[PPT] - Three Types of Symmetry Bret Benesh College of St. Benedict/St. PowerPoint Presentation

SLIDE 1

Introduction Examples of Symmetry Summary

Three Types of Symmetry

Bret Benesh

College of St. Benedict/St. John’s University Department of Mathematics

Concordia College Mathematics Colloquium September 8, 2009

Bret Benesh Three Types of Symmetry

SLIDE 2

Introduction Examples of Symmetry Summary

Outline

1

Introduction

2

Examples of Symmetry Geometry Permutations Linear Algebra

3

Summary

Bret Benesh Three Types of Symmetry

SLIDE 3

Introduction Examples of Symmetry Summary

Definition of Symmetry

1

1Symmetry by Stuant63, Shared under a Creative Commons License Bret Benesh Three Types of Symmetry

SLIDE 4

Introduction Examples of Symmetry Summary

Definition of Symmetry

1

Definition: A symmetry is an action that preserves some specified structure.

1Symmetry by Stuant63, Shared under a Creative Commons License Bret Benesh Three Types of Symmetry

SLIDE 5

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Outline

1

Introduction

2

Examples of Symmetry Geometry Permutations Linear Algebra

3

Summary

Bret Benesh Three Types of Symmetry

SLIDE 6

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Introduction to Geometry

2

Definition: A symmetry is an action that preserves some specified structure.

2Lyra... by Daveybot, Shared under a Creative Commons License Bret Benesh Three Types of Symmetry

SLIDE 7

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Facts about the Icosahedron

20 faces

Bret Benesh Three Types of Symmetry

SLIDE 8

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Facts about the Icosahedron

20 faces 30 edges

Bret Benesh Three Types of Symmetry

SLIDE 9

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Facts about the Icosahedron

20 faces 30 edges 12 vertices

Bret Benesh Three Types of Symmetry

SLIDE 10

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Facts about the Icosahedron

20 faces 30 edges 12 vertices 20 faces × 3 edges/face = 60 symmetries

Bret Benesh Three Types of Symmetry

SLIDE 11

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Types of Icosahedron Symmetry

1 “Identity"

Bret Benesh Three Types of Symmetry

SLIDE 12

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Types of Icosahedron Symmetry

1 “Identity" 15 Edge-Edge rotations

Bret Benesh Three Types of Symmetry

SLIDE 13

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Types of Icosahedron Symmetry

1 “Identity" 15 Edge-Edge rotations 20 Face-Face rotations

Bret Benesh Three Types of Symmetry

SLIDE 14

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Types of Icosahedron Symmetry

1 “Identity" 15 Edge-Edge rotations 20 Face-Face rotations 24 Vertex-Vertex rotations

Bret Benesh Three Types of Symmetry

SLIDE 15

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Types of Icosahedron Symmetry

1 “Identity" 15 Edge-Edge rotations 20 Face-Face rotations 24 Vertex-Vertex rotations (60 total)

Bret Benesh Three Types of Symmetry

SLIDE 16

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Outline

1

Introduction

2

Examples of Symmetry Geometry Permutations Linear Algebra

3

Summary

Bret Benesh Three Types of Symmetry

SLIDE 17

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Introduction to Permutations

3

Definition: A symmetry is an action that preserves some specified structure.

3Arashiyama... by Jpellgen, Shared under a Creative Commons License Bret Benesh Three Types of Symmetry

SLIDE 18

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Number of Symmetries

5 · 4 · 3 · 2 · 1 = 5! = 120 symmetries 1 Identity

Bret Benesh Three Types of Symmetry

SLIDE 19

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Number of Symmetries

5 · 4 · 3 · 2 · 1 = 5! = 120 symmetries 1 Identity 10 of type (a b)

Bret Benesh Three Types of Symmetry

SLIDE 20

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Number of Symmetries

5 · 4 · 3 · 2 · 1 = 5! = 120 symmetries 1 Identity 10 of type (a b) 15 of type (a b)(c d)

Bret Benesh Three Types of Symmetry

SLIDE 21

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Number of Symmetries

5 · 4 · 3 · 2 · 1 = 5! = 120 symmetries 1 Identity 10 of type (a b) 15 of type (a b)(c d) 20 of type (a b c)

