Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups
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Symmetry Groups
Parker Glynn-Adey
August 15, 2019
Parker Glynn-Adey Symmetry Groups
Symmetries
DefinitionA symmetry is a transformation which leaves a figure unchanged.
Parker Glynn-Adey Symmetry Groups
Triangle
Parker Glynn-Adey Symmetry Groups
Triangle Reflection
Parker Glynn-Adey Symmetry Groups
Triangle Rotation
Parker Glynn-Adey Symmetry Groups
Reflection
1 2 3↓ ↓ ↓1 3 2
Parker Glynn-Adey Symmetry Groups
Reflection
1 2 3↓ ↓ ↓1 3 2
Parker Glynn-Adey Symmetry Groups
Reflect Twice
1 2 3↓ ↓ ↓1 3 2
1 2 3↓ ↓ ↓1 3 2
=
1 2 3↓ ↓ ↓1 2 3
One reflection, performed twice, cancels out.
Parker Glynn-Adey Symmetry Groups
Reflect Twice
1 2 3↓ ↓ ↓1 3 2
1 2 3↓ ↓ ↓1 3 2
=
1 2 3↓ ↓ ↓1 2 3
One reflection, performed twice, cancels out.
Parker Glynn-Adey Symmetry Groups
Reflect Twice
1 2 3↓ ↓ ↓1 3 2
1 2 3↓ ↓ ↓1 3 2
=
1 2 3↓ ↓ ↓1 2 3
One reflection, performed twice, cancels out.
Parker Glynn-Adey Symmetry Groups
Reflect Twice
1 2 3↓ ↓ ↓1 3 2
1 2 3↓ ↓ ↓1 3 2
=
1 2 3↓ ↓ ↓1 2 3
One reflection, performed twice, cancels out.
Parker Glynn-Adey Symmetry Groups
Rotation
1 2 3↓ ↓ ↓3 1 2
Parker Glynn-Adey Symmetry Groups
Rotation
1 2 3↓ ↓ ↓3 1 2
Parker Glynn-Adey Symmetry Groups
Rotate Thrice
1 2 3↓ ↓ ↓3 1 2
1 2 3↓ ↓ ↓3 1 2
1 2 3↓ ↓ ↓3 1 2
︸ ︷︷ ︸
=
1 2 3↓ ↓ ↓3 1 2
1 2 3↓ ↓ ↓2 3 1
=
1 2 3↓ ↓ ↓1 2 3
A rotation of a triangle, performed three times, cancels out.
Parker Glynn-Adey Symmetry Groups
Rotate Thrice
1 2 3↓ ↓ ↓3 1 2
1 2 3↓ ↓ ↓3 1 2
1 2 3↓ ↓ ↓3 1 2
︸ ︷︷ ︸
=
1 2 3↓ ↓ ↓3 1 2
1 2 3↓ ↓ ↓2 3 1
=
1 2 3↓ ↓ ↓1 2 3
A rotation of a triangle, performed three times, cancels out.
Parker Glynn-Adey Symmetry Groups
Rotate Thrice
1 2 3↓ ↓ ↓3 1 2
1 2 3↓ ↓ ↓3 1 2
1 2 3↓ ↓ ↓3 1 2
︸ ︷︷ ︸
=
1 2 3↓ ↓ ↓3 1 2
1 2 3↓ ↓ ↓2 3 1
=
1 2 3↓ ↓ ↓1 2 3
A rotation of a triangle, performed three times, cancels out.
Parker Glynn-Adey Symmetry Groups
Rotate Thrice
1 2 3↓ ↓ ↓3 1 2
1 2 3↓ ↓ ↓3 1 2
1 2 3↓ ↓ ↓3 1 2
︸ ︷︷ ︸
=
1 2 3↓ ↓ ↓3 1 2
1 2 3↓ ↓ ↓2 3 1
=
1 2 3↓ ↓ ↓1 2 3
A rotation of a triangle, performed three times, cancels out.
Parker Glynn-Adey Symmetry Groups
Rotate Thrice
1 2 3↓ ↓ ↓3 1 2
1 2 3↓ ↓ ↓3 1 2
1 2 3↓ ↓ ↓3 1 2
︸ ︷︷ ︸
=
1 2 3↓ ↓ ↓3 1 2
1 2 3↓ ↓ ↓2 3 1
=
1 2 3↓ ↓ ↓1 2 3
A rotation of a triangle, performed three times, cancels out.
Parker Glynn-Adey Symmetry Groups
Symmetries
What is a symmetry? What defines a symmetry?
Parker Glynn-Adey Symmetry Groups
Symmetries
Parker Glynn-Adey Symmetry Groups
Symmetries
Parker Glynn-Adey Symmetry Groups
Mystery Symmetry
Symmetries have exactly one in-arrow and one out-arrow per number.
Parker Glynn-Adey Symmetry Groups
Mystery Symmetry
Symmetries have exactly one in-arrow and one out-arrow per number.
Parker Glynn-Adey Symmetry Groups
Mystery Symmetry
Symmetries have exactly one in-arrow and one out-arrow per number.
Parker Glynn-Adey Symmetry Groups
Mystery Symmetry
Symmetries have exactly one in-arrow and one out-arrow per number.
Parker Glynn-Adey Symmetry Groups
Mystery Symmetry
Every symmetry is a collection of cycles.
