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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 Introduction Examples of Symmetry Summary Outline 1 Introduction 2 Examples of Symmetry Geometry Permutations Linear Algebra 3 Summary Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Definition of Symmetry 1 1 Symmetry by Stuant63, Shared under a Creative Commons License Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Definition of Symmetry 1 Definition: A symmetry is an action that preserves some specified structure. 1 Symmetry by Stuant63, Shared under a Creative Commons License Bret Benesh Three Types of Symmetry 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 Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra Introduction to Geometry 2 Definition: A symmetry is an action that preserves some specified structure. 2 Lyra... by Daveybot, Shared under a Creative Commons License Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra Facts about the Icosahedron 20 faces Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra Facts about the Icosahedron 20 faces 30 edges Bret Benesh Three Types of Symmetry 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 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 Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra Types of Icosahedron Symmetry 1 “Identity" Bret Benesh Three Types of Symmetry 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 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 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 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 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 Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra Introduction to Permutations 3 Definition: A symmetry is an action that preserves some specified structure. 3 Arashiyama... by Jpellgen, Shared under a Creative Commons License Bret Benesh Three Types of Symmetry 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 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 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 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 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 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 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 Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra “Factoring" Permutations 96 = 2 · 2 · 2 · 2 · 2 · 3 Bret Benesh Three Types of Symmetry 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 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 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 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 0 trans: 1 trans: 2 trans: 2 trans: 3 trans: 3 trans: 4 trans: 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 0 trans: 1 trans: 2 trans: 2 trans: 3 trans: 3 trans: 4 trans: EVEN ODD EVEN EVEN ODD ODD EVEN 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 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. 4 Matrix Code by David Asch, Shared under a Creative Commons License Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra New Number System F4 + 0 1 a b 0 0 1 a b 1 1 0 b a a a b 0 1 b b a 1 0 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra New Number System F4 + 0 1 a b 0 0 1 a b 1 1 0 b a a a b 0 1 b b a 1 0 × 0 1 a b 0 0 0 0 0 1 0 1 a b a 0 a b 1 b 0 b 1 a Bret Benesh Three Types of Symmetry 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 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 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 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 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 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 Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a a 1 a a a 1 1 a 0 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a a 1 a 1 a a 1 =a b 1 a 0 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a a 1 a 1 a a 1 =a (so l1 goes to l4 ) b 1 a 0 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a a 1 a a 1 a 1 1 a 1 =a (so l1 goes to l4 ) b a 0 a a a 1 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a a 1 a a 1 a 1 1 a 1 =a (so l1 goes to l4 ) b a 0 a a 1 = a 1 0 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a a 1 a a 1 a 1 1 a 1 =a (so l1 goes to l4 ) b a 0 a a 1 = (so l4 goes to l1 ) a 1 0 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a a 1 a a 1 a 1 a 1 1 a 1 =a (so l1 goes to l4 ) b a 0 a a 1 = (so l4 goes to l1 ) a 1 0 a 1 a 1 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a a 1 a a 1 a 1 a 1 1 a 1 =a (so l1 goes to l4 ) b a 0 a a 1 = (so l4 goes to l1 ) a 1 0 0 a 1 =b 1 a 1 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a a 1 a a 1 a 1 a 1 1 a 1 =a (so l1 goes to l4 ) b a 0 a a 1 = (so l4 goes to l1 ) a 1 0 0 a 1 =b (so l2 goes to l5 ) 1 a 1 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a a 1 a a 1 a 1 a 1 a 1 1 a 1 =a (so l1 goes to l4 ) b a 0 a a 1 = (so l4 goes to l1 ) a 1 0 0 a 1 =b (so l2 goes to l5 ) 1 a 1 a 0 a 1 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a a 1 a a 1 a 1 a 1 a 1 a 1 a 0 a a a 1 a 1 a 1 a 0 a 1 1 =a (so l1 goes to l4 ) b 1 = (so l4 goes to l1 ) 0 0 =b (so l2 goes to l5 ) 1 1 =a 1 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a a 1 a a 1 a 1 a 1 a 1 a 1 a 0 a a a 1 a 1 a 1 a 0 a 1 1 =a (so l1 goes to l4 ) b 1 = (so l4 goes to l1 ) 0 0 =b (so l2 goes to l5 ) 1 1 =a (so l5 goes to l2 ) 1 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a a 1 a a 1 a 1 a 1 a 1 a 1 a 1 a 0 a a a 1 a 1 a 1 a 0 a 1 a 1 a a 1 =a (so l1 goes to l4 ) b 1 = (so l4 goes to l1 ) 0 0 =b (so l2 goes to l5 ) 1 1 =a (so l5 goes to l2 ) 1 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a a 1 a a 1 a 1 a 1 a 1 a 1 a 1 a 0 a a a 1 a 1 a 1 a 0 a 1 a 1 a a 1 =a (so l1 goes to l4 ) b 1 = (so l4 goes to l1 ) 0 0 =b (so l2 goes to l5 ) 1 1 =a (so l5 goes to l2 ) 1 1 = a Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a a 1 a a 1 a 1 a 1 a 1 a 1 a 1 a 0 a a a 1 a 1 a 1 a 0 a 1 a 1 a a 1 =a (so l1 goes to l4 ) b 1 = (so l4 goes to l1 ) 0 0 =b (so l2 goes to l5 ) 1 1 =a (so l5 goes to l2 ) 1 1 = (so l3 goes to l3 ) a Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a a 1 a a 1 a 1 a 1 a 1 a 1 a 1 a 0 a a a 1 a 1 a 1 a 0 a 1 a 1 a a 1 =a (so l1 goes to l4 ) b 1 = (so l4 goes to l1 ) 0 0 =b (so l2 goes to l5 ) 1 1 =a (so l5 goes to l2 ) 1 1 = (so l3 goes to l3 ) a (l1 l4 )(l2 l5 ) Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a 1 0 b a 1 1 0 b 0 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a 1 0 b 1 a 1 1 =a 0 0 b 0 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a 1 0 b 1 a 1 1 =a (so l1 goes to l1 ) 0 0 b 0 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a 1 0 b a 0 a 0 1 1 1 =a (so l1 goes to l1 ) 0 b 0 1 1 b 1 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a 1 0 b a 0 a 0 1 1 1 =a (so l1 goes to l1 ) 0 b 0 1 1 1 =b b 1 1 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a 1 0 b a 0 a 0 1 1 1 =a (so l1 goes to l1 ) 0 b 0 1 1 1 =b (so l2 goes to l2 ) b 1 1 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a 1 0 b a 0 a 0 a 0 1 1 1 =a (so l1 goes to l1 ) 0 b 0 1 1 1 =b (so l2 goes to l2 ) b 1 1 1 1 b a Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a 1 0 b a 0 a 0 a 0 1 1 1 =a (so l1 goes to l1 ) 0 b 0 1 1 1 =b (so l2 goes to l2 ) b 1 1 0 1 1 = 1 b a Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a 1 0 b a 0 a 0 a 0 1 1 1 =a (so l1 goes to l1 ) 0 b 0 1 1 1 =b (so l2 goes to l2 ) b 1 1 0 1 1 = (so l3 goes to l5 ) 1 b a Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a 1 0 b a 0 a 0 a 0 a 0 1 1 1 =a (so l1 goes to l1 ) 0 b 0 1 1 1 =b (so l2 goes to l2 ) b 1 1 0 1 1 = (so l3 goes to l5 ) 1 b a 1 0 b 1 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a 1 0 b a 0 a 0 a 0 a 0 1 1 b 0 1 1 b 1 1 1 b a 1 0 b 1 1 =a (so l1 goes to l1 ) 0 1 =b (so l2 goes to l2 ) 1 0 = (so l3 goes to l5 ) 1 1 = b Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a 1 0 b a 0 a 0 a 0 a 0 1 1 b 0 1 1 b 1 1 1 b a 1 0 b 1 1 =a (so l1 goes to l1 ) 0 1 =b (so l2 goes to l2 ) 1 0 = (so l3 goes to l5 ) 1 1 = (so l5 goes to l4 ) b Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a 1 0 b a 0 a 0 a 0 a 0 a 0 1 1 b 0 1 1 b 1 1 1 b a 1 0 b 1 1 1 b b 1 =a (so l1 goes to l1 ) 0 1 =b (so l2 goes to l2 ) 1 0 = (so l3 goes to l5 ) 1 1 = (so l5 goes to l4 ) b Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a 1 0 b a 0 a 0 a 0 a 0 a 0 1 1 b 0 1 1 b 1 1 1 b a 1 0 b 1 1 1 b b 1 =a (so l1 goes to l1 ) 0 1 =b (so l2 goes to l2 ) 1 0 = (so l3 goes to l5 ) 1 1 = (so l5 goes to l4 ) b 1 = a Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a 1 0 b a 0 a 0 a 0 a 0 a 0 1 1 b 0 1 1 b 1 1 1 b a 1 0 b 1 1 1 b b 1 =a (so l1 goes to l1 ) 0 1 =b (so l2 goes to l2 ) 1 0 = (so l3 goes to l5 ) 1 1 = (so l5 goes to l4 ) b 1 = (so l4 goes to l3 ) a Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra a 1 0 b a 0 a 0 a 0 a 0 a 0 1 1 b 0 1 1 b 1 1 1 b a 1 0 b 1 1 1 b b 1 =a (so l1 goes to l1 ) 0 1 =b (so l2 goes to l2 ) 1 0 = (so l3 goes to l5 ) 1 1 = (so l5 goes to l4 ) b 1 = (so l4 goes to l3 ) a (l3 l5 l4 ) Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary 1 1 b a Geometry Permutations Linear Algebra 1 1 b a 1 0 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary 1 1 b a Geometry Permutations Linear Algebra 1 1 b a 1 1 = b 0 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary 1 1 b a Geometry Permutations Linear Algebra 1 1 b a 1 1 = (so l1 goes to l4 ) b 0 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary 1 1 b a 1 b 1 b Geometry Permutations Linear Algebra 1 1 1 = (so l1 goes to l4 ) b a 0 1 1 a b Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary 1 1 b a 1 b 1 b Geometry Permutations Linear Algebra 1 1 1 = (so l1 goes to l4 ) b a 0 1 1 1 =a a b 1 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary 1 1 b a 1 b 1 b Geometry Permutations Linear Algebra 1 1 1 = (so l1 goes to l4 ) b a 0 1 1 1 =a (so l4 goes to l2 ) a b 1 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary 1 1 b a 1 b 1 b 1 b Geometry Permutations Linear Algebra 1 1 1 = (so l1 goes to l4 ) b a 0 1 1 1 =a (so l4 goes to l2 ) a b 1 1 1 a 1 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary 1 1 b a 1 b 1 b 1 b Geometry Permutations Linear Algebra 1 1 1 = (so l1 goes to l4 ) b a 0 1 1 1 =a (so l4 goes to l2 ) a b 1 0 1 1 = 1 a 1 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary 1 1 b a 1 b 1 b 1 b Geometry Permutations Linear Algebra 1 1 1 = (so l1 goes to l4 ) b a 0 1 1 1 =a (so l4 goes to l2 ) a b 1 0 1 1 = (so l2 goes to l5 ) 1 a 1 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary 1 1 b a 1 b 1 b 1 b 1 b Geometry Permutations Linear Algebra 1 1 1 = (so l1 goes to l4 ) b a 0 1 1 1 =a (so l4 goes to l2 ) a b 1 0 1 1 = (so l2 goes to l5 ) 1 a 1 1 0 a 1 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary 1 1 b a 1 b 1 b 1 b 1 b Geometry Permutations Linear Algebra 1 1 a 0 1 1 a b 1 1 a 1 1 0 a 1 1 = (so l1 goes to l4 ) b 1 =a (so l4 goes to l2 ) 1 0 = (so l2 goes to l5 ) 1 1 = a Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary 1 1 b a 1 b 1 b 1 b 1 b Geometry Permutations Linear Algebra 1 1 a 0 1 1 a b 1 1 a 1 1 0 a 1 1 = (so l1 goes to l4 ) b 1 =a (so l4 goes to l2 ) 1 0 = (so l2 goes to l5 ) 1 1 = (so l5 goes to l3 ) a Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary 1 1 b a 1 b 1 b 1 b 1 b 1 b Geometry Permutations Linear Algebra 1 1 a 0 1 1 a b 1 1 a 1 1 0 a 1 1 1 a a 1 = (so l1 goes to l4 ) b 1 =a (so l4 goes to l2 ) 1 0 = (so l2 goes to l5 ) 1 1 = (so l5 goes to l3 ) a Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary 1 1 b a 1 b 1 b 1 b 1 b 1 b Geometry Permutations Linear Algebra 1 1 a 0 1 1 a b 1 1 a 1 1 0 a 1 1 1 a a 1 = (so l1 goes to l4 ) b 1 =a (so l4 goes to l2 ) 1 0 = (so l2 goes to l5 ) 1 1 = (so l5 goes to l3 ) a 1 =b 0 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary 1 1 b a 1 b 1 b 1 b 1 b 1 b Geometry Permutations Linear Algebra 1 1 a 0 1 1 a b 1 1 a 1 1 0 a 1 1 1 a a 1 = (so l1 goes to l4 ) b 1 =a (so l4 goes to l2 ) 1 0 = (so l2 goes to l5 ) 1 1 = (so l5 goes to l3 ) a 1 =b (so l3 goes to l1 ) 0 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary 1 1 b a 1 b 1 b 1 b 1 b 1 b