles differ? Which one is the most
symmetric? Least symmetric?
Example 11.1 Symmetries of a Triangle
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Even without a formal understanding of what symmetry
is, most people would answer that the equilateral
triangle in (c) is the most symmetric and the scalene
triangle in (a) is the least symmetric. This is in fact
correct, but why?
Example 11.1 Symmetries of a Triangle
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Think of an imaginary observer–say a tiny (but very
observant) ant–standing at the vertices of each of the
triangles, looking toward the opposite side. In the case
of the scalene triangle (a), the view from each vertex is
different.
Example 11.1 Symmetries of a Triangle
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In the case of the isosceles triangle (b), the view from
vertices B and C is the same, but the view from vertex A
is different. In the case of the equilateral triangle (c),
the view is the same from each of the three vertices.
Example 11.1 Symmetries of a Triangle
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Let’s say, for starters, that symmetry is a property of an
object that looks the same to an observer standing at
different vantage points. This is still pretty vague but a
start nonetheless. Now instead of talking about an
observer moving around to different vantage points
think of the object itself moving–forget the observer.
Thus, we might think of symmetry as having to do with
ways to move an object so that when all the moving
is done, the object looks exactly as it did before.
Symmetry Again
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The act of taking an object and moving it from some
starting position to some ending position without
altering its shape or size is called a rigid motion (and
sometimes an isometry). If, in the process of moving
the object, we stretch it, tear it, or generally alter its
shape or size, the motion is not a rigid motion. Since in
a rigid motion the size and shape of an object are not
altered, distances between points are preserved:
Symmetry - Rigid Motion
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The distance between any two points X and Y in the
starting position is the same as the distance between
the same two points in the ending position. In (a), the
motion does not change the shape of the object;
only its position in space has changed. In (b), both
position and shape have changed.
Symmetry - Rigid Motion
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In defining rigid motions we are completely result
oriented. We are only concerned with the net effect
of the motion–where the object started and where
the object ended. What happens during the “trip” is
irrelevant.
Symmetry - Rigid Motion
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This implies that a rigid motion is completely defined
by the starting and ending positions of the object
being moved, and two rigid motions that move an
object from the same starting position to the same
ending position are equivalent rigid motions–never
mind the details of how they go about it.
Symmetry - Rigid Motion
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Because rigid motions are defined strictly in terms of
their net effect, there is a surprisingly small number of
scenarios. In the case of two-dimensional objects in a
plane, there are only four possibilities: A rigid motion is
equivalent to (1) a reflection, (2) a rotation, (3) a
translation, or (4) a glide reflection. We will call these
four types of rigid motions the basic rigid motions of
the plane.
Symmetry - Rigid Motion
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A rigid motion of the plane–let’s call it M–moves each
point in the plane from its starting position P to an
ending position P´, also in the plane. (From here on we will use script letters such as M and N to denote
rigid motions, which should eliminate any possible
confusion between the point M and the rigid motion
M.) We will call the point P´ the image of the point P
under the rigid motion M and describe this informally
by saying that M moves P to P´.Symmetry - Rigid Motion
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We will also stick to the convention that the image
point has the same label as the original point but with
a prime symbol added.
It may happen that a point P is moved back to itself
under M, in which case we call P a fixed point of the
rigid motion M.
Symmetry - Rigid Motion
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Chapter 11:The Mathematics
of Symmetry
11.2 Reflections
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A reflection in the plane is a rigid motion that moves
an object into a new position that is a mirror image of
the starting position. In two dimensions, the “mirror” is a line called the axis of reflection.
From a purely geometric point of view a reflection
can be defined by showing how it moves a generic
point P in the plane.
Reflection
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The image of any point P is found by drawing a line
through P perpendicular to the axis l and finding the
point on the opposite side of l at the same distance
as P from l.
Reflection
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Points on the axis itself are fixed points of the
reflection.
Reflection
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The following figures show three cases of reflection of a
triangle ABC. In all cases the original triangle ABC is
shaded in blue and the reflected triangle A´B´C´ is shaded in red.
