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

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra Lines Through Origin in F 4 Only consider matrices of determinant 1 l 1 : y = 0 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra Lines Through Origin in F 4 Only consider matrices of determinant 1 l 1 : y = 0 l 2 : y = x Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra Lines Through Origin in F 4 Only consider matrices of determinant 1 l 1 : y = 0 l 2 : y = x l 3 : y = ax Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra Lines Through Origin in F 4 Only consider matrices of determinant 1 l 1 : y = 0 l 2 : y = x l 3 : y = ax l 4 : y = bx Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra Lines Through Origin in F 4 Only consider matrices of determinant 1 l 1 : y = 0 l 2 : y = x l 3 : y = ax l 4 : y = bx l 5 : x = 0 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a a � 1 a � a � � 1 � a 1 a 0 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a a � 1 a � a � � 1 � � 1 � a = a 1 a 0 b Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a a � 1 a � a � � 1 � � 1 � a = a (so l 1 goes to l 4 ) 1 a 0 b Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a a � 1 a � a � � 1 � � 1 � a = a (so l 1 goes to l 4 ) 1 a 0 b � a � � a � a 1 a 1 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a a � 1 a � a � � 1 � � 1 � a = a (so l 1 goes to l 4 ) 1 a 0 b � a � � a � � 1 � a = 1 a 1 0 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a a � 1 a � a � � 1 � � 1 � a = a (so l 1 goes to l 4 ) 1 a 0 b � a � � a � � 1 � a = (so l 4 goes to l 1 ) 1 a 1 0 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a a � 1 a � a � � 1 � � 1 � a = a (so l 1 goes to l 4 ) 1 a 0 b � a � � a � � 1 � a = (so l 4 goes to l 1 ) 1 a 1 0 � a � � 1 � a 1 a 1 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a a � 1 a � a � � 1 � � 1 � a = a (so l 1 goes to l 4 ) 1 a 0 b � a � � a � � 1 � a = (so l 4 goes to l 1 ) 1 a 1 0 � a � � 1 � � 0 � a = b 1 a 1 1 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a a � 1 a � a � � 1 � � 1 � a = a (so l 1 goes to l 4 ) 1 a 0 b � a � � a � � 1 � a = (so l 4 goes to l 1 ) 1 a 1 0 � a � � 1 � � 0 � a = b (so l 2 goes to l 5 ) 1 a 1 1 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a a � 1 a � a � � 1 � � 1 � a = a (so l 1 goes to l 4 ) 1 a 0 b � a � � a � � 1 � a = (so l 4 goes to l 1 ) 1 a 1 0 � a � � 1 � � 0 � a = b (so l 2 goes to l 5 ) 1 a 1 1 � a � � 0 � a 1 a 1 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a a � 1 a � a � � 1 � � 1 � a = a (so l 1 goes to l 4 ) 1 a 0 b � a � � a � � 1 � a = (so l 4 goes to l 1 ) 1 a 1 0 � a � � 1 � � 0 � a = b (so l 2 goes to l 5 ) 1 a 1 1 � a � � 0 � � 1 � a = a 1 a 1 1 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a a � 1 a � a � � 1 � � 1 � a = a (so l 1 goes to l 4 ) 1 a 0 b � a � � a � � 1 � a = (so l 4 goes to l 1 ) 1 a 1 0 � a � � 1 � � 0 � a = b (so l 2 goes to l 5 ) 1 a 1 1 � a � � 0 � � 1 � a = a (so l 5 goes to l 2 ) 1 a 1 1 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a a � 1 a � a � � 1 � � 1 � a = a (so l 1 goes to l 4 ) 1 a 0 b � a � � a � � 1 � a = (so l 4 goes to l 1 ) 1 a 1 0 � a � � 1 � � 0 � a = b (so l 2 goes to l 5 ) 1 a 1 1 � a � � 0 � � 1 � a = a (so l 5 goes to l 2 ) 1 a 1 1 � a � � 1 � a 1 a a Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a a � 1 a � a � � 1 � � 1 � a = a (so l 1 goes to l 4 ) 1 a 0 b � a � � a � � 1 � a = (so l 4 goes to l 1 ) 1 a 1 0 � a � � 1 � � 0 � a = b (so l 2 goes to l 5 ) 1 a 1 1 � a � � 0 � � 1 � a = a (so l 5 goes to l 2 ) 1 a 1 1 � a � � 1 � � 1 � a = 1 a a a Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a a � 1 a � a � � 1 � � 1 � a = a (so l 1 goes to l 4 ) 1 a 0 b � a � � a � � 1 � a = (so l 4 goes to l 1 ) 1 a 1 0 � a � � 1 � � 0 � a = b (so l 2 goes to l 5 ) 1 a 1 1 � a � � 0 � � 1 � a = a (so l 5 goes to l 2 ) 1 a 1 1 � a � � 1 � � 1 � a = (so