Geometry Permutations Linear Algebra 1 1 a 0 1 1 a b 1 1 a 1 1 0 a 1 1 1 a a 1 = (so l1 goes to l4 ) b 1 =a (so l4 goes to l2 ) 1 0 = (so l2 goes to l5 ) 1 1 = (so l5 goes to l3 ) a 1 =b (so l3 goes to l1 ) 0 (l1 l4 l2 l5 l3 ) Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra Linear Algebra Summary In fact...there are Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra Linear Algebra Summary In fact...there are 1 0 1 matrix that acts as the identity ( ) 0 1 Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra Linear Algebra Summary In fact...there are 1 0 1 matrix that acts as the identity ( ) 0 1 a a 15 matrices that act like : (la lb )(lc ld ) 1 a Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra Linear Algebra Summary In fact...there are 1 0 1 matrix that acts as the identity ( ) 0 1 a a 15 matrices that act like : (la lb )(lc ld ) 1 a a 1 20 matrices that act like : (la lb lc ) 0 b Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry Permutations Linear Algebra Linear Algebra Summary In fact...there are 1 0 1 matrix that acts as the identity ( ) 0 1 a a 15 matrices that act like : (la lb )(lc ld ) 1 a a 1 20 matrices that act like : (la lb lc ) 0 b 1 1 24 matrices that act like : (la lb lc ld le ) b a Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Permutations v. Linear Algebra Permutations 1 Identity EVEN 10 of type (a b) ODD 15 of type (a b)(c d) EVEN 20 of type (a b c) EVEN 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 Linear Algebra 1 Identity matrix 15 matrices: (la lb )(lc ld ) 20 matrices: (la lb lc ) 24 matrices: (la lb lc ld le ) Three Types of Symmetry Introduction Examples of Symmetry Summary Permutations v. Linear Algebra Permutations 1 Identity EVEN 10 of type (a b) ODD 15 of type (a b)(c d) EVEN 20 of type (a b c) EVEN 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 Linear Algebra 1 Identity matrix 15 matrices: (la lb )(lc ld ) 20 matrices: (la lb lc ) 24 matrices: (la lb lc ld le ) Three Types of Symmetry Introduction Examples of Symmetry Summary Even Permutations v. Linear Algebra Even Permutations 1 Identity 15 of type (a b)(c d) 20 of type (a b c) 24 of type (a b c d e) Linear Algebra 1 Identity matrix 15 matrices: (la lb )(lc ld ) 20 matrices: (la lb lc ) 24 matrices: (la lb lc ld le ) Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Even Permutations v. Linear Algebra Even Permutations 1 Identity 15 of type (a b)(c d) 20 of type (a b c) 24 of type (a b c d e) Linear Algebra 1 Identity matrix 15 matrices: (la lb )(lc ld ) 20 matrices: (la lb lc ) 24 matrices: (la lb lc ld le ) Same thing! Bret Benesh Three Types of Symmetry 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 Introduction Examples of Symmetry Summary Geometry vs. Even Permutations vs. Linear Algebra Geometry 1 Identity 15 Edge-Edge 20 Face-Face 24 Vertex-Vertex Even Permutations 1 Identity 15 (a b)(c d) 20 (a b c) 24 (a b c d e) Bret Benesh Linear Algebra 1 Identity matrix 15 matrices: (la lb )(lc ld ) 20 matrices: (la lb lc ) 24 matrices: (la lb lc ld le ) Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry vs. Even Permutations vs. Linear Algebra Geometry 1 Identity 15 (c1 c2 )(c3 c4 ) 20 (c1 c2 c3 ) 24 (c1 c2 c3 c4 c5 ) Even Permutations 1 Identity 15 (a b)(c d) 20 (a b c) 24 (a b c d e) Bret Benesh Linear Algebra 1 Identity matrix 15 matrices: (la lb )(lc ld ) 20 matrices: (la lb lc ) 24 matrices: (la lb lc ld le ) Three Types of Symmetry Introduction Examples of Symmetry Summary Geometry vs. Even Permutations vs. Linear Algebra Geometry 1 Identity 15 (c1 c2 )(c3 c4 ) 20 (c1 c2 c3 ) 24 (c1 c2 c3 c4 c5 ) Even Permutations 1 Identity 15 (a b)(c d) 20 (a b c) 24 (a b c d e) Linear Algebra 1 Identity matrix 15 matrices: (la lb )(lc ld ) 20 matrices: (la lb lc ) 24 matrices: (la lb lc ld le ) Same things! Bret Benesh Three Types of Symmetry Introduction Examples of Symmetry Summary Thank you! 5 Bret Benesh College of St. Benedict St. Joseph, MN [email protected] 5 Questioned... by Eleaf, Shared under a Creative Commons License Bret Benesh Three Types of Symmetry
