Symmetries
Reflection Symmetry
Rotation Symmetry
Superheroes
Regular Polygons
Symmetries in the Plane
Kevin Gerstle
University of Iowa
Department of Mathematics
November 3, 2014
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Symmetries
Reflection Symmetry
Rotation Symmetry
Superheroes
Regular Polygons
Symmetries
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Symmetries
Reflection Symmetry
Rotation Symmetry
Superheroes
Regular Polygons
Symmetries
Definition:
A symmetry of a plane figure S is any rigid motion T of the
plane that maps S back to itself
(ie: the starting figure S is the same as the ending figure T(S) )
eg:
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Symmetries
Reflection Symmetry
Rotation Symmetry
Superheroes
Regular Polygons
Examples!
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Symmetries
Reflection Symmetry
Rotation Symmetry
Superheroes
Regular Polygons
Reflection symmetry/line symmetry/bilateral symmetry
Definition:
A reflection symmetry S is a symmetry of a figure via some
reflection
Definition:
A line of symmetry is a line of reflection that produces a
symmetry
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Symmetries
Reflection Symmetry
Rotation Symmetry
Superheroes
Regular Polygons
Reflection symmetry/line symmetry/bilateral symmetry
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Symmetries
Reflection Symmetry
Rotation Symmetry
Superheroes
Regular Polygons
WARNING!
Not all lines in the plane will produce reflection symmetries of a
given figure.
eg:
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Symmetries
Reflection Symmetry
Rotation Symmetry
Superheroes
Regular Polygons
WARNING!
Not all lines in the plane will produce reflection symmetries of a
given figure.
eg:
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Symmetries
Reflection Symmetry
Rotation Symmetry
Superheroes
Regular Polygons
Rotation symmetry/turn symmetry
Definition:
An n◦ rotation symmetry is a symmetry of a figure produced by
rotating the figure n◦ clockwise around some point called the
center of symmetry
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Symmetries
Reflection Symmetry
Rotation Symmetry
Superheroes
Regular Polygons
Rotation symmetry/turn symmetry
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Symmetries
Reflection Symmetry
Rotation Symmetry
Superheroes
Regular Polygons
NB: A figure with m◦ rotation symmetry must automatically have
2m◦ symmetry, 3m◦ symmetry,. . . , in general having km◦ rotation
symmetry for any counting number k.
eg:
has 72◦ rotation symmetry; hence it must have 72 × 2 = 144◦
rotation symmetry, 72 × 3 = 216◦ rotation symmetry, etc.
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Symmetries
Reflection Symmetry
Rotation Symmetry
Superheroes
Regular Polygons
Point symmetry
Definition:
A figure has point symmetry if it has 180◦ rotation symmetry
about some point.
EX: Which of the following figures have point symmetry?
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Symmetries
Reflection Symmetry
Rotation Symmetry
Superheroes
Regular Polygons
??????
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Symmetries
Reflection Symmetry
Rotation Symmetry
Superheroes
Regular Polygons
??????
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Symmetries
Reflection Symmetry
Rotation Symmetry
Superheroes
Regular Polygons
??????
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Symmetries
Reflection Symmetry
Rotation Symmetry
Superheroes
Regular Polygons
Regular Polygons
(HINT: A 0◦ rotation is still a rotation symmetry)
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Symmetries
Reflection Symmetry
Rotation Symmetry
Superheroes
Regular Polygons
Regular Polygons
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Symmetries
Reflection Symmetry
Rotation Symmetry
Superheroes
Regular Polygons
Regular Polygons
What is the pattern?
ie: How many symmetries exist of a regular n-gon?
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Symmetries
Reflection Symmetry
Rotation Symmetry
Superheroes
Regular Polygons
Regular Polygons
What is the pattern?
ie: How many symmetries exist of a regular n-gon?
2n
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