Bret Benesh Three Types of Symmetry

SLIDE 22

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Number of Symmetries

5 · 4 · 3 · 2 · 1 = 5! = 120 symmetries 1 Identity 10 of type (a b) 15 of type (a b)(c d) 20 of type (a b c) 20 of type (a b c)(d e)

Bret Benesh Three Types of Symmetry

SLIDE 23

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Number of Symmetries

5 · 4 · 3 · 2 · 1 = 5! = 120 symmetries 1 Identity 10 of type (a b) 15 of type (a b)(c d) 20 of type (a b c) 20 of type (a b c)(d e) 30 of type (a b c d)

Bret Benesh Three Types of Symmetry

SLIDE 24

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Number of Symmetries

5 · 4 · 3 · 2 · 1 = 5! = 120 symmetries 1 Identity 10 of type (a b) 15 of type (a b)(c d) 20 of type (a b c) 20 of type (a b c)(d e) 30 of type (a b c d) 24 of type (a b c d e)

Bret Benesh Three Types of Symmetry

SLIDE 25

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

“Factoring" Permutations

96 = 2 · 2 · 2 · 2 · 2 · 3

Bret Benesh Three Types of Symmetry

SLIDE 26

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

“Factoring" Permutations

96 = 2 · 2 · 2 · 2 · 2 · 3 “Primes" for permutations are called “transpositions." (1 2) (1 3) (1 4) (1 5) (2 3) (2 4) (2 5) (3 4) (3 5) (4 5)

Bret Benesh Three Types of Symmetry

SLIDE 27

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Number of Symmetries

5 · 4 · 3 · 2 · 1 = 5! = 120 symmetries 1 Identity 10 of type (a b) 15 of type (a b)(c d) 20 of type (a b c) 20 of type (a b c)(d e) 30 of type (a b c d) 24 of type (a b c d e)

Bret Benesh Three Types of Symmetry

SLIDE 28

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Number of Symmetries

5 · 4 · 3 · 2 · 1 = 5! = 120 symmetries 1 Identity = () 10 of type (a b) = (a b) 15 of type (a b)(c d) = (a b)(c d) 20 of type (a b c) = (a c)(a b) 20 of type (a b c)(d e) = (a c)(a b)(d e) 30 of type (a b c d) = (a d)(a c)(a b) 24 of type (a b c d e) = (a e)(a d)(a c)(a b)

Bret Benesh Three Types of Symmetry

SLIDE 29

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Number of Symmetries

5 · 4 · 3 · 2 · 1 = 5! = 120 symmetries 1 Identity = () 0 trans: 10 of type (a b) = (a b) 1 trans: 15 of type (a b)(c d) = (a b)(c d) 2 trans: 20 of type (a b c) = (a c)(a b) 2 trans: 20 of type (a b c)(d e) = (a c)(a b)(d e) 3 trans: 30 of type (a b c d) = (a d)(a c)(a b) 3 trans: 24 of type (a b c d e) = (a e)(a d)(a c)(a b) 4 trans:

Bret Benesh Three Types of Symmetry

SLIDE 30

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Number of Symmetries

5 · 4 · 3 · 2 · 1 = 5! = 120 symmetries 1 Identity = () 0 trans: EVEN 10 of type (a b) = (a b) 1 trans: ODD 15 of type (a b)(c d) = (a b)(c d) 2 trans: EVEN 20 of type (a b c) = (a c)(a b) 2 trans: EVEN 20 of type (a b c)(d e) = (a c)(a b)(d e) 3 trans: ODD 30 of type (a b c d) = (a d)(a c)(a b) 3 trans: ODD 24 of type (a b c d e) = (a e)(a d)(a c)(a b) 4 trans: EVEN

Bret Benesh Three Types of Symmetry

SLIDE 31

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Outline

1

Introduction

2

Examples of Symmetry Geometry Permutations Linear Algebra

3

Summary

Bret Benesh Three Types of Symmetry

SLIDE 32

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Introduction to Linear Algebra

4

Definition: A symmetry is an action that preserves some specified structure.