Example 11.2 Reflections of a Triangle
In this figure the axis of
Reflection I does not
intersect the triangle
ABC.
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In this figure, the axis of reflection l cuts through the
triangle ABC–here the points where l intersects the
triangle are fixed points of the triangle.
Example 11.2 Reflections of a Triangle
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In this figure, the reflected triangle A´B´C´ falls on top of the original triangle ABC. The vertex B is a fixed point
of the triangle, but the vertices A and C swap positions
under the reflection.
Example 11.2 Reflections of a Triangle
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The following are simple but useful properties of a
reflection.
Properties of Reflections
Property 1
If we know the axis of reflection, we can find the
image of any point P under the reflection (just drop a
perpendicular to the axis through P and find the point
on the other side of the axis that is at an equal
distance). Essentially a reflection is completely
determined by its axis l.
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Property 2
If we know a point P and its image P´ under the reflection (and assuming P´ is different from P), we can find the axis l of the reflection (it is the
perpendicular bisector of the segment PP´). Once we have the axis l of the reflection, we can find the
image of any other point (property 1).
Properties of Reflections
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Property 3
The fixed points of a reflection are all the points on
the axis of reflection l.
Property 4
Reflections are improper rigid motions, meaning that
they change the left-right and clockwise-
counterclockwise orientations of objects. This property
is the reason a left hand reflected in a mirror looks like
a right hand and the hands of a clock reflected in a
mirror move counterclockwise.
Properties of Reflections
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Property 5
If P´ is the image of P under a reflection, then (P´)´ = P (the image of the image is the original point). Thus,
when we apply the same reflection twice, every point
ends up in its original position and the rigid motion is
equivalent to not having moved the object at all.
Properties of Reflections
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A rigid motion that is equivalent to not moving the
object at all is called the identitymotion. At first blush
it may seem somewhat silly to call the identity motion
a motion (after all, nothing moves), but there are very
good mathematical reasons to do so, and we will
soon see how helpful this convention is for studying
and classifying symmetries.
Identity Motion
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■ A reflection is completely determined by its axis l.■ A reflection is completely determined by a single point-image pair P and P´ (as long as P´≠ P).■ A reflection has infinitely many fixed points (all points on l).■ A reflection is an improper rigid motion.■When the same reflection is applied twice, we get the identitymotion.
PROPERTIES OF REFLECTIONS
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Chapter 11:The Mathematics
of Symmetry
11.3 Rotations
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Informally, a rotation in the plane is a rigid motion that
pivots or swings an object around a fixed point O. A
rotation is defined by two pieces of information: (1)
the rotocenter (the point O that acts as the center of
the rotation) and (2) the angle of rotation (actually
the measure of an angle indicating the amount of
rotation). In addition, it is necessary to specify the
direction (clockwise or counterclockwise) associated
with the angle of rotation.
Rotation
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The figure illustrates geometrically how a clockwise
rotation with rotocenter O and angle of rotation
moves a point P to the point P´. Rotation
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The following illustrates three cases of rotation of a
triangle ABC. In all cases the original triangle ABC is
shaded in blue and the
Example 11.3 Rotations of a Triangle
reflected triangle A´B´C´is shaded in red. The rotocenter O lies outside the triangle ABC. The 90º clockwise rotation moved
the triangle from the “12 o’clock position” to the “3 o’clock position.”
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The rotocenter O is at the center of the triangle ABC.
The 180º rotation turns the triangle “upside down.” For obvious reasons, a 180º rotation is often called a half-
turn. (With half turns the result is
Example 11.3 Rotations of a Triangle
the same whether we rotate
clockwise or counter-
clockwise, so it is unnecessary
to specify a direction.)
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The 360º rotation moves every point back to its original
position–from the rigid motion point of view it’s as if the triangle had not moved.
Example 11.3 Rotations of a Triangle
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The following are some important properties of a
rotation.
Properties of Rotations
Property 1
A 360º rotation is equivalent to a 0º rotation, and a 0º
rotation is just the identity motion. (The expression
“going around full circle” is the well-known colloquial
version of this property.)