l 3 goes to l 3 ) 1 a a a Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a a � 1 a � a � � 1 � � 1 � a = a (so l 1 goes to l 4 ) 1 a 0 b � a � � a � � 1 � a = (so l 4 goes to l 1 ) 1 a 1 0 � a � � 1 � � 0 � a = b (so l 2 goes to l 5 ) 1 a 1 1 � a � � 0 � � 1 � a = a (so l 5 goes to l 2 ) 1 a 1 1 � a � � 1 � � 1 � a = (so l 3 goes to l 3 ) 1 a a a ( l 1 l 4 )( l 2 l 5 ) Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a 1 � 0 b � a � � 1 � 1 0 b 0 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a 1 � 0 b � a � � 1 � � 1 � 1 = a 0 b 0 0 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a 1 � 0 b � a � � 1 � � 1 � 1 = a (so l 1 goes to l 1 ) 0 b 0 0 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a 1 � 0 b � a � � 1 � � 1 � 1 = a (so l 1 goes to l 1 ) 0 b 0 0 � a � � 1 � 1 0 b 1 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a 1 � 0 b � a � � 1 � � 1 � 1 = a (so l 1 goes to l 1 ) 0 b 0 0 � a � � 1 � � 1 � 1 = b 0 b 1 1 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a 1 � 0 b � a � � 1 � � 1 � 1 = a (so l 1 goes to l 1 ) 0 b 0 0 � a � � 1 � � 1 � 1 = b (so l 2 goes to l 2 ) 0 b 1 1 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a 1 � 0 b � a � � 1 � � 1 � 1 = a (so l 1 goes to l 1 ) 0 b 0 0 � a � � 1 � � 1 � 1 = b (so l 2 goes to l 2 ) 0 b 1 1 � a � � 1 � 1 0 b a Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a 1 � 0 b � a � � 1 � � 1 � 1 = a (so l 1 goes to l 1 ) 0 b 0 0 � a � � 1 � � 1 � 1 = b (so l 2 goes to l 2 ) 0 b 1 1 � a � � 1 � � 0 � 1 = 0 b a 1 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a 1 � 0 b � a � � 1 � � 1 � 1 = a (so l 1 goes to l 1 ) 0 b 0 0 � a � � 1 � � 1 � 1 = b (so l 2 goes to l 2 ) 0 b 1 1 � a � � 1 � � 0 � 1 = (so l 3 goes to l 5 ) 0 b a 1 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a 1 � 0 b � a � � 1 � � 1 � 1 = a (so l 1 goes to l 1 ) 0 b 0 0 � a � � 1 � � 1 � 1 = b (so l 2 goes to l 2 ) 0 b 1 1 � a � � 1 � � 0 � 1 = (so l 3 goes to l 5 ) 0 b a 1 � a � � 0 � 1 0 b 1 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a 1 � 0 b � a � � 1 � � 1 � 1 = a (so l 1 goes to l 1 ) 0 b 0 0 � a � � 1 � � 1 � 1 = b (so l 2 goes to l 2 ) 0 b 1 1 � a � � 1 � � 0 � 1 = (so l 3 goes to l 5 ) 0 b a 1 � a � � 0 � � 1 � 1 = 0 b 1 b Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a 1 � 0 b � a � � 1 � � 1 � 1 = a (so l 1 goes to l 1 ) 0 b 0 0 � a � � 1 � � 1 � 1 = b (so l 2 goes to l 2 ) 0 b 1 1 � a � � 1 � � 0 � 1 = (so l 3 goes to l 5 ) 0 b a 1 � a � � 0 � � 1 � 1 = (so l 5 goes to l 4 ) 0 b 1 b Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a 1 � 0 b � a � � 1 � � 1 � 1 = a (so l 1 goes to l 1 ) 0 b 0 0 � a � � 1 � � 1 � 1 = b (so l 2 goes to l 2 ) 0 b 1 1 � a � � 1 � � 0 � 1 = (so l 3 goes to l 5 ) 0 b a 1 � a � � 0 � � 1 � 1 = (so l 5 goes to l 4 ) 0 b 1 b � a � � 1 � 1 0 b b Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a 1 � 0 b � a � � 1 � � 1 � 1 = a (so l 1 goes to l 1 ) 0 b 0 0 � a � � 1 � � 1 � 1 = b (so l 2 goes to l 2 ) 0 b 1 1 � a � � 1 � � 0 � 1 = (so l 3 goes to l 5 ) 0 b a 1 � a � � 0 � � 1 � 1 = (so l 5 goes to l 4 ) 0 b 1 b � a � � 1 � � 1 � 1 = 0 b b a Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a 1 � 0 b � a � � 1 � � 1 � 1 = a (so l 1 goes to l 1 ) 0 b 0 0 � a � � 1 � � 1 � 1 = b (so l 2 goes to l 2 ) 0 b 1 1 � a � � 1 � � 0 � 1 = (so l 3 goes to l 5 ) 0 b a 1 � a � � 0 � � 1 � 1 = (so l 5 goes to l 4 ) 0 b 1 b � a � � 1 � � 1 � 1 = (so l 4 goes to l 3 ) 0 b b a Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � a 1 � 0 b � a � � 1 � � 1 � 1 = a (so l 1 goes to l 1 ) 0 b 0 0 � a � � 1 � � 1 � 1 = b (so l 2 goes to l 2 ) 0 b 1 1 � a � � 1 � � 0 � 1 = (so l 3 goes to l 5 ) 0 b a 1 � a � � 0 � � 1 � 1 = (so l 5 goes to l 4 ) 0 b 1 b � a � � 1 � � 1 � 1 = (so l 4 goes to l 3 ) 0 b b a ( l 3 l 5 l 4 ) Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � 1 1 � b a � 1 � � 1 � 1 b a 0 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � 1 1 � b a � 1 � � 1 � � 1 � 1 = b a 0 b Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � 1 1 � b a � 1 � � 1 � � 1 � 1 = (so l 1 goes to l 4 ) b a 0 b Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � 1 1 � b a � 1 � � 1 � � 1 � 1 = (so l 1 goes to l 4 ) b a 0 b � 1 � � 1 � 1 b a b Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � 1 1 � b a � 1 � � 1 � � 1 � 1 = (so l 1 goes to l 4 ) b a 0 b � 1 � � 1 � � 1 � 1 = a b a b 1 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � 1 1 � b a � 1 � � 1 � � 1 � 1 = (so l 1 goes to l 4 ) b a 0 b � 1 � � 1 � � 1 � 1 = a (so l 4 goes to l 2 ) b a b 1 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � 1 1 � b a � 1 � � 1 � � 1 � 1 = (so l 1 goes to l 4 ) b a 0 b � 1 � � 1 � � 1 � 1 = a (so l 4 goes to l 2 ) b a b 1 � 1 � � 1 � 1 b a 1 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � 1 1 � b a � 1 � � 1 � � 1 � 1 = (so l 1 goes to l 4 ) b a 0 b � 1 � � 1 � � 1 � 1 = a (so l 4 goes to l 2 ) b a b 1 � 1 � � 1 � � 0 � 1 = b a 1 1 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � 1 1 � b a � 1 � � 1 � � 1 � 1 = (so l 1 goes to l 4 ) b a 0 b � 1 � � 1 � � 1 � 1 = a (so l 4 goes to l 2 ) b a b 1 � 1 � � 1 � � 0 � 1 = (so l 2 goes to l 5 ) b a 1 1 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � 1 1 � b a � 1 � � 1 � � 1 � 1 = (so l 1 goes to l 4 ) b a 0 b � 1 � � 1 � � 1 � 1 = a (so l 4 goes to l 2 ) b a b 1 � 1 � � 1 � � 0 � 1 = (so l 2 goes to l 5 ) b a 1 1 � 1 � � 0 � 1 b a 1 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � 1 1 � b a � 1 � � 1 � � 1 � 1 = (so l 1 goes to l 4 ) b a 0 b � 1 � � 1 � � 1 � 1 = a (so l 4 goes to l 2 ) b a b 1 � 1 � � 1 � � 0 � 1 = (so l 2 goes to l 5 ) b a 1 1 � 1 � � 0 � � 1 � 1 = b a 1 a Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � 1 1 � b a � 1 � � 1 � � 1 � 1 = (so l 1 goes to l 4 ) b a 0 b � 1 � � 1 � � 1 � 1 = a (so l 4 goes to l 2 ) b a b 1 � 1 � � 1 � � 0 � 1 = (so l 2 goes to l 5 ) b a 1 1 � 1 � � 0 � � 1 � 1 = (so l 5 goes to l 3 ) b a 1 a Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � 1 1 � b a � 1 � � 1 � � 1 � 1 = (so l 1 goes to l 4 ) b a 0 b � 1 � � 1 � � 1 � 1 = a (so l 4 goes to l 2 ) b a b 1 � 1 � � 1 � � 0 � 1 = (so l 2 goes to l 5 ) b a 1 1 � 1 � � 0 � � 1 � 1 = (so l 5 goes to l 3 ) b a 1 a � 1 � � 1 � 1 b a a Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � 1 1 � b a � 1 � � 1 � � 1 � 1 = (so l 1 goes to l 4 ) b a 0 b � 1 � � 1 � � 1 � 1 = a (so l 4 goes to l 2 ) b a b 1 � 1 � � 1 � � 0 � 1 = (so l 2 goes to l 5 ) b a 1 1 � 1 � � 0 � � 1 � 1 = (so l 5 goes to l 3 ) b a 1 a � 1 � � 1 � � 1 � 1 = b b a a 0 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � 1 1 � b a � 1 � � 1 � � 1 � 1 = (so l 1 goes to l 4 ) b a 0 b � 1 � � 1 � � 1 � 1 = a (so l 4 goes to l 2 ) b a b 1 � 1 � � 1 � � 0 � 1 = (so l 2 goes to l 5 ) b a 1 1 � 1 � � 0 � � 1 � 1 = (so l 5 goes to l 3 ) b a 1 a � 1 � � 1 � � 1 � 1 = b (so l 3 goes to l 1 ) b a a 0 Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra � 1 1 � b a � 1 � � 1 � � 1 � 1 = (so l 1 goes to l 4 ) b a 0 b � 1 � � 1 � � 1 � 1 = a (so l 4 goes to l 2 ) b a b 1 � 1 � � 1 � � 0 � 1 = (so l 2 goes to l 5 ) b a 1 1 � 1 � � 0 � � 1 � 1 = (so l 5 goes to l 3 ) b a 1 a � 1 � � 1 � � 1 � 1 = b (so l 3 goes to l 1 ) b a a 0 ( l 1 l 4 l 2 l 5 l 3 ) Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary Linear Algebra Linear Algebra Summary In fact...there are Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary 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 Geometry Examples of Symmetry Permutations Summary 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 : ( l a l b )( l c l d ) 1 a Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary 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 : ( l a l b )( l c l d ) 1 a � a � 1 : ( l a l b l c ) 20 matrices that act like 0 b Bret Benesh Three Types of Symmetry