4Matrix Code by David Asch, Shared under a Creative Commons License Bret Benesh Three Types of Symmetry

SLIDE 33

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

New Number System F4

+ 1 a b 1 a b 1 1 b a a a b 1 b b a 1

Bret Benesh Three Types of Symmetry

SLIDE 34

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

New Number System F4

+ 1 a b 1 a b 1 1 b a a a b 1 b b a 1 × 1 a b 1 1 a b a a b 1 b b 1 a

Bret Benesh Three Types of Symmetry

SLIDE 35

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Lines Through Origin in F4

Only consider matrices of determinant 1

Bret Benesh Three Types of Symmetry

SLIDE 36

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Lines Through Origin in F4

Only consider matrices of determinant 1 l1 : y = 0

Bret Benesh Three Types of Symmetry

SLIDE 37

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Lines Through Origin in F4

Only consider matrices of determinant 1 l1 : y = 0 l2 : y = x

Bret Benesh Three Types of Symmetry

SLIDE 38

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Lines Through Origin in F4

Only consider matrices of determinant 1 l1 : y = 0 l2 : y = x l3 : y = ax

Bret Benesh Three Types of Symmetry

SLIDE 39

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Lines Through Origin in F4

Only consider matrices of determinant 1 l1 : y = 0 l2 : y = x l3 : y = ax l4 : y = bx

Bret Benesh Three Types of Symmetry

SLIDE 40

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Lines Through Origin in F4

Only consider matrices of determinant 1 l1 : y = 0 l2 : y = x l3 : y = ax l4 : y = bx l5 : x = 0

Bret Benesh Three Types of Symmetry

SLIDE 41

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a a 1 a

a

a 1 a 1

Bret Benesh

Three Types of Symmetry

SLIDE 42

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a a 1 a

a

a 1 a 1

= a

1 b

Bret Benesh

Three Types of Symmetry

SLIDE 43

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a a 1 a

a

a 1 a 1

= a

1 b

(so l1 goes to l4)

Bret Benesh Three Types of Symmetry

SLIDE 44

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a a 1 a

a

a 1 a 1

= a

1 b

(so l1 goes to l4)

a a 1 a a 1

Bret Benesh

Three Types of Symmetry

SLIDE 45

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a a 1 a

a

a 1 a 1

= a

1 b

(so l1 goes to l4)

a a 1 a a 1

=

1

Bret Benesh

Three Types of Symmetry

SLIDE 46

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a a 1 a

a

a 1 a 1

= a

1 b

(so l1 goes to l4)

a a 1 a a 1

=

1

(so l4 goes to l1)

Bret Benesh Three Types of Symmetry

SLIDE 47

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a a 1 a

a

a 1 a 1

= a

1 b

(so l1 goes to l4)

a a 1 a a 1

=

1

(so l4 goes to l1)

a a 1 a 1 1

Bret Benesh

Three Types of Symmetry

SLIDE 48

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a a 1 a

a

a 1 a 1

= a

1 b

(so l1 goes to l4)

a a 1 a a 1

=

1

(so l4 goes to l1)

a a 1 a 1 1

= b

1

Bret Benesh

Three Types of Symmetry

SLIDE 49

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a a 1 a

a

a 1 a 1

= a

1 b

(so l1 goes to l4)

a a 1 a a 1

=

1

(so l4 goes to l1)

a a 1 a 1 1

= b

1

(so l2 goes to l5)

Bret Benesh Three Types of Symmetry

SLIDE 50

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a a 1 a

a

a 1 a 1

= a

1 b

(so l1 goes to l4)

a a 1 a a 1

=

1

(so l4 goes to l1)

a a 1 a 1 1

= b

1

(so l2 goes to l5)

a a 1 a 1

Bret Benesh

Three Types of Symmetry

SLIDE 51

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a a 1 a

a

a 1 a 1

= a

1 b

(so l1 goes to l4)

a a 1 a a 1

=

1

(so l4 goes to l1)

a a 1 a 1 1

= b

1

(so l2 goes to l5)

a a 1 a 1

= a

1 1

Bret Benesh

Three Types of Symmetry

SLIDE 52

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a a 1 a

a

a 1 a 1

= a

1 b

(so l1 goes to l4)

a a 1 a a 1

=

1

(so l4 goes to l1)

a a 1 a 1 1

= b

1

(so l2 goes to l5)

a a 1 a 1

= a

1 1

(so l5 goes to l2)