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From property 1 we can conclude that all rotations
can be described using an angle of rotation between
0º and 360º. For angles larger than 360º we divide the
angle by 360º and just use the remainder (a clockwise
rotation by 759º is equivalent to a clockwise rotation
by 39º).
In addition, we can describe a rotation using
clockwise or counterclockwise orientations (a
clockwise rotation by 39º is equivalent to a
counterclockwise rotation by 321º).
Properties of Rotations
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Property 2
When an object is rotated, the left-right and
clockwise-counterclockwise orientations are
preserved (a rotated left hand remains a left hand,
and the hands of a rotated clock still move in the
clockwise direction). We will describe this fact by
saying that a rotation is a proper rigid motion. Any
motion that preserves the left-right and clockwise-
counterclockwise orientations of objects is called a
proper rigid motion.
Properties of Rotations
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A common misconception is to confuse a 180º
rotation with a reflection, but we can see that they
are very different from just observing that the
reflection is an improper rigid motion, whereas the
180º rotation is a proper rigid motion.
Properties of Rotations
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Property 3
In every rotation, the rotocenter is a fixed point, and
except for the case of the identity (where all points
are fixed points) it is the only fixed point.
Properties of Rotations
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Property 4
Unlike a reflection, a rotation cannot be determined
by a single point-image pair
P and P´ it takes a second point-image pair Q and
Properties of Rotations
Q´ to nail down the rotation. The reason is that infinitely many rotations can move Pto P´. Any point located on the perpendicular bisector of the segment PP´ can be a rotocenter for such a rotation.
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Given a second pair of points Q and Q´ we can identify the rotocenter O as the point where the
perpendicular bisectors of PP´Properties of Rotations
and QQ´meet. Once we have identified the rotocenter O, the angle of rotation α is given by the measure of angle POP´ (or for that matter QOQ´ –they are the same).
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Note: In the special case
where PP´ and QQ´ happen to have the same
perpendicular bisector, the
rotocenter O is the intersection
of PQ and PQ´.Properties of Rotations
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■ A 360º rotation is equivalent to the identitymotion. ■ A rotation is a proper rigid motion.■ A rotation that is not the identity motion has only one fixed point, its rotocenter. ■ A rotation is completely determined by twopoint-image pairs P, P´ and Q,Q´.PROPERTIES OF ROTATIONS
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Chapter 11:The Mathematics
of Symmetry
11.4 Translations
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A translation consists of essentially dragging an object
in a specified direction and by a specified amount
(the length of the translation). The two pieces of
information (direction and length of the translation)
are combined in the form of a vector of translation
(usually denoted by v). The vector of translation is
represented by an arrow–the arrow points in the
direction of translation and the length of the arrow is
the length of the translation.
Translation
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A very good illustration of a translation in a two-
dimensional plane is the dragging of the cursor on a
computer screen. Regardless of what happens in
between, the net result when you drag an icon on
your screen is a translation in a specific direction and
by a specific length.
Translation
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This figure illustrates the translation of a triangle ABC.
Two “different” arrows are shown in the figure, but they
both have the same length and direction, so they
describe the same vector of translation v.
Example 11.4 Translation of a Triangle
As long as the arrow points in
the proper direction and has
the right length, the placement
of the arrow in the picture is
immaterial.
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The following are some important properties of a
translation.
Properties of Translations
Property 1
If we are given a point P and its image P´ under a translation, the arrow joining P to P´ gives the vector of the translation. Once we know the vector of the
translation, we know where the translation moves any
other point. Thus, a single point-image pair P and P´ is all we need to completely determine the translation.
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Property 2
In a translation, every point gets moved some
distance and in some direction, so a translation has
no fixed points.
Property 3
When an object is translated, left-right and clockwise-
counterclockwise orientations are preserved: A
translated left hand is still a left hand, and the hands
of a translated clock still move in the clock-wise
direction. In other words, translations are proper rigid
motions.
Properties of Translations
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Property 4
The effect of a translation with vector v can be
undone by a translation of the same length but in the
opposite direction. The vector for this opposite
translation can be conveniently described as –v. Thus,
a translation with vector v followed with a translation
with vector –v is equivalent to the identity motion.
