Math 103 Contemporary Math

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# Math 103 Contemporary Math - PowerPoint PPT Presentation

Math 103 Contemporary Math. Tuesday, February 8, 2005. Review from last class. FAPP video on Tilings of the plane. . Symmetry Ideas. Reflective symmetry: BI LATERAL SYMMETRY T  C  O   0    I   A  Folding line: &quot;axis of symmetry&quot; The &quot;flip.“ The &quot;mirror.&quot; . R(P) = P': A Transformation.

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### Math 103 Contemporary Math

Tuesday, February 8, 2005

Review from last class

FAPP video on Tilings of the plane.

Symmetry Ideas

Reflective symmetry: BI LATERAL SYMMETRY

T  C  O   0    I   A

• Folding line: "axis of symmetry"
• The "flip.“
• The "mirror."
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).

Definition
• We say F has a reflective symmetry wrt a line lif  there is a reflection  R about the line l where  R(P)=P' is still an element of Ffor every P in F....
• i.e.. R (F) = F.
• l is called the axis of symmetry.
• Examples of reflective symmetry:Squares...  People
Rotational Symmetry
• Center of rotation. "rotational pole" (usually O) and angle/direction of rotation.
• The "spin.“
R(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 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.
Why are there only six?
• Before: AAfter : A  or    B  or     CSuppose 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 BBy an analysis of a "tree" we count there are exactly and only 6 possibilities for where the vertices can be transformed.
Tree Analysis

Identity

B

C

A

Reflection

C

B

C

Reflection

A

B

A

Rotation

C

B

Rotation

A

C

Reflection

A

B

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”
• 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.
Activity
• 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?