Section 6.1 Symmetries 1 - Mathematics & Statisticsfarlow/sec61.pdf · crystalline structure of a...

Section 6.1 Symmetries 1

Section 6.1 Section 6.1 Section 6.1 Section 6.1 Symmetries Symmetries Symmetries Symmetries and Algebraic Systemsand Algebraic Systemsand Algebraic Systemsand Algebraic Systems

Purpose of SectionPurpose of SectionPurpose of SectionPurpose of Section: To introduce the idea of a symmetry of an object in the

plane, which will act as an introduction to the study of algebraic structures, in

particular algebraic groups.

AbstractionAbstractionAbstractionAbstraction and Abstract Algebra and Abstract Algebra and Abstract Algebra and Abstract Algebra

The ability to abstract is a unique feature of human thought, an ability

not shared by “lower forms” of living creatures. The ability to capture the

essence of what we see and experience is so engrained in our mental

processes, we never give it a second thought. If the human mind did not have

the capability to abstract commonalities in daily living, we would live in a

different world altogether. Suppose we lacked the capacity to grasp the

“essence” of what makes up a chair. We would be forced to call every chair

by a different name in order to communicate to others what we are referring.

The statement “the chair in the living room” would have no meaning unless we

knew what exact chair was being referred. Parents point to a picture of a

dog in a picture book and tell their infant, “dog”, and it is a proud moment for

the parents when the child sees a dog in the yard and says, “dog!”

The concept of number is a crowning achievement of human’s ability to

abstract the essence of size of sets. It is not necessary to talk about “three

people”, “three days”, “three dogs” and so on. We have abstracted among

those things the commonality of “threeness” so there is no need to say add

“three goats plus five goats, or “three cats plus five cats”, we simply say

three plus five.

The current chapter is a glimpse into some ideas of what is called

“abstract algebra”. Before defining what we mean by an “abstract algebra”,

you should realize you have already studied some “abstract algebras” whether

you know it or not, one being the integers. The integers are an abstract

algebra, although you probably have never called them “abstract” or an

“algebra.” The integers are a set of objects, called integers, and binary

operations of addition, subtraction, and multiplication, defined on the integers,

along with a collection of axioms which the operations obey. Abstract algebra

“abstracts” the essence of the integers and other mathematical structures, and

says; “let’s not study just this or that, let’s study all things which have certain

properties. (Not all that different from the infant when it says “dog,” realizing

there are more dogs than in the picture book.) Abstract mathematics allows

one to think about attributes and relationships, and not focus on specific

objects that possess those attributes and relationships.

The benefits of abstraction are many; it uncovers deep relationships

between different areas of mathematics by allowing one to “rise up” above the


nuances of a particular area and see things from a broader point of view, like

seeing the forest instead of the trees, so to speak. The main disadvantage of

abstraction is that abstract concepts are more difficult to learn and require

more “mathematical maturity” before they can be understood and appreciated.

In summary, abstract algebra studies general mathematical structures with

given properties, the most important structures being groups, rings, and fields.

Before we start our formal discussion of algebraic groups in the next section, we

motivate their study with the introduction of symmetries.

SymmeSymmeSymmeSymmetries tries tries tries

We are all familiar with symmetrical objects, which we generally think

of as objects of beauty, and although you may not be prepared to give a

precise mathematical definition of symmetry, you know one when you see one.

Most people would say a square is more symmetrical than a rectangle, and a

hexagon more symmetrical than a square, and a circle is the most symmetrical

object of all.

Regular patterns and symmetries are known to all cultures and

societies. Although we generally think of symmetry in terms of geometric

objects, we can also include physical objects as well, like an atom, the

crystalline structure of a mineral, a plant, an animal, the solar system, or even

the universe. The concept of symmetry also embodies processes like

chemical reactions, the scattering of elementary particles, a musical score, the

evolution of the solar system, and even mathematical equations. In physics

symmetry1 has to do with the invariance (i.e. unchanging) of natural laws

under space and time transformations. A physical law having space/time

symmetries establishes that the law is independent of translating, rotating, or

reflecting the coordinates of the system. The symmetries of a physical

system are fundamental to how the system acts and behaves. And once the

word symmetry is mentioned, a mathematician thinks of a mathematical

structure called a group.

