Math 103 Contemporary Math

Math 103 Contemporary Math

Tuesday, February 8, 2005

Review from last classReview from last class

FAPP video on Tilings of the plane.

Symmetry IdeasSymmetry IdeasReflective symmetry: BI LATERAL SYMMETRY

T  C  O   0    I   A  • Folding line: "axis of symmetry"

– The "flip.“– The "mirror."

R(P) = P': A Transformation

R(P) = P': A Transformation

Before: P .... After : P' If P is on the line (axis), then R(P)=P.

"P remains fixed by the reflection."

If P is not on the axis, then the line PP' is perpendicular to the axis and if Q is the point of intersection of PP' with the axis then m(PQ) = m(P'Q).

DefinitionDefinition• We say F has a reflective symmetry wrt

a line l if  there is a reflection  R about the line l where  R(P)=P' is still an element of F for every P in F....

• i.e.. R (F) = F. • l is called the axis of symmetry.

• Examples of reflective symmetry:Squares...  People

Rotational SymmetryRotational Symmetry

• Center of rotation. "rotational pole" (usually O) and angle/direction of rotation.

• The "spin.“

R(P) = P' : A transformationR(P) = P' : A transformation

• If O is the center then R(O) = O.

• If the angle is 360 then R(P) = P for all P.... called the identity transformation.

• If the angle is between 0 and 360 then only the center remains fixed.

• For any point P the angle POP'  is the same.

• Examples of rotational symmetry.

Single Figure SymmetriesSingle Figure Symmetries

• Now... what about finding all the reflective and rotational symmetries of a single figure?

• Symmetries of playing card.... • Classify the cards having the same symmetries.

Notice symmetry of clubs, diamonds, hearts, spades.

• Organization of markers.

Symmetries of an equilateral triangleSymmetries of an equilateral triangle

Why are there only six? Why are there only six?

• Before: A                              After : A  or    B  or     C

Suppose I know where A goes:

What about B?  If A -> A     Before: B   After: B or C

                          If  A ->B     Before:B    After: A or C

                          If  A ->C     Before: B   After: A or B

By an analysis of a "tree" we count there are exactly and only 6 possibilities for where the vertices can be transformed.

Tree AnalysisTree Analysis

A

B

C

B

C

A

C

A

B

C

BC

A

B

A

Identity

Reflection

Reflection

Rotation

Rotation

Reflection

What about combining transformations to give new symmetries

What about combining transformations to give new symmetries

Think of a symmetry as a transformation:Example: V will mean reflection across the line that is the vertical

altitude of the equilateral triangle.

Then let's consider a second symmetry, R=R120, which will rotate the equilateral triangle counterclockwise about its center O by 120 degrees.

We now can think of first performing V to the figure and then performing R to the figure.   We will denote this V*R... meaning V followed by R.[Note that order can make a difference here, and there is an alternative  convention for this notation that would reverse the order and say that R*V means V followed by R.]

Does the resulting transformation V*R also leave the equilateral  covering the same position in which it started?

Symmetry “Products”Symmetry “Products”

• V*R     =   ?

• If so it is also a symmetry.... which of the six is it?

• What about other products? 

• This gives a  "product" for symmetries.If S and R are any symmetries of a figure then S*R is also a symmetry of the figure.

A "multiplication" table for Symmetries

A "multiplication" table for Symmetries

* Id R120 R240 V G=R1 H=R2

Id Id

R120 R240 Id

R240 Id

V Id

G=R1 Id

H=R2 Id

ActivityActivity• Do Activity. • This shows that R240*V = ? • This "multiplicative" structure  is called the Group of

symmetries of the equilateral triangle.

Given any figure we can talk about the group of its symmetries.Does a figure always have at least one symmetry? .....

Yes... The Identity symmetry.Such a symmetry is called the trivial symmetry.

So we can compare objects for symmetries.... how many?Does the multiplication table for the symmetries look the same in some sense?