Symmetry:

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Symmetry

• Wallpaper, Tiles, and Escher
• Richard Kramer
• Feb. 8, 2001

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• Rigid Motions on a plane
• Symmetry of Finite Objects
• The Algebra of a Pentagon
• First Diversion Symmetry in Physics
• Symmetry of Infinite Objects
• Strips Ribbon Science
• 17 Types of Symmetry on a plane
• Tiling
• Escher Bending the Boundaries

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Rigid Motions on a plane

• Definition
• The distance between any two points is the same
  afterwards.
• Types
• Do Nothing
• Translation
• Rotation
• Reflection
• Glide Reflection

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• Symmetry of Finite Objects
• The Algebra of a Pentagon
• First Diversion Symmetry in Physics

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A symmetry of an of an object is the result of
applying a rigid motion such that it looks the
same.
An object is finite if there exists no
translation of it that results in a symmetry.
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The Algebra of a Pentagon

• Definitions
• 1 do nothing
• r 1/5 rotation counterclockwise
• m reflection across the vertical axis
• ab doing b and then doing a
• r2 rr
• rarb r ab
• Discoveries
• mm 1
• rmr m
• rrrrr r5 1

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• Group Axioms
• If a and b are members of the group, then so
  is ab.
• There exists an 1 such that for any a, 1a
  a1 a.
• For any a, there exists a b such that ab 1.
• For each a, we will call its b, a-1.
• For any a, b, and c, (ab)c a(bc).

• For a Pentagon combinations of r, m, and 1
  are a group
• 1 mm r5
• m-1 m
• (rn)-1 r5-n
• (mrn)-1 mrn

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Diversion 1

• Symmetry in Physics

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1 (a)(b)(c)(d)(e) r (abcde) m (a)(be)(cd)
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v f e 2
(v f) (e)
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• Symmetry of Infinite Objects
• Strips Ribbon Science
• 17 Types of Symmetry on a plane
• Tiling
• A Finite Object Legal Moves on a Grid
• 3 Possible Grids
• Examples
• Second Diversion Penrose Tiles

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Strips
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Seventeen Kinds of Symmetry
0 monorhombic
monoscopic
00 monoglide
0 monotropic
2222 discopic
222 dirhombic
22 digyro
220 diglide
2222 ditropic
333 triscopic
33 trigyro
333 tritropic
442 tetrascopic
42 tetragyro
442 tetratropic
632 hexascopic
632 hexatropic
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scopic all mirrors tropic no mirrors gyro
mirrors gyration points glide glide
reflection no reflection rhombic glide
reflection reflection
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• Tiling
• Choose a grid
• Get a basic shape
• Define a set of legal moves such that
• everything is covered
• there is no overlap

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Diversion 2
Penrose Tiles
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Escher Bending the Boundaries
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