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

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Introduction Examples of Symmetry Summary Three Types of Symmetry Bret Benesh College of St. Benedict/St. Johns University Department of Mathematics Concordia College Mathematics Colloquium September 8, 2009 Bret Benesh Three Types of

  1. 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

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. 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

  26. 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

  27. 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

  28. 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

  29. 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

  30. 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

  31. 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

  32. 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

  33. 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

  34. 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

  35. 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

  36. 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

  37. 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

  38. 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

  39. 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

  40. 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

  41. 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

  42. 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

  43. 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

  44. 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

  45. 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

  46. 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

  47. 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

  48. 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

  49. 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

  50. 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

  51. 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

  52. 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

  53. 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

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

  55. 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

  56. 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

  57. 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

  58. 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

  59. 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

  60. 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

  61. 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

  62. 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

  63. 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

  64. 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

  65. 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

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