SymmetriesSymmetriesSymmetriesSymmetries in Two Dimensions in Two Dimensions in Two Dimensions in Two Dimensions

For a single (bounded) figure in two dimensions there are two types of

symmetries2. There is symmetry across a line in which one side of the object

is the mirror image of its other half. This type of symmetry is called line line line line

symmetrysymmetrysymmetrysymmetry (or reflectivereflectivereflectivereflective or mirrormirrormirrormirror symmetry). Figure 1 shows an isosceles

triangle with a line symmetry through its vertical median.

1 For the seminal work on symmetries, see Symmetry by Hermann Weyl (Princeton Univ Press)

1980. 2We do not include translation symmetries here since we are considering only single

geometric objects.


Line Symmetry

Figure 1

The drawings of insects in Figure 2 have a bilateral symmetry. Other figures

are symmetric through more than one line, such as a square, which has four

lines (or axes) of symmetry; its horizontal and vertical midlines, and its two

diagonal lines. A regular n -gon has n lines of symmetry and a circle has an

infinite number.

Figures with One Line Symmetry

Figure 2

A second type of symmetry of a figure in the plane is rrrrotational otational otational otational (or

radialradialradialradial) symmetrysymmetrysymmetrysymmetry. An object has rotational symmetry if the object is repeated

when rotated certain degrees about a central point. The objects in Figure 3

have rotational symmetry about a point but no line symmetries.

Rotational Symmetries but No Line Symmetry

Figure 3

The flower shaped figure at the left in Figure 3 repeats itself when rotated 0,

120, 240 degrees about its center point so we say the figure has rotational

symmetry of degree 3. The letter Z has rotational symmetry 2, and the two

figures at the right have, respectively, rotational symmetries 3 and 4.


Some objects have both reflective and rotational symmetries as

illustrated by the regular polygons3 in Figure 4. The equilateral triangle has 3

rotational symmetries (rotations of 0 ,120� � and 240� about a center point) and

3 reflective symmetries through median lines passing through the three

vertices. A regular polygon with n vertices has n rotation symmetries (each

rotation 360 / n degrees) and n lines of symmetry.

3 Recall that a regular polygon is a polygon whose lengths of sides and angles are the same. Common

ones are the equilateral triange, square, pentagon, and so on.


Equal Equal Equal Equal Number Number Number Number of Rotationof Rotationof Rotationof Rotation and and and and Line SLine SLine SLine Symmetrymmetrymmetrymmetriesiesiesies

FigureFigureFigureFigure RotationRotationRotationRotation

SymmetriesSymmetriesSymmetriesSymmetries

LineLineLineLine

SymmetriesSymmetriesSymmetriesSymmetries

3 rotations

0 ,120 ,240� � �

3 line

symmetries

4 rotations

0 ,90 ,180 ,270� � � �

4 line

symmetries

5 rotations

0 ,72 ,144 ,216 ,288� � � � �

5 line

symmetries

Infinite number

of rotation symmetries

Infinite number of

line symmetries

Figures Having Both Rotational and Reflective Symmetries

Figure 4

Margin Note:Margin Note:Margin Note:Margin Note: Many letters of the alphabet, such as A,B,C,D,… , have various

rotation and line symmetries.

SymmetrSymmetrSymmetrSymmetry Transformations y Transformations y Transformations y Transformations

Although you can think of symmetries as a property of an object, which it

it, there is another interpretation which is more beneficial for our purposes.

For us a symmetry is a function or what we often will call a mapping or

transformation.


Definition Definition Definition Definition A symmetrysymmetrysymmetrysymmetry of an object, say A , in the plane is a distance distance distance distance

preserving mappingpreserving mappingpreserving mappingpreserving mapping f (called an isometryisometryisometryisometry) that maps the object A onto itself4.

That is, if f is a symmetry map, then ( )f A A= .

Rotation Symmetry of 180 degrees

NoteNoteNoteNote: If an object A in the plane is bounded (i.e. it is inside some circle with

finite radius) then a translation of the object merely sliding it in some direction

is not a symmetry since it alters the location of the figure. For unbounded

figures such as wallpaper designs, floor tiles, and so on are often symmetries.

