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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.

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math 103 contemporary math

Math 103 Contemporary Math

Tuesday, February 8, 2005

review from last class
Review from last class

FAPP video on Tilings of the plane.

symmetry ideas
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
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
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
Rotational Symmetry
  • Center of rotation. "rotational pole" (usually O) and angle/direction of rotation.
  • The "spin.“
r p p a transformation1
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
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
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
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
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.
activity
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?
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