Bret Benesh Three Types of Symmetry

SLIDE 53

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a a 1 a

a

a 1 a 1

= a

1 b

(so l1 goes to l4)

a a 1 a a 1

=

1

(so l4 goes to l1)

a a 1 a 1 1

= b

1

(so l2 goes to l5)

a a 1 a 1

= a

1 1

(so l5 goes to l2)

a a 1 a 1 a

Bret Benesh

Three Types of Symmetry

SLIDE 54

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a a 1 a

a

a 1 a 1

= a

1 b

(so l1 goes to l4)

a a 1 a a 1

=

1

(so l4 goes to l1)

a a 1 a 1 1

= b

1

(so l2 goes to l5)

a a 1 a 1

= a

1 1

(so l5 goes to l2)

a a 1 a 1 a

=

1 a

Bret Benesh

Three Types of Symmetry

SLIDE 55

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a a 1 a

a

a 1 a 1

= a

1 b

(so l1 goes to l4)

a a 1 a a 1

=

1

(so l4 goes to l1)

a a 1 a 1 1

= b

1

(so l2 goes to l5)

a a 1 a 1

= a

1 1

(so l5 goes to l2)

a a 1 a 1 a

=

1 a

(so l3 goes to l3)

Bret Benesh Three Types of Symmetry

SLIDE 56

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a a 1 a

a

a 1 a 1

= a

1 b

(so l1 goes to l4)

a a 1 a a 1

=

1

(so l4 goes to l1)

a a 1 a 1 1

= b

1

(so l2 goes to l5)

a a 1 a 1

= a

1 1

(so l5 goes to l2)

a a 1 a 1 a

=

1 a

(so l3 goes to l3)

(l1 l4)(l2 l5)

Bret Benesh Three Types of Symmetry

SLIDE 57

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a 1 0 b

a

1 b 1

Bret Benesh

Three Types of Symmetry

SLIDE 58

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a 1 0 b

a

1 b 1

= a

1

Bret Benesh

Three Types of Symmetry

SLIDE 59

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a 1 0 b

a

1 b 1

= a

1

(so l1 goes to l1)

Bret Benesh Three Types of Symmetry

SLIDE 60

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a 1 0 b

a

1 b 1

= a

1

(so l1 goes to l1)

a 1 b 1 1

Bret Benesh

Three Types of Symmetry

SLIDE 61

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a 1 0 b

a

1 b 1

= a

1

(so l1 goes to l1)

a 1 b 1 1

= b

1 1

Bret Benesh

Three Types of Symmetry

SLIDE 62

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a 1 0 b

a

1 b 1

= a

1

(so l1 goes to l1)

a 1 b 1 1

= b

1 1

(so l2 goes to l2)

Bret Benesh Three Types of Symmetry

SLIDE 63

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a 1 0 b

a

1 b 1

= a

1

(so l1 goes to l1)

a 1 b 1 1

= b

1 1

(so l2 goes to l2)

a 1 b 1 a

Bret Benesh

Three Types of Symmetry

SLIDE 64

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a 1 0 b

a

1 b 1

= a

1

(so l1 goes to l1)

a 1 b 1 1

= b

1 1

(so l2 goes to l2)

a 1 b 1 a

=

1

Bret Benesh

Three Types of Symmetry

SLIDE 65

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a 1 0 b

a

1 b 1

= a

1

(so l1 goes to l1)

a 1 b 1 1

= b

1 1

(so l2 goes to l2)

a 1 b 1 a

=

1

(so l3 goes to l5)

