Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey...

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Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Transcript of Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey...

Page 1: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Symmetry Groups

Parker Glynn-Adey

August 15, 2019

Parker Glynn-Adey Symmetry Groups

Page 2: Symmetry Groups - pgadey.ca · 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

Page 3: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Triangle

Parker Glynn-Adey Symmetry Groups

Page 4: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Triangle Reflection

Parker Glynn-Adey Symmetry Groups

Page 5: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Triangle Rotation

Parker Glynn-Adey Symmetry Groups

Page 6: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Reflection

1 2 3↓ ↓ ↓1 3 2

Parker Glynn-Adey Symmetry Groups

Page 7: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Reflection

1 2 3↓ ↓ ↓1 3 2

Parker Glynn-Adey Symmetry Groups

Page 8: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 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

Page 9: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 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

Page 10: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 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

Page 11: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 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

Page 12: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Rotation

1 2 3↓ ↓ ↓3 1 2

Parker Glynn-Adey Symmetry Groups

Page 13: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Rotation

1 2 3↓ ↓ ↓3 1 2

Parker Glynn-Adey Symmetry Groups

Page 14: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 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

Page 15: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 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

Page 16: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 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

Page 17: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 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

Page 18: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 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

Page 19: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Symmetries

What is a symmetry? What defines a symmetry?

Parker Glynn-Adey Symmetry Groups

Page 20: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Symmetries

Parker Glynn-Adey Symmetry Groups

Page 21: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Symmetries

Parker Glynn-Adey Symmetry Groups

Page 22: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Mystery Symmetry

Symmetries have exactly one in-arrow and one out-arrow per number.

Parker Glynn-Adey Symmetry Groups

Page 23: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Mystery Symmetry

Symmetries have exactly one in-arrow and one out-arrow per number.

Parker Glynn-Adey Symmetry Groups

Page 24: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Mystery Symmetry

Symmetries have exactly one in-arrow and one out-arrow per number.

Parker Glynn-Adey Symmetry Groups

Page 25: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Mystery Symmetry

Symmetries have exactly one in-arrow and one out-arrow per number.

Parker Glynn-Adey Symmetry Groups

Page 26: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Mystery Symmetry

Every symmetry is a collection of cycles.

Parker Glynn-Adey Symmetry Groups

Page 27: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Mystery Symmetry

Every symmetry is a collection of cycles.

Parker Glynn-Adey Symmetry Groups

Page 28: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 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

Page 29: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 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

Page 30: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 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

Page 31: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 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

Page 32: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 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

Page 33: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 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

Page 34: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 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

Page 35: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 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

Page 36: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 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

Page 37: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 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

Page 38: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 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

Page 39: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Rotation as a Cycle

(1 2 3)

Parker Glynn-Adey Symmetry Groups

Page 40: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Rotation as a Cycle

(1 2 3)

Parker Glynn-Adey Symmetry Groups

Page 41: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Reflection as a Cycle

(1)(2 3)

Parker Glynn-Adey Symmetry Groups

Page 42: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Reflection as a Cycle

(1)(2 3)

Parker Glynn-Adey Symmetry Groups

Page 43: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Mystery Symmetry (Revisited)

(1 4)(2 5 3)(6) = (2 5 3)(4 1) = . . .

Parker Glynn-Adey Symmetry Groups

Page 44: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Mystery Symmetry (Revisited)

(1 4)

(2 5 3)(6) = (2 5 3)(4 1) = . . .

Parker Glynn-Adey Symmetry Groups

Page 45: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Mystery Symmetry (Revisited)

(1 4)(2 5 3)

(6) = (2 5 3)(4 1) = . . .

Parker Glynn-Adey Symmetry Groups

Page 46: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Mystery Symmetry (Revisited)

(1 4)(2 5 3)(6)

= (2 5 3)(4 1) = . . .

Parker Glynn-Adey Symmetry Groups

Page 47: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Mystery Symmetry (Revisited)

(1 4)(2 5 3)(6) = (2 5 3)(4 1)

= . . .

Parker Glynn-Adey Symmetry Groups

Page 48: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Mystery Symmetry (Revisited)

(1 4)(2 5 3)(6) = (2 5 3)(4 1) = . . .

Parker Glynn-Adey Symmetry Groups

Page 49: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 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

Page 50: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 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

Page 51: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Si8O12(CH3)8 – Octamethylsilsesquioxane

Parker Glynn-Adey Symmetry Groups

Page 52: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Cube and Octohedron

Parker Glynn-Adey Symmetry Groups

Page 53: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Octohedron

Parker Glynn-Adey Symmetry Groups

Page 54: Symmetry Groups - pgadey.ca · Symmetry Groups Parker Glynn-Adey August 15, 2019 Parker Glynn-Adey Symmetry Groups

Cube

Parker Glynn-Adey Symmetry Groups