We now look at some simple geometric shapes in the plane.

Symmetries of a RectangleSymmetries of a RectangleSymmetries of a RectangleSymmetries of a Rectangle

Figure 5 shows a rectangle in which the length and width are not the

same in which we have labeled the corners as , , ,A B C D to help us visualize

how the rectangle is rotated and reflected.

Figure 5

You can convince yourself that the rectangle has two rotation symmetries of

0� , 180� , and two line (or flip) symmetries where the lines of symmetry are

the horizontal and vertical midlines. These four symmetries are illustrated in

Figure 6.

4 Symmetries in three dimensions are defined analogously.


MotionMotionMotionMotion SymbolSymbolSymbolSymbol FirstFirstFirstFirst

PositionPositionPositionPosition

FinalFinalFinalFinal

PositionPositionPositionPosition

No Motion e

Rotate 180� 180R

Flip over

Horizontal Median H

Flip over

Vertical Median V

The Four Symmetries of a Rectangle

Figure 6

So what do these symmetries have to do with algebraic structures? Since a

symmetry is a transformation (i.e. a function) which maps the points of an

object back into itself, we can define the product of two transformations as

performing one symmetry followed by the other (i.e. a composition of

functions). It is clear that since each symmetry leaves the object unchanged,

so does the product of two symmetries. In other words, the product formed in

this manner is also a symmetry. For example, if we perform a 180� rotation5,

denoted by 180R , followed by H a flip through the horizontal midline, the

result, as illustrated in Figure 7, is the same as performing the single

symmetry V , a flip through the vertical midline. In other words, we have

computed the product 180R H V= .

5 It is our convention that all rotations are done counterclockwise.


Product of Symmetries 180R H V=

Figure 7

Note that the “do nothing” symmetry e , called the identity symmetryidentity symmetryidentity symmetryidentity symmetry, and is

analogous to 1 in normal multiplication of integers. Figure 8 shows the

product of a few symmetries.

180 180 180

180 180

eH He H

Ve Ve V

R e eR R

ee e

R R e

VV e

HH e

= =

= =

= =

=

=

=

=

Products of Typical Symmetries

Figure 8

Note also that two successive operations of any of the symmetries 180,, ,e R V H

results in returning to the original position. For that reason we say each of

these symmetries is its own inverseinverseinverseinverse, which are expressed in Figure 9.


1

1

1

180 180 180 180

1

HH e H H

VV e V V

R R e R R

ee e e e

−

−

−

−

= ⇒ =

= ⇒ =

= ⇒ =

= ⇒ =

Inverses of Symmetries

Figure 9

We can now make a multiplication table for the symmetries of a rectangle,

which is shown in Figure 10. The product of any two symmetries lies at the

intersection of the row and column of the symmetries. For example, the

intersection of the row labeled 180R and column labeled H is V , which means

180R H V= . The borders of the cells containing the identity symmetry e are

darkened as an aid in reading the table.

e 180R H V

e e 180R H V

180R 180R e V H

H H V e 180R

V V H 180R e

Multiplication Table for Symmetries of a Rectangle

Figure 10

Note: Note: Note: Note:

1. Every row and column of the multiplication table contains one and exactly

one of the four symmetries. It is like a Latin square.

2. In this example the main diagonal contains the identity symmetry e , which

means the product of each symmetry times itself is the identity, or equivalently

each symmetry is equal to its inverse.

3. The table is symmetric about the main diagonal which means the

multiplication of symmetries is commutativecommutativecommutativecommutative. i.e. AB BA= just like multiplication

of numbers in arithmetic. We would call this a commutativecommutativecommutativecommutative algebraic system.

4. The four symmetries 180,, ,e R V H of the rectangle along with the product

operation form what is called an algebraic groupalgebraic groupalgebraic groupalgebraic group....


We have seen how the symmetries of a rectangle form an algebraic

structure of four elements, complete with algebraic identity, and where (in this

example) every element has an inverse. Observe how this system is

analogous to the integers with the operation of addition, except there are an

infinite number of integers (and in the case of integers, only 0 is its own

inverse).

Margin Note:Margin Note:Margin Note:Margin Note: There are objects with no line or rotational symmetries. Can

you draw some?