Bret Benesh Three Types of Symmetry

SLIDE 66

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a 1 0 b

a

1 b 1

= a

1

(so l1 goes to l1)

a 1 b 1 1

= b

1 1

(so l2 goes to l2)

a 1 b 1 a

=

1

(so l3 goes to l5)

a 1 b 1

Bret Benesh

Three Types of Symmetry

SLIDE 67

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a 1 0 b

a

1 b 1

= a

1

(so l1 goes to l1)

a 1 b 1 1

= b

1 1

(so l2 goes to l2)

a 1 b 1 a

=

1

(so l3 goes to l5)

a 1 b 1

=

1 b

Bret Benesh

Three Types of Symmetry

SLIDE 68

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a 1 0 b

a

1 b 1

= a

1

(so l1 goes to l1)

a 1 b 1 1

= b

1 1

(so l2 goes to l2)

a 1 b 1 a

=

1

(so l3 goes to l5)

a 1 b 1

=

1 b

(so l5 goes to l4)

Bret Benesh Three Types of Symmetry

SLIDE 69

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a 1 0 b

a

1 b 1

= a

1

(so l1 goes to l1)

a 1 b 1 1

= b

1 1

(so l2 goes to l2)

a 1 b 1 a

=

1

(so l3 goes to l5)

a 1 b 1

=

1 b

(so l5 goes to l4)

a 1 b 1 b

Bret Benesh

Three Types of Symmetry

SLIDE 70

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a 1 0 b

a

1 b 1

= a

1

(so l1 goes to l1)

a 1 b 1 1

= b

1 1

(so l2 goes to l2)

a 1 b 1 a

=

1

(so l3 goes to l5)

a 1 b 1

=

1 b

(so l5 goes to l4)

a 1 b 1 b

=

1 a

Bret Benesh

Three Types of Symmetry

SLIDE 71

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a 1 0 b

a

1 b 1

= a

1

(so l1 goes to l1)

a 1 b 1 1

= b

1 1

(so l2 goes to l2)

a 1 b 1 a

=

1

(so l3 goes to l5)

a 1 b 1

=

1 b

(so l5 goes to l4)

a 1 b 1 b

=

1 a

(so l4 goes to l3)

Bret Benesh Three Types of Symmetry

SLIDE 72

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

a 1 0 b

a

1 b 1

= a

1

(so l1 goes to l1)

a 1 b 1 1

= b

1 1

(so l2 goes to l2)

a 1 b 1 a

=

1

(so l3 goes to l5)

a 1 b 1

=

1 b

(so l5 goes to l4)

a 1 b 1 b

=

1 a

(so l4 goes to l3)

(l3 l5 l4)

Bret Benesh Three Types of Symmetry

SLIDE 73

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

1 1 b a

1

1 b a 1

Bret Benesh

Three Types of Symmetry

SLIDE 74

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

1 1 b a

1

1 b a 1

=

1 b

Bret Benesh

Three Types of Symmetry

SLIDE 75

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

1 1 b a

1

1 b a 1

=

1 b

(so l1 goes to l4)

Bret Benesh Three Types of Symmetry

SLIDE 76

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

1 1 b a

1

1 b a 1

=

1 b

(so l1 goes to l4)

1 1 b a 1 b

Bret Benesh

Three Types of Symmetry

SLIDE 77

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

1 1 b a

1

1 b a 1

=

1 b

(so l1 goes to l4)

1 1 b a 1 b

= a

1 1

Bret Benesh

Three Types of Symmetry

SLIDE 78

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

1 1 b a

1

1 b a 1

=

1 b

(so l1 goes to l4)

1 1 b a 1 b

= a

1 1

(so l4 goes to l2)

Bret Benesh Three Types of Symmetry

SLIDE 79

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

1 1 b a

1

1 b a 1

=

1 b

(so l1 goes to l4)

1 1 b a 1 b

= a

1 1

(so l4 goes to l2)

1 1 b a 1 1

Bret Benesh

Three Types of Symmetry

SLIDE 80

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

1 1 b a

1

1 b a 1

=

1 b

(so l1 goes to l4)

1 1 b a 1 b

= a

1 1

(so l4 goes to l2)

1 1 b a 1 1

=

1

Bret Benesh

Three Types of Symmetry

SLIDE 81

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

1 1 b a

1

1 b a 1

=

1 b

(so l1 goes to l4)

1 1 b a 1 b

= a

1 1

(so l4 goes to l2)

1 1 b a 1 1

=

1

(so l2 goes to l5)

