Math 103 Contemporary Math

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Math 103 Contemporary Math

• Tuesday, February 8, 2005

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Review from last class

• FAPP video on Tilings of the plane.

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Symmetry Ideas

• Reflective symmetry BI LATERAL SYMMETRY
• T  C  O   0    I   A 
• Folding line "axis of symmetry"
• The "flip.
• The "mirror."

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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).

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Definition

• 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 symmetrySquares...
   People

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Rotational Symmetry

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

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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.

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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.

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Symmetries of an equilateral triangle
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Why are there only six?

• Before A                              After A
   or    B  or     CSuppose I know where A
  goesWhat about B?  If A -gt A     Before B  
  After B or C                          If  A
  -gtB     BeforeB    After A or C             
              If  A -gtC     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.

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Tree Analysis
Identity
B
C
A
Reflection
C
B
C
Reflection
A
B
A
Rotation
C
B
Rotation
A
C
Reflection
A
B
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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,
  RR120, 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 VR...
  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 RV means V
  followed by R.Does the resulting
  transformation VR also leave the equilateral 
  covering the same position in which it started?

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Symmetry Products

• VR        ?
• 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 SR is also
  a symmetry of the figure.

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A "multiplication" table for Symmetries
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Activity

• Do Activity.
• This shows that R240V ?
• 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?