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Transcript
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