Parker Glynn-Adey Symmetry Groups
Mystery Symmetry
Every symmetry is a collection of cycles.
Parker Glynn-Adey Symmetry Groups
Cycle Notation
DefinitionThe cycle (1 2 3) maps 1
→ 2 → 3 → 1 → 2 → 3 → . . .
Observe: (1 2 3) = (2 3 1) = (3 1 2) are the same cycle.
Parker Glynn-Adey Symmetry Groups
Cycle Notation
DefinitionThe cycle (1 2 3) maps 1 → 2
→ 3 → 1 → 2 → 3 → . . .
Observe: (1 2 3) = (2 3 1) = (3 1 2) are the same cycle.
Parker Glynn-Adey Symmetry Groups
Cycle Notation
DefinitionThe cycle (1 2 3) maps 1 → 2 → 3
→ 1 → 2 → 3 → . . .
Observe: (1 2 3) = (2 3 1) = (3 1 2) are the same cycle.
Parker Glynn-Adey Symmetry Groups
Cycle Notation
DefinitionThe cycle (1 2 3) maps 1 → 2 → 3 → 1
→ 2 → 3 → . . .
Observe: (1 2 3) = (2 3 1) = (3 1 2) are the same cycle.
Parker Glynn-Adey Symmetry Groups
Cycle Notation
DefinitionThe cycle (1 2 3) maps 1 → 2 → 3 → 1 → 2
→ 3 → . . .
Observe: (1 2 3) = (2 3 1) = (3 1 2) are the same cycle.
Parker Glynn-Adey Symmetry Groups
Cycle Notation
DefinitionThe cycle (1 2 3) maps 1 → 2 → 3 → 1 → 2 → 3
→ . . .
Observe: (1 2 3) = (2 3 1) = (3 1 2) are the same cycle.
Parker Glynn-Adey Symmetry Groups
Cycle Notation
DefinitionThe cycle (1 2 3) maps 1 → 2 → 3 → 1 → 2 → 3 → . . .
Observe: (1 2 3) = (2 3 1) = (3 1 2) are the same cycle.
Parker Glynn-Adey Symmetry Groups
Cycle Notation
DefinitionThe cycle (1 2 3) maps 1 → 2 → 3 → 1 → 2 → 3 → . . .
Observe: (1 2 3)
= (2 3 1) = (3 1 2) are the same cycle.
Parker Glynn-Adey Symmetry Groups
Cycle Notation
DefinitionThe cycle (1 2 3) maps 1 → 2 → 3 → 1 → 2 → 3 → . . .
Observe: (1 2 3) = (2 3 1)
= (3 1 2) are the same cycle.
Parker Glynn-Adey Symmetry Groups
Cycle Notation
DefinitionThe cycle (1 2 3) maps 1 → 2 → 3 → 1 → 2 → 3 → . . .
Observe: (1 2 3) = (2 3 1) = (3 1 2)
are the same cycle.
Parker Glynn-Adey Symmetry Groups
Cycle Notation
DefinitionThe cycle (1 2 3) maps 1 → 2 → 3 → 1 → 2 → 3 → . . .
Observe: (1 2 3) = (2 3 1) = (3 1 2) are the same cycle.
Parker Glynn-Adey Symmetry Groups
Rotation as a Cycle
(1 2 3)
Parker Glynn-Adey Symmetry Groups
Rotation as a Cycle
(1 2 3)
Parker Glynn-Adey Symmetry Groups
Reflection as a Cycle
(1)(2 3)
Parker Glynn-Adey Symmetry Groups
Reflection as a Cycle
(1)(2 3)
Parker Glynn-Adey Symmetry Groups
Mystery Symmetry (Revisited)
(1 4)(2 5 3)(6) = (2 5 3)(4 1) = . . .
Parker Glynn-Adey Symmetry Groups
Mystery Symmetry (Revisited)
(1 4)
(2 5 3)(6) = (2 5 3)(4 1) = . . .
Parker Glynn-Adey Symmetry Groups
Mystery Symmetry (Revisited)
(1 4)(2 5 3)
(6) = (2 5 3)(4 1) = . . .
Parker Glynn-Adey Symmetry Groups
Mystery Symmetry (Revisited)
(1 4)(2 5 3)(6)
= (2 5 3)(4 1) = . . .
Parker Glynn-Adey Symmetry Groups
Mystery Symmetry (Revisited)
(1 4)(2 5 3)(6) = (2 5 3)(4 1)
= . . .
Parker Glynn-Adey Symmetry Groups
Mystery Symmetry (Revisited)
(1 4)(2 5 3)(6) = (2 5 3)(4 1) = . . .
Parker Glynn-Adey Symmetry Groups
The Fundamental Theorem
TheoremAny symmetry can be written as a product of disjoint cycles.This product is unique up to re-arranging the terms, and cyclicallyrewriting terms.
Parker Glynn-Adey Symmetry Groups
The Symmetries of the Two Sided Triangle
{(), (1 2 3), (1 3 2), (2 3), (1 3), (1 2)}
Parker Glynn-Adey Symmetry Groups
Si8O12(CH3)8 – Octamethylsilsesquioxane
Parker Glynn-Adey Symmetry Groups
Cube and Octohedron
Parker Glynn-Adey Symmetry Groups
Octohedron
Parker Glynn-Adey Symmetry Groups
Cube
Parker Glynn-Adey Symmetry Groups
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