Symmetries Symmetries Symmetries Symmetries of of of of an an an an Equilateral TriangleEquilateral TriangleEquilateral TriangleEquilateral Triangle

We now examine a figure that is “more symmetric” than the rectangle,

the equilateral triangle. The equilateral triangle6 as drawn in Figure 11 has 6

symmetries, three rotational symmetries in which the triangle is rotated

0 ,120 ,240� � � about its center, and 3 line symmetries in which the triangle is

reflected through lines passing through the vertices as drawn as dotted line

segments.

Six Symmetries of an Equilateral Triangle

Figure 11

Denoting these symmetry mappings as 120 240, , , , ,v nw ne

e R R F F F , where e is the

identity (do nothing) symmetry, 120 240,R R are (counterclockwise) rotations of

120 ,240� � respectively, v

F denotes the flip through the vertical median, nw

F

denotes the flip around the northwest median, and ne

F is the flip around the

northeast median. Figure 12 illustrates the effect of these symmetries.

6 Recall that an equilateral triangle is a triangle with the three sides (or three angles) the same.


Motion Symbol First

Position

Final

Position

No Motion e

Rotate 120�

Counterclockwise 120R

Rotate 240�

Counterclockwise 240R

Flip over the

Vertical Axis vF

Flip over the

Northeast Axis neF

Flip over the

Northwest Axis nwF

Symmetries of an Equilateral Triangle

Figure 12


As we did for the rectangle, we can compute the multiplication, or Cayley

table, for these symmetries. See Figure 13.

e 120R 240R v

F ne

F nw

F

e e 120R 240R v

F ne

F nw

F

120R 120R 240R e neF

nwF

vF

240R 240R e 120R nw

F v

F ne

F

vF

vF

nwF

neF e 240R 120R

neF

neF

vF

nwF 120R e 240R

nwF

nwF

neF

vF 240R 120R e

Multiplication Table for Symmetries of an Equilateral Triangle

Figure 13

We have drawn the borders around the identity symmetry e so the table can

be more easily read and interpreted. For easier reading, we have shaded the

“northeast” and “northwest” blocks of products involving the three flip

symmetries.

Example 1Example 1Example 1Example 1 ( ( ( (Commutative Operations)Commutative Operations)Commutative Operations)Commutative Operations)

Are the symmetry operations for the equilateral triangle commutative?

In other words, does the order in which the operations are performed make a

difference in the outcome7?

SolutionSolutionSolutionSolution

To determine if the symmetries are commutative, we examine the

products in the multiplication table in Figure 11 to see if they are symmetric

around the diagonal elements. In this case the table is not symmetric for all

symmetries so the symmetry operations are not commutative. Note that

nw ne ne nwF F F F≠ although one product is commutative; 120 240 240 120R R R R e= = .

Example 2Example 2Example 2Example 2 (Finding Inverse Symmetries) (Finding Inverse Symmetries) (Finding Inverse Symmetries) (Finding Inverse Symmetries)

What are the inverses of each symmetry operation?

SolutionSolutionSolutionSolution

Note that 2 2 2 2

v ne nwe F F F e= = = = which means the symmetries , , ,

v ne nve F F F

are their own inverses, that is 1 1,v v

e e F F− −= = , 1

ne neF F

−= , 1

nv nvF F

−= .

However, note that 1

120 240R R− = and 1

240 120R R− = . The fact there is one and only

7 Not all mathematical operations are commutative. Examples are matrix multiplication and the cross

product of vectors. In daily life not all operations are commutative either. Going outside and emptying the

garbage is one example.


one identity symmetry e in every row and column means that each symmetry

has exactly one inverse symmetry.

The collection 120 240, , , , ,v nw ne

e R R F F F of symmetries of the equilateral

triangle along with the product operation defines the dihedral groupdihedral groupdihedral groupdihedral group of order 6

and denoted by 6D . In general the dihedral group of order 2nD represents the

symmetries of a regular n -gon.