Introduction Geometry Examples of Symmetry Permutations Summary 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 : ( l a l b )( l c l d ) 1 a � a � 1 : ( l a l b l c ) 20 matrices that act like 0 b � 1 � 1 24 matrices that act like : ( l a l b l c l d l e ) b a Bret Benesh Three Types of Symmetry

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: ( l a l b )( l c l d ) 15 of type ( a b )( c d ) EVEN 20 matrices: ( l a l b l c ) 20 of type ( a b c ) EVEN 24 matrices: ( l a l b l c l d l e ) 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

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: ( l a l b )( l c l d ) 15 of type ( a b )( c d ) EVEN 20 matrices: ( l a l b l c ) 20 of type ( a b c ) EVEN 24 matrices: ( l a l b l c l d l e ) 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

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: ( l a l b )( l c l d ) 20 of type ( a b c ) 20 matrices: ( l a l b l c ) 24 of type ( a b c d e ) 24 matrices: ( l a l b l c l d l e ) Bret Benesh Three Types of Symmetry

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: ( l a l b )( l c l d ) 20 of type ( a b c ) 20 matrices: ( l a l b l c ) 24 of type ( a b c d e ) 24 matrices: ( l a l b l c l d l e ) 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 Even Permutations Linear Algebra 1 Identity 1 Identity 1 Identity matrix 15 Edge-Edge 15 ( a b )( c d ) 15 matrices: ( l a l b )( l c l d ) 20 Face-Face 20 ( a b c ) 20 matrices: ( l a l b l c ) 24 Vertex-Vertex 24 ( a b c d e ) 24 matrices: ( l a l b l c l d l e ) Bret Benesh Three Types of Symmetry

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 ( c 1 c 2 )( c 3 c 4 ) 15 ( a b )( c d ) 15 matrices: ( l a l b )( l c l d ) 20 ( c 1 c 2 c 3 ) 20 ( a b c ) 20 matrices: ( l a l b l c ) 24 ( c 1 c 2 c 3 c 4 c 5 ) 24 ( a b c d e ) 24 matrices: ( l a l b l c l d l e ) Bret Benesh Three Types of Symmetry