Properties of Translations
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■ A translation is completely determined by a single point-image pair P and P´. ■ A translation has no fixed points.■ A translation is a proper rigid motion. ■ When a translation with vector v is followed with a translation with vector –v we get to the identitymotion.
PROPERTIES OF TRANSLATIONS
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Chapter 11:The Mathematics
of Symmetry
11.5 Glide Reflections
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A glide reflection is a rigid motion obtained by
combining a translation (the glide) with a reflection.
Moreover, the axis of reflection must be parallel to the
direction of translation. Thus, a glide reflection is
described by two things: the vector of the translation
v and the axis of the reflection l, and these two must
be parallel.
Glide Reflection
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The footprints left behind by
someone walking on soft
sand are a classic example of
a glide reflection: right and
left footprints are images of
each other under the
combined effects of a
reflection and a translation.
Glide Reflection
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The figures on the next two slides illustrate the result of
applying the glide reflection with vector v and axis l to
the triangle ABC. We can do this in two different ways,
but the final result will be the same.
Example 11.4 Glide Reflection of a
Triangle
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The translation is applied first,
moving triangle ABC to
the intermediate position
A*B*C*. The reflection is then
applied to A*B*C* giving the
final position A´B´C´.Example 11.4 Glide Reflection of a
Triangle
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If we apply the reflection first,
the triangle ABC gets moved
to the intermediate position
A*B*C* and then translated
to the final position A´B´C´.Example 11.4 Glide Reflection of a
Triangle
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Property 1
A glide reflection is completely determined by two
point-image pairs P, P´ and Q, Q´. Given a point-image pair P and P´ under a glide reflection, we do not have enough information to determine the glide
reflection, but we do know that the axis lmust pass
through the midpoint of the line segment PP´.Properties of Glide Reflections
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Given a second point-image pair Q and Q´, we can determine the axis of the reflection: It is the line
passing through the points M (midpoint of the line
segment PP´) and N (midpoint of the line segment QQ´).Properties of Glide Reflections
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Once we find the axis of reflection l, we can find the
image of one of the points–say P´– under the reflection. This gives the intermediate point P*, and
the vector that moves P to P* is the vector of
translation v.
Properties of Glide Reflections
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Properties of Glide Reflections
In the event that the
midpoints of PP´ and QQ´are the same point M, we
can still find the axis l by
drawing a line
perpendicular to the line
PQ passing through the
common midpoint M.
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Properties of Glide Reflections
Property 2
A fixed point of a glide reflection would have to be a
point that ends up exactly where it started after it is
first translated and then reflected. This cannot
happen because the translation moves every point
and the reflection cannot undo the action of the
translation. It follows that a glide reflection has no
fixed points.
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Property 3
A glide reflection is a combination of a proper rigid
motion (the translation) and an improper rigid motion
(the reflection). Since the translation preserves left-
right and clockwise-counterclockwise orientations but
the reflection reverses them, the net result under a
glide reflection is that orientations are reversed. Thus,
a glide reflection is an improper rigid motion.
Properties of Glide Reflections
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Property 4
To undo the effects of a glide reflection, we need a
second glide reflection in the opposite direction. To
be more precise, if we move an object under a glide
reflection with vector of translation v and axis of
reflection l and then follow it with another glide
reflection with vector of translation –v and axis of
reflection still l, we get the identity motion. It is as if the
object was not moved at all.
Properties of Glide Reflections
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■ A guide reflection has no fixed points.■ A guide reflection is an improper rigid motion. ■ A guide reflection is completely determined by two point-image pairs P, P´ and Q,Q´. ■ When a guide reflection with vector v and axis of reflection l is followed with a translation with vector –v and the same axis of reflection l we get to the identitymotion.