Pure Mathematics:Pure Mathematics:Pure Mathematics:Pure Mathematics: The story goes how Abraham Lincoln, failing to convince his

Cabinet how their reasoning was faulty, ask them, “How many legs does a cow

have?” When they said four, he then continued, “Well then, if a cow’s tail was

a leg, how many legs does it have?” When they said five, obviously, Lincoln

said, “That’s where you are wrong. Just calling a tail a leg doesn’t make it a

let.” This story may be true in the real world, but in the world of pure mathematics it is wrong. If you call a cow’s tail a leg, then it is a leg. In pure

mathematics, we do not need to know what the things we are working with are,

only the rules with govern them. We only need to know the axioms.


ProblemsProblemsProblemsProblems

1. Determine the number of line and rotational symmetries of the following

letters.

a) A l) L w) W

b) B m) M x) X

c) C n) N y) Y

d) D o) O z) Z

e) E p) P

f) F q Q

g) G r) R

h) H s) S

i) I t) T

j) J u) U

k) K v) V

2. Draw a figure that has the following symmetries.

a) no rotational and no line symmetries

b) 1 rotational and no line symmetries

c) no rotational and 1 line symmetry

d) 1 rotational and 1 line symmetry

e) 2 rotational and no line symmetries

f) no rotational and 2 line symmetries

g) 4 rotational and no line symmetries

3. (Multiplication Tables)(Multiplication Tables)(Multiplication Tables)(Multiplication Tables) For the following objects determine the

symmetries and compute the multiplication table for the symmetries. Find the

inverse of each symmetry.

a)


b)

c)

d)

e)

f)


g)

4. (Common Sy(Common Sy(Common Sy(Common Symmetries)mmetries)mmetries)mmetries)8888 The following logos have an equal number of line

and rotation symmetries. These are the symmetries of a regular n gon.

Symmetries of this type are called dihedral symmetriesdihedral symmetriesdihedral symmetriesdihedral symmetries. Find the rotational

and reflection symmetries of the following figures. Hint: Don’t forget the

identity mapping which is a rotation of zero degrees.

a)

b)

c)

d)

e)

8 We thank Annalisa Crannel of Franklin and Marshall College for

providing these logos.


5. (Symmetries (Symmetries (Symmetries (Symmetries of Solutions of of Solutions of of Solutions of of Solutions of Differential Equations) Differential Equations) Differential Equations) Differential Equations) The solutions of the

differential equation dy dx y= is the one-parameter family xy ce= where c is

an arbitrary constant. Show that the transformation ,x x h y y′ ′= + = where

h is an arbitrary real number maps the family of solutions back into the family

of solutions, and hence is a symmetry transformation of the differential

equation.

6. (Find the Symmetries)(Find the Symmetries)(Find the Symmetries)(Find the Symmetries) Draw a geometric object whose symmetries

, , ,a b c d have the following multiplication tables.

a)

e a

e e a

a a e

b)

e a b

e e a b

a a b e

b b e a

c)

e a b c

e e a b c

a a b c e

b b c e a

c c e a b

d)

e a b c d

e e a b c d

a a b c d e

b b c d e a

c c d e a b

d d e a b c

6 (Symmetries of a Parallelogram)(Symmetries of a Parallelogram)(Symmetries of a Parallelogram)(Symmetries of a Parallelogram) Describe the symmetries of a

parallelogram that is neither a rhombus nor a rectangle.


7. (Symmetries of an Ellipse)(Symmetries of an Ellipse)(Symmetries of an Ellipse)(Symmetries of an Ellipse) Describe the symmetries of an ellipse that is not

a circle.

8. (Multiplication Tables)(Multiplication Tables)(Multiplication Tables)(Multiplication Tables) Find the multiplication table for the symmetries of the

star in Figure 15. Once you determine the pattern for the table, it goes fairly

fast.

Figure 14

9. (Representation of (Representation of (Representation of (Representation of 2D with Matrices) with Matrices) with Matrices) with Matrices) Show that the matrices

1 0 1 0 0 1 0 1

, , ,0 1 0 1 1 0 1 0

e A B C− −

= = = = − −

satisfy the following multiplication table of the dihedral group 2D where the

group operation is defined as matrix multiplication. This means the group can

be represented by matrices where the group operation is matrix multiplication.

� e A B C

e e A B C

A A e C B

B B C e A

C C B A e

Dihedral Multiplication Table

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