Lecture 19: Introduction To Topology COMPSCI/MATH 290-04 Chris Tralie, Duke University 3/24/2016 COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Announcements ⊲ Group Assignment 2 Due Wednesday 3/30 ⊲ First project milestone Friday 4/8/2016 ⊲ Welcome to unit 3! COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Table of Contents ◮ The Euler Characteristic ⊲ Spherical Polytopes / Platonic Solids ⊲ Fundamental Polygons, Tori ⊲ Connected Sums, Genus COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Graphs Review COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Planar Graphs COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
The Euler Characteristic χ = V − E + F COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
The Euler Characteristic χ = V − E + F Planar graphs? COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
The Euler Characteristic χ = V − E + F = 2 Planar graphs? COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
The Euler Characteristic: Proof COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Table of Contents ⊲ The Euler Characteristic ◮ Spherical Polytopes / Platonic Solids ⊲ Fundamental Polygons, Tori ⊲ Connected Sums, Genus COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Regular Polygons COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Stereographic Projection http://www.ics.uci.edu/˜eppstein/junkyard/euler/ COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Regular Polyhedra (Platonic Solids) The Tetrahedron: 4 Vertices, 4 Faces, Triangle Faces COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Regular Polyhedra (Platonic Solids) The Cube: 8 Vertices, 6 Faces, Square Faces COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Regular Polyhedra (Platonic Solids) The Octahedron: 6 Vertices, 8 Faces, Triangle Faces COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Regular Polyhedra (Platonic Solids) The Dodecahedron: 20 Vertices, 12 Faces, Pentagonal Faces COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Regular Polyhedra (Platonic Solids) The Icosahedron: 12 Vertices, 20 Faces, Triangle Faces COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Constructing The Tetrahedron COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Constructing The Icosahedron COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Platonic Solids: Is This it?? Let p be the number of sides per face, q be the degree of each vertex COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Platonic Solids: Is This it?? Let p be the number of sides per face, q be the degree of each vertex pF = 2 E = qV COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Platonic Solids: Is This it?? Let p be the number of sides per face, q be the degree of each vertex pF = 2 E = qV Combine with V − E + F = 2 2 E q − E + 2 E p = 2 COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Platonic Solids: Is This it?? Let p be the number of sides per face, q be the degree of each vertex pF = 2 E = qV Combine with V − E + F = 2 2 E q − E + 2 E p = 2 1 q + 1 p = 1 2 + 1 E COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Platonic Solids: Is This it?? Let p be the number of sides per face, q be the degree of each vertex pF = 2 E = qV Combine with V − E + F = 2 2 E q − E + 2 E p = 2 1 q + 1 p = 1 2 + 1 E ⇒ 1 q + 1 p > 1 = 2 COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Flattening To Plane We don’t need convex polygons, as long as they are “sphere-like” COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Flattening To Plane We don’t need convex polygons, as long as they are “sphere-like” COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Flattening To Plane COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Flattening To Plane COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Flattening To Plane COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Table of Contents ⊲ The Euler Characteristic ⊲ Spherical Polytopes / Platonic Solids ◮ Fundamental Polygons, Tori ⊲ Connected Sums, Genus COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
The Torus COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Constructing Torus Show Video COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Torus Fundamental Polygon COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Torus Fundamental Polygon ◮ What is the Euler characteristic of a torus? COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Intermezzo: Rhythm And Tori / Grateful Dead COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Table of Contents ⊲ The Euler Characteristic ⊲ Spherical Polytopes / Platonic Solids ⊲ Fundamental Polygons, Tori ◮ Connected Sums, Genus COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