Symmetry

1
Symmetry
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What is Symmetry?

• The word symmetry comes from Greek and means of
  like measure.
• It conveys the idea of correspondence,
  equivalence, or balance.
• Many people think of symmetry in terms of
  reflection.
• But there is much more to symmetry than that.

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Understanding Symmetry

• Consider the objects below.
• Which would you describe as having the most
  symmetry?
• The least?

A
B
B
A
A
B
C
D
C
D
A
B
C
D
D
C
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Describing Symmetry

• We say that an object has symmetry if it looks
  the same from two or more different vantage
  points.
• Another way of looking at this is if we can move
  the object so that it looks exactly the same.
• We will use this idea to study the symmetry of
  two-dimensional objects.
• So lets study how we can move objects.

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Rigid Motions

• A rigid motion is the act of moving an object
  without changing its shape or size.
• Under a rigid motion a circle is still a circle,
  a square is still a square, and the dimensions
  remain the same.
• The distance between two points will stay the
  same.

A rigid motion.
Not a rigid motion.
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Equivalent Rigid Motions

• When studying rigid motions we are not concerned
  as much with what happens in the process as the
  end result.
• If the end result of two different rigid motions
  is the same, we say these motions are equivalent.
  
• In the plane, all rigid motions are equivalent to
  one of four basic types
• reflection
• rotation
• translation
• glide reflection

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Studying Rigid Motions

• A rigid motion moves every point in the plane
  from its starting position to an ending position.
  
• If we call the starting point P, we will call the
  ending point P'.
• If P and P' are the same point (i.e. P is not
  really moved), we call P a fixed point of the
  motion.
• Since a rigid motion does not change shapes,
  distances or angles, we can understand the effect
  of a rigid motion by knowing how it affects only
  a few points.

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Reflections
Reflections
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Reflections

• A reflection creates a mirror image of the
  original object.
• A reflection moves every point an equal distance
  to the other side of a given line, called the
  axis of reflection.

A
A'
B
B'
C
C'
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Reflections

• To understand a reflection, all we need to know
  is how it moves one point P.
• Every point is reflected perpendicular to the
  axis of reflection and is moved to equal distance
  on the other side of the axis.
• That is, the distances from the axis of
  reflection to P and P' are the same.
• And the axis of reflection is perpendicular to
  the line joining these two points.

P'
P