PROPERTIES OF GUIDE REFLECTIONS
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Chapter 11:The Mathematics
of Symmetry
11.6 Symmetries and
Symmetry Types
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With an understanding of the four basic rigid motions
and their properties, we can now look at the concept
of symmetry in a much more precise way. Here,
finally, is a good definition of symmetry, one that
probably would not have made much sense at the
start of this chapter:
Symmetry
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A symmetry of an object (or shape) is any rigid motion that moves the object back onto itself.
SYMMETRY
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You observe the position of an object, and then, while
you are not looking, the object is moved. If you can’t tell that the object was moved, the rigid motion is a
symmetry. It is important to note that this does not
necessarily force the rigid motion to be the identity
motion. Individual points may be moved to different
positions, even though the whole object is moved back
into itself. And, of course, the identity motion is itself a
symmetry, one possessed by every object and that
from now on we will call simply the identity.
One Way to Think of Symmetry
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Since there are only four basic kinds of rigid motions of
two-dimensional objects in two-dimensional space,
there are also only four possible types of symmetries:
reflection symmetries, rotation symmetries, translation
symmetries, and glide reflection symmetries.
Four Types of Symmetry
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What are the possible rigid motions that move the
square in Fig. 11-17(a) back onto itself?
Example 11.6 The Symmetries of a
Square
Fig. 11-17(a)
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First, there are reflection symmetries. For example, if we
use the line l1 as the axis of reflection, the square falls
back into itself with
Example 11.6 The Symmetries of a
Square
points A and B
interchanging places and
C and D interchanging
places. It is not hard to
think of three other
reflection symmetries, with
axes l2, l3, and l4 as shown.
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The square has rotation symmetries as well. Using the
center of the square O as the rotocenter, we can
rotate the square by an angle of 90º. This moves the A
to B, B to C, C to D and D to A.
Example 11.6 The Symmetries of a
Square
Likewise, rotations with
rotocenter O and angles of
180º, 270º, and 360º,
respectively, are also
symmetries of the square.
Notice that the 360º rotation is
just the identity.
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All in all, we have easily found eight symmetries for
the square. Four of them are reflections, and the
other four are rotations. Could there be more? What if
we combined one of the reflections with one of the
rotations? A symmetry combined with another
symmetry, after all, has to be itself a symmetry. It turns
out that the eight symmetries we listed are all there
are–no matter how we combine them we always end
up with one of the eight.
8 Symmetries of the Square
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Let’s now consider the symmetries of the shape shown–a two-dimensional version of a boat propeller (or a
ceiling fan if you prefer) with four blades.
Example 11.7 The Symmetries of a
Propeller
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Once again, we have a shape with four reflection
symmetries, the axes of reflection are l1, l2, l3, and l4.
Example 11.7 The Symmetries of a
Propeller
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There are four rotation symmetries with rotocenter O
and angles of 90º, 180º, 270º, and 360º, respectively.
And, just as with the square, there are no other possible
symmetries.
Example 11.7 The Symmetries of a
Propeller
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An important lesson lurks behind Examples 11.6 and
11.7: Two different-looking objects can have exactly
the same set of symmetries. A good way to think
about this is that the square and the propeller, while
certainly different objects, are members of the same “symmetry family” and carry exactly the same symmetry genes. Formally, we will say that two
objects or shapes are of the same symmetry type if
they have exactly the same set of symmetries.
Objects with the Same Symmetries
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The symmetry type for the square, the propeller, and
each of the objects shown is called D4 (shorthand for
four reflections plus four rotations).
Objects with the Same Symmetries
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Let’s consider now the propeller shown. This object is only slightly different from the one in Example 11.7, but
from the symmetry point of view the difference is
significant.
Example 11.8 The Symmetry of Type Z4
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Here we still have the four rotation symmetries (90º, 180º,
270º, and 360º), but there are no reflection symmetries!
This makes sense because the individual blades of the
propeller have no reflection symmetry.
Example 11.8 The Symmetry of Type Z4
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As can be seen here, a vertical reflection is not a
symmetry, and neither are any of the other reflections.
This object belongs to a new symmetry family called Z4
(shorthand for the symmetry type of objects having four
rotations only).
Example 11.8 The Symmetry of Type Z4
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Here is one last propeller example. Every once in a
while a propeller looks like the one here, which is kind of
a cross between the previous two examples–only
opposite blades are the same.
