Lecture 19: Introduction To Topology
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1 Chris Tralie, Duke University 3/24/2016
2 Announcements Group Assignment 2 Due Wednesday 3/30 First project milestone Friday 4/8/2016 Welcome to unit 3!
3 Table of Contents The Euler Characteristic Spherical Polytopes / Platonic Solids Fundamental Polygons, Tori Connected Sums, Genus
4 Graphs Review
5 Planar Graphs
6 The Euler Characteristic χ = V E + F
7 The Euler Characteristic Planar graphs? χ = V E + F
8 The Euler Characteristic Planar graphs? χ = V E + F = 2
9 The Euler Characteristic: Proof
10 Table of Contents The Euler Characteristic Spherical Polytopes / Platonic Solids Fundamental Polygons, Tori Connected Sums, Genus
11 Regular Polygons
12 Stereographic Projection eppstein/junkyard/euler/
13 Regular Polyhedra (Platonic Solids) The Tetrahedron: 4 Vertices, 4 Faces, Triangle Faces
14 Regular Polyhedra (Platonic Solids) The Cube: 8 Vertices, 6 Faces, Square Faces
15 Regular Polyhedra (Platonic Solids) The Octahedron: 6 Vertices, 8 Faces, Triangle Faces
16 Regular Polyhedra (Platonic Solids) The Dodecahedron: 20 Vertices, 12 Faces, Pentagonal Faces
17 Regular Polyhedra (Platonic Solids) The Icosahedron: 12 Vertices, 20 Faces, Triangle Faces
18 Constructing The Tetrahedron
19 Constructing The Icosahedron
20 Platonic Solids: Is This it?? Let p be the number of sides per face, q be the degree of each vertex
21 Platonic Solids: Is This it?? Let p be the number of sides per face, q be the degree of each vertex pf = 2E = qv
22 Platonic Solids: Is This it?? Let p be the number of sides per face, q be the degree of each vertex pf = 2E = qv Combine with V E + F = 2 2E q E + 2E p = 2
23 Platonic Solids: Is This it?? Let p be the number of sides per face, q be the degree of each vertex pf = 2E = qv Combine with V E + F = 2 2E q E + 2E p = 2 1 q + 1 p = E
24 Platonic Solids: Is This it?? Let p be the number of sides per face, q be the degree of each vertex pf = 2E = qv Combine with V E + F = 2 2E q E + 2E p = 2 1 q + 1 p = E = 1 q + 1 p > 1 2
25 Flattening To Plane We don t need convex polygons, as long as they are sphere-like
26 Flattening To Plane We don t need convex polygons, as long as they are sphere-like
27 Flattening To Plane
28 Flattening To Plane
29 Flattening To Plane
30 Table of Contents The Euler Characteristic Spherical Polytopes / Platonic Solids Fundamental Polygons, Tori Connected Sums, Genus
31 The Torus
32 Constructing Torus Show Video
33 Torus Fundamental Polygon
34 Torus Fundamental Polygon What is the Euler characteristic of a torus?
35 Intermezzo: Rhythm And Tori / Grateful Dead
36 Table of Contents The Euler Characteristic Spherical Polytopes / Platonic Solids Fundamental Polygons, Tori Connected Sums, Genus
37 Duplicating Spheres What s the euler characteristic of two spheres?
38 Duplicating Tori What s the euler characteristic of two tori?
39 Connected Sum T 1 #T 1 = T 2
40 Connected Sum T 1 #T 1 = T 2 What is the Euler characteristic?
41 Connected Sum: g Tori What is the Euler characteristic of T N = T 1 #T 1 #... #T 1 g times?
42 Connected Sum: g Tori What is the Euler characteristic of T N = T 1 #T 1 #... #T 1 g times? χ = 2 2g
43 Connected Sum: g Tori What is the Euler characteristic of T N = T 1 #T 1 #... #T 1 g times? g is known as the genus χ = 2 2g
44 Connected Sum with Spheres What is the connected sum of a sphere with a sphere?
45 Connected Sum with Spheres What is the connected sum of a torus with a sphere?
46 Euler Characteristic: Homology χ = β 0 β 1 + β 2 β 0 : Number of connected components β 1 : Number of independent loops/cycles β 2 Number of independent voids
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