Bret Benesh Three Types of Symmetry

SLIDE 82

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

1 1 b a

1

1 b a 1

=

1 b

(so l1 goes to l4)

1 1 b a 1 b

= a

1 1

(so l4 goes to l2)

1 1 b a 1 1

=

1

(so l2 goes to l5)

1 1 b a 1

Bret Benesh

Three Types of Symmetry

SLIDE 83

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

1 1 b a

1

1 b a 1

=

1 b

(so l1 goes to l4)

1 1 b a 1 b

= a

1 1

(so l4 goes to l2)

1 1 b a 1 1

=

1

(so l2 goes to l5)

1 1 b a 1

=

1 a

Bret Benesh

Three Types of Symmetry

SLIDE 84

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

1 1 b a

1

1 b a 1

=

1 b

(so l1 goes to l4)

1 1 b a 1 b

= a

1 1

(so l4 goes to l2)

1 1 b a 1 1

=

1

(so l2 goes to l5)

1 1 b a 1

=

1 a

(so l5 goes to l3)

Bret Benesh Three Types of Symmetry

SLIDE 85

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

1 1 b a

1

1 b a 1

=

1 b

(so l1 goes to l4)

1 1 b a 1 b

= a

1 1

(so l4 goes to l2)

1 1 b a 1 1

=

1

(so l2 goes to l5)

1 1 b a 1

=

1 a

(so l5 goes to l3)

1 1 b a 1 a

Bret Benesh

Three Types of Symmetry

SLIDE 86

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

1 1 b a

1

1 b a 1

=

1 b

(so l1 goes to l4)

1 1 b a 1 b

= a

1 1

(so l4 goes to l2)

1 1 b a 1 1

=

1

(so l2 goes to l5)

1 1 b a 1

=

1 a

(so l5 goes to l3)

1 1 b a 1 a

= b

1

Bret Benesh

Three Types of Symmetry

SLIDE 87

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

1 1 b a

1

1 b a 1

=

1 b

(so l1 goes to l4)

1 1 b a 1 b

= a

1 1

(so l4 goes to l2)

1 1 b a 1 1

=

1

(so l2 goes to l5)

1 1 b a 1

=

1 a

(so l5 goes to l3)

1 1 b a 1 a

= b

1

(so l3 goes to l1)

Bret Benesh Three Types of Symmetry

SLIDE 88

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

1 1 b a

1

1 b a 1

=

1 b

(so l1 goes to l4)

1 1 b a 1 b

= a

1 1

(so l4 goes to l2)

1 1 b a 1 1

=

1

(so l2 goes to l5)

1 1 b a 1

=

1 a

(so l5 goes to l3)

1 1 b a 1 a

= b

1

(so l3 goes to l1)

(l1 l4 l2 l5 l3)

Bret Benesh Three Types of Symmetry

SLIDE 89

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Linear Algebra Summary

In fact...there are

Bret Benesh Three Types of Symmetry

SLIDE 90

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Linear Algebra Summary

In fact...there are 1 matrix that acts as the identity ( 1 1

)

Bret Benesh Three Types of Symmetry

SLIDE 91

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Linear Algebra Summary

In fact...there are 1 matrix that acts as the identity ( 1 1

)

15 matrices that act like a a 1 a

: (la lb)(lc ld)

Bret Benesh Three Types of Symmetry

SLIDE 92

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Linear Algebra Summary

In fact...there are 1 matrix that acts as the identity ( 1 1

)

15 matrices that act like a a 1 a

: (la lb)(lc ld)

20 matrices that act like a 1 b

: (la lb lc)

Bret Benesh Three Types of Symmetry

SLIDE 93

Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra

Linear Algebra Summary

In fact...there are 1 matrix that acts as the identity ( 1 1

)

15 matrices that act like a a 1 a

: (la lb)(lc ld)

20 matrices that act like a 1 b

: (la lb lc)

24 matrices that act like 1 1 b a

: (la lb lc ld le)