Example 11.9 The Symmetry of Type Z2
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This figure has no reflection symmetries, and a 90º
rotation won’t work either. Example 11.9 The Symmetry of Type Z2
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The only symmetries of this shape are a 180º rotation
(turn it upside down and it looks the same!) and the
360º rotation (the identity).
Example 11.9 The Symmetry of Type Z2
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An object having only two rotation symmetries (the
identity and a 180º rotation symmetry) is said to be of
symmetry type Z2. Here are a few additional examples
of shapes and objects with symmetry type Z2.
Example 11.9 The Symmetry of Type Z2
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One of the most common symmetry types occurring in
nature is that of objects having a single reflection
symmetry plus a single rotation symmetry (the identity).
This symmetry type is called D1. Notice that it doesn’t matter if the axis of reflection is vertical, horizontal, or
slanted.
Example 11.10 The Symmetry of Type D1
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Many objects and shapes are informally considered to
have no symmetry at all, but this is a little misleading,
since every object has at least the identity symmetry.
Objects whose only symmetry is the identity are said to
have symmetry type Z1.
Example 11.11 The Symmetry of Type Z1
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In everyday language, certain objects and shapes are
said to be “highly symmetric” when they have lots of rotation and reflection symmetries. Here two very
different looking snowflakes, but from the symmetry
point of view they are the same:
Example 11.12 Objects with Lots of
Symmetry
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All snowflakes have six reflection symmetries and six
rotation symmetries. Their symmetry type is D6. (Try to
find the six axes of reflection symmetry and the six
angles of rotation symmetry.)
Example 11.12 Objects with Lots of
Symmetry
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This figure shows a decorative ceramic plate. It has nine
reflection symmetries and nine rotation symmetries,
and, as you may have guessed, its symmetry type is
called D9.
Example 11.12 Objects with Lots of
Symmetry
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Finally, we have an architectural blueprint of the dome
of the Sports Palace in Rome, Italy. The design has 36
reflection and 36 rotation symmetries (symmetry type
D36 ).
Example 11.12 Objects with Lots of
Symmetry
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In each of the objects in Example 11.12, the number
of reflections matches the number of rotations. This
was also true in Examples 11.6, 11.7, and 11.10.
Coincidence? Not at all. When a finite object or
shape has both reflection and rotation symmetries,
the number of rotation symmetries (which includes
the identity) has to match the number of reflection
symmetries! Any finite object or shape with exactly N
reflection symmetries and N rotation symmetries is
said to have symmetry type DN.
Properties of Glide Reflections
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Are there two-dimensional objects with infinitely many
symmetries? The answer is yes–circles. A circle has
infinitely many reflection symmetries (any line passing
through the center of the circle can serve as an axis) as
well as infinitely many rotation symmetries (use the
center of the circle as a rotocenter and any angle of
rotation will work). We call the symmetry type of the
circle D∞ (the ∞ is the mathematical symbol for “infinity”).Example 11.13 The Symmetry Type D∞
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We now know that if a finite two-dimensional shape has
rotations and reflections, it must have exactly the same
number of each. In this case, the shape belongs to the
D family of symmetries, specifically, it has symmetry type
DN. However, we also saw in Examples 11.8, 11.9, and 11.11 shapes that have rotations, but no reflections. In
this case, we used the notation ZN to describe the
symmetry type, with the subscript N indicating the
actual number of rotations.
Example 11.14 Shapes with Rotations, but
No Reflections
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The figure on the left shows a flower with five petals that
has symmetry type Z5. The figure on the right shows an
airplane turbine with 24 blades that has symmetry type
Z24. Notice that the absence of reflections is the result of
some twist or bump on the petals or blades.
