Symmetry

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Defining Symmetry

• Now we are ready to define what we mean
  mathematically by symmetry.
• A symmetry of an object is a rigid motion that
  moves the object back onto itself.
• That is, after the rigid motion (without
  labeling) we could not tell the original from the
  image.
• Of course every object is moved back onto itself
  by the identity motion, the interesting objects
  are those that exhibit other symmetry.
• We will look at some examples and classify
  objects by the amount and type of symmetry they
  exhibit.

3
Symmetry An Example

• We will begin by examining the symmetries of a
  square.
• A square exhibits several symmetries.
• If we put the center of rotation at the center of
  the square, any rotation by a multiple of 90º
  produces a symmetry.

C
D
B
C
A
B
A
B
A
D
By 90º . . .
. . . 180º . . .
center of rotation
. . . 270º . . .
C
D
B
C
A
B
A
D
C
D
. . . or 360º.
4

• We can also find several reflection symmetries

A
A
B
A
B
A
B
A
B
A
B
B
C
D
D
C
Horizontally . . .
. . . vertically . . .
. . . or along either diagonal.
C
C
D
B
A
C
D
C
D
C
D
D
D
C
A
B

• So we find a total of 4 rotations (including
  the identity motion) and 4 reflections that move
  the square back onto itself.
• We say then that a square is of symmetry type
  D4, or dihedral 4.

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Symmetry Type DN

• The symmetry type D4 is so named because the
  object has 4 rotations and 4 reflections.
• We will find that any object that has
  reflectional symmetry(ies) will have the same
  number of rotational symmetries (if we include
  the identity motion).
• An object that has both reflection and rotation
  symmetries is of type DN where N indicates the
  number of rotations and reflections.
• Any object of type DN must have the same number
  of reflection and rotation symmetries.
• Lets look at some more examples.

6
More Examples

• We find that the cross below has the same
  symmetries as a square . . .

rotations by 90, 180, 270, and 360 . . .
and reflections horizontally, vertically, and
along both diagonals.
So the cross is also of symmetry type D4.
7
A minor change

• Now suppose we change the cross a little bit, say
  cut off some corners a little.

Now we no longer have any reflection symmetry.
If we reflect it any direction it just doesnt
quite match up.
But we do still have all 4 rotations.
So this object is of a different type.
When an object has only rotational symmetry, we
say it is of type ZN, where N is the number of
rotations.
So this object is of type Z4.
8
Identify and classify the following shapes as to
symmetry type.
Click for hints or to get the final answer.
D3
D5
D8
D8
D2
Z2
9
Midterm 2

• Wednesday, May 11, 1130 am - 1220 pm
• Ch. 9, 10, 11.1-7

Suggested Problems Chapter 9 5, 7, 8,
12, 15, 16, (see if you can prove these using
the recursive definition) 19, 25, 27, 29,
31, 52 (for b, rewrite FN in terms of FN-1 and
FN1) Chapter 10 3, 5, 6, 13, 17,
19, 23, 25, 31, 33, 35,
37, 39, 41, 45, 51, 53, 61, 62 Chapter 11 1,
3, 5, 9, 11, 13, 15, 19, 21, 23, 29, 31, 33,
35, 37