Bret Benesh Three Types of Symmetry

SLIDE 94

Introduction Examples of Symmetry Summary

Permutations v. Linear Algebra

Permutations Linear Algebra 1 Identity EVEN 1 Identity matrix 10 of type (a b) ODD 15 matrices: (la lb)(lc ld) 15 of type (a b)(c d) EVEN 20 matrices: (la lb lc) 20 of type (a b c) EVEN 24 matrices: (la lb lc ld le) 20 of type (a b c)(d e) ODD 30 of type (a b c d) ODD 24 of type (a b c d e) EVEN

Bret Benesh Three Types of Symmetry

SLIDE 95

Introduction Examples of Symmetry Summary

Permutations v. Linear Algebra

Permutations Linear Algebra 1 Identity EVEN 1 Identity matrix 10 of type (a b) ODD 15 matrices: (la lb)(lc ld) 15 of type (a b)(c d) EVEN 20 matrices: (la lb lc) 20 of type (a b c) EVEN 24 matrices: (la lb lc ld le) 20 of type (a b c)(d e) ODD 30 of type (a b c d) ODD 24 of type (a b c d e) EVEN

Bret Benesh Three Types of Symmetry

SLIDE 96

Introduction Examples of Symmetry Summary

Even Permutations v. Linear Algebra

Even Permutations Linear Algebra 1 Identity 1 Identity matrix 15 of type (a b)(c d) 15 matrices: (la lb)(lc ld) 20 of type (a b c) 20 matrices: (la lb lc) 24 of type (a b c d e) 24 matrices: (la lb lc ld le)

Bret Benesh Three Types of Symmetry

SLIDE 97

Introduction Examples of Symmetry Summary

Even Permutations v. Linear Algebra

Even Permutations Linear Algebra 1 Identity 1 Identity matrix 15 of type (a b)(c d) 15 matrices: (la lb)(lc ld) 20 of type (a b c) 20 matrices: (la lb lc) 24 of type (a b c d e) 24 matrices: (la lb lc ld le) Same thing!

Bret Benesh Three Types of Symmetry

SLIDE 98

Introduction Examples of Symmetry Summary

Types of Icosahedron Symmetry

1 “Identity" 15 Edge-Edge rotations 20 Face-Face rotations 24 Vertex-Vertex rotations (60 total)

Bret Benesh Three Types of Symmetry

SLIDE 99

Introduction Examples of Symmetry Summary

Geometry vs. Even Permutations vs. Linear Algebra

Geometry Even Permutations Linear Algebra 1 Identity 1 Identity 1 Identity matrix 15 Edge-Edge 15 (a b)(c d) 15 matrices: (la lb)(lc ld) 20 Face-Face 20 (a b c) 20 matrices: (la lb lc) 24 Vertex-Vertex 24 (a b c d e) 24 matrices: (la lb lc ld le)

Bret Benesh Three Types of Symmetry

SLIDE 100

Introduction Examples of Symmetry Summary

Geometry vs. Even Permutations vs. Linear Algebra

Geometry Even Permutations Linear Algebra 1 Identity 1 Identity 1 Identity matrix 15 (c1 c2)(c3 c4) 15 (a b)(c d) 15 matrices: (la lb)(lc ld) 20 (c1 c2 c3) 20 (a b c) 20 matrices: (la lb lc) 24 (c1 c2 c3 c4 c5) 24 (a b c d e) 24 matrices: (la lb lc ld le)

Bret Benesh Three Types of Symmetry

SLIDE 101

Introduction Examples of Symmetry Summary

Geometry vs. Even Permutations vs. Linear Algebra

Geometry Even Permutations Linear Algebra 1 Identity 1 Identity 1 Identity matrix 15 (c1 c2)(c3 c4) 15 (a b)(c d) 15 matrices: (la lb)(lc ld) 20 (c1 c2 c3) 20 (a b c) 20 matrices: (la lb lc) 24 (c1 c2 c3 c4 c5) 24 (a b c d e) 24 matrices: (la lb lc ld le) Same things!

Bret Benesh Three Types of Symmetry

SLIDE 102

Introduction Examples of Symmetry Summary

Thank you!

5

Bret Benesh College of St. Benedict

St. Joseph, MN

bbenesh@csbsju.edu

5Questioned... by Eleaf, Shared under a Creative Commons License Bret Benesh Three Types of Symmetry