Example 11.14 Shapes with Rotations, but
No Reflections
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We are now in a position to classify the possible
symmetries of any finite two- dimensional shape or
object. (The word finite is in there for a reason, which
will become clear in the next section.) The possibilities
boil down to a surprisingly short list of symmetry types:
Types of Symmetries
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DN
This is the symmetry type of shapes with both rotation
and reflection symmetries.The subscript N (N = 1, 2, 3,
etc.) denotes the number of reflection symmetries,
which is always equal to the number of rotation
symmetries. (The rotations are an automatic
consequence of the reflections–an object can’t have reflection symmetries without having an equal
number of rotation symmetries.)
Types of Symmetries
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ZN
This is the symmetry type of shapes with rotation
symmetries only. The subscript N (N = 1, 2, 3, etc.) denotes the number of rotation symmetries.
D∞
This is the symmetry type of a circle and of circular
objects such as rings and washers, the only possible
two-dimensional shapes or objects with an infinite
number of rotations and reflections.
Types of Symmetries
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Chapter 11:The Mathematics
of Symmetry
11.7 Patterns
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We will formally define a pattern as an infinite “shape”
consisting of an infinitely repeating basic design
called the motif of the pattern. The reason we have
“shape” in quotation marks is that a pattern is really
an abstraction–in the real world there are no infinite
objects as such, although the idea of an infinitely
repeating motif is familiar to us from such everyday
objects as pottery, tile designs, and textiles.
Pattern
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Here are a few examples:
Pattern
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Just like finite shapes, patterns can be classified by
their symmetries. The classification of patterns
according to their symmetry type is of fundamental
importance in the study of molecular and crystal
organization in chemistry, so it is not surprising that
some of the first people to seriously investigate the
symmetry types of patterns were crystallographers.
Symmetry in Patterns
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Archaeologists and anthropologists have also found
that analyzing the symmetry types used by a
particular culture in their textiles and pottery helps
them gain a better understanding of that culture.
We will briefly discuss the symmetry types of border
and wallpaper patterns.
Symmetry in Patterns
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Border patterns (also called linear patterns) are
patterns in which a basic motif repeats itself
indefinitely in a single direction, as in an architectural
frieze, a ribbon, or the border design of a ceramic
pot.
Border Patterns
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The most common direction in a border pattern
(what we will call the direction of the pattern) is
horizontal, but in general a border pattern can be in
any direction (vertical, slanted 45º, etc.).
Border Patterns
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A border pattern always has translation symmetries–
they come with the territory. There is a basic
translation symmetry v (v moves each copy of the
motif one unit to the right), the opposite translation –v
and any multiple of v or –v.
Border Patterns - Translations
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A border pattern can have (a) no reflection
symmetry, (b) horizontal reflection symmetry only, (c)
vertical reflection symmetries only, or (d) both
horizontal and vertical reflection symmetries.
Border Patterns - Reflections
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In this last case the border pattern automatically
picks up a half-turn symmetry as well. In terms of
reflection symmetries, these figures illustrate the only
four possibilities in a border pattern.
Border Patterns - Reflections
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Like with any other object, the identity (i.e., a
360º rotation) is a rotation symmetry of every border
pattern, so every border pattern has at least one
rotation symmetry. The only other possible rotation
symmetry of a border pattern is a half-turn
(180º rotation). Clearly, no other angle of rotation can
take a horizontal pattern and move it back onto itself.
Border Patterns - Rotations
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Thus, in terms of rotation symmetry there are two kinds
of border patterns: those whose only rotation
symmetry is the identity (a) and those having half-turn
symmetry in addition to the identity (b).
Border Patterns - Rotations
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A border pattern can have a glide reflection
symmetry, but there is only one way this can happen:
The axis of reflection has to be a line along the center
of the pattern, and the reflection part of the glide
reflection is not by itself a symmetry of the pattern.
This means that a border pattern having horizontal
reflection symmetry such as the one shown is not
considered to have glide reflection symmetry.
Border Patterns - Glide Reflections
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On the other hand, the border pattern shown does not
have horizontal reflection symmetry (the footprints do
not fall back onto other footprints), but a glide by the
vector w combined with a reflection along the axis l
result in an honest-to-goodness glide reflection
symmetry. An important property of the glide reflection
symmetry is that the vector w is always half the length
of the basic translation symmetry v.
