unit8symmetry -...

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Chapter 11:The Mathematics

of Symmetry

11.1 Rigid Motions

Excursions in Modern Mathematics, 7e: 1.1 - 2Copyright © 2010 Pearson Education, Inc.Copyright © 2014 Pearson Education. All rights reserved. 11.1-2

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


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


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?



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.



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.


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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


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


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.



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.



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.



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.


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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


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.




of Symmetry

11.2 Reflections


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


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


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.


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.



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.



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.


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).



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.


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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.



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


■ 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



of Symmetry

11.3 Rotations


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


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.”


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


the same whether we rotate

clockwise or counter-

clockwise, so it is unnecessary

to specify a direction.)


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.



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.)


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).



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.


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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.



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.



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


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.


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).


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


■ 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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of Symmetry

11.4 Translations


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


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


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.


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.


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.


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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.



■ 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



of Symmetry

11.5 Glide Reflections


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


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


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´BĆ´.Example 11.4 Glide Reflection of a

Triangle


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´BĆ´.Example 11.4 Glide Reflection of a

Triangle


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


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


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



In the event that the

midpoints of PP´ and QQáre 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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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.


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.



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.



■ 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



of Symmetry

11.6 Symmetries and

Symmetry Types


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


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


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


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)


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


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.


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.


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


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.


Propeller


Once again, we have a shape with four reflection

symmetries, the axes of reflection are l1, l2, l3, and l4.


Propeller


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.


Propeller


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


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


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.



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).



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.



This figure has no reflection symmetries, and a 90º

rotation won’t work either. Example 11.9 The Symmetry of Type Z2


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).


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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.



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


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.



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


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.)


Symmetry


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.


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 ).


Symmetry


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.



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∞


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


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


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


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



of Symmetry

11.7 Patterns


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


Here are a few examples:

Pattern


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


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


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


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


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


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


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


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


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.



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.



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.



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.



■ 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


■ 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.



■ 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.



Summary of the seven border pattern 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


Every wallpaper pattern has translation symmetry in at least two different (nonparallel) directions.

Wallpaper Patterns - Translations


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


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


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


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!

unit8symmetry -...

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