Border Patterns - Glide Reflections
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This border pattern has a vertical reflection symmetry
as well as a glide reflection symmetry. In these cases a
half-turn symmetry (rotocenter O) comes free in the
bargain.
Border Patterns - Glide Reflections
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1. The identity: All border patterns have it.
2. Translations: All border patterns have them. There are a basic translation v, the opposite translation –v, and any multiples of these.
3. Horizontal reflection: Some patterns have it, some don’t. There is only one possible horizontal axis of reflection, and it must run through the middle of the pattern.
Symmetries of Border Patterns
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4. Vertical reflections: Some patterns have them, some don’t. Vertical axes of reflection (i.e., axes perpendicular to the direction of the pattern) can run through the middle of a motif or between two motifs.
5. Half-turns: Some patterns have them, some don’t. Rotocenters must be located at the center of a motif or between two motifs.
Symmetries of Border Patterns
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6. Glide reflections: Some patterns have them, some don’t. Neither the reflection nor the glide can be symmetries on their own. The length of the glide w is half that of the basic translation The axis of the reflection runs through the middle of the pattern v.
Symmetries of Border Patterns
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■ 11. This symmetry type represents border patterns that have no symmetries other than the identity and translation symmetry.■ 1m. This symmetry type represents border patterns with just a horizontal reflection symmetry.■ m1. This symmetry type represents border patterns with just a vertical reflection symmetry.
Border Patterns - Symmetry Families
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■ mm. This symmetry type represents border patterns with both a horizontal and a vertical reflection symmetry. When both of these symmetries are present, there is also half-turn symmetry.■ 12. This symmetry type represents border patterns with only a half-turn symmetry.■ 1g. This symmetry type represents border patterns with only a glide reflection symmetry.
Border Patterns - Symmetry Families
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■ mg. This symmetry type represents border patterns with a vertical reflection and a glide reflection symmetry. When both of these symmetries are present, there is also a half-turn symmetry.
Border Patterns - Symmetry Families
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Summary of the seven border pattern symmetry families.
Border Patterns - Symmetry Families
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Wallpaper patterns are patterns that fill the plane by
repeating a motif indefinitely along several (two or
more) nonparallel directions. Typical examples of such
patterns can be found in wallpaper (of course),
carpets, and textiles.
With wallpaper patterns things get a bit more
complicated, so we will skip the details. The possible
symmetries of a wallpaper pattern are as follows:
Wallpaper Patterns
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Every wallpaper pattern has translation symmetry in at least two different (nonparallel) directions.
Wallpaper Patterns - Translations
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A wallpaper pattern can have (a) no reflections, (b)
reflections in only one direction, (c) reflections in two
nonparallel directions, (d) reflections in three
nonparallel directions, (e) reflections in four nonparallel
directions, and (f) reflections in six nonparallel
directions. There are no other possibilities. Note that
particularly conspicuous in its absence is the case of
reflections in exactly five different directions.
Wallpaper Patterns - Reflections
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In terms of rotation symmetries, a wallpaper pattern
can have (a) the identity only, (b) two rotations
(identity and 180º), (c) three rotations (identity, 120º,
and 240º), (d) four rotations (identity, 90º, 180º, and
270º), and (e) six rotations (identity, 60º, 120º, 180º, 240º,
and 300º). There are no other possibilities. Once again,
note that a wallpaper pattern cannot have exactly
five different rotations.
Wallpaper Patterns - Rotations
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A wallpaper pattern can have (a) no glide reflections,
(b) glide reflections in only one direction, (c) glide
reflections in two nonparallel directions, (d) glide
reflections in three nonparallel directions, (e) glide
reflections in four nonparallel directions, and (f) glide
reflections in six nonparallel directions. There are no
other possibilities.
Wallpaper Patterns - Glide Reflections
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In the early 1900s, it was shown mathematically that
there are only 17 possible symmetry types for wallpaper
patterns. This is quite a surprising fact–it means that the
hundreds and thousands of wallpaper patterns one
can find at a decorating store all fall into just 17
different symmetry families.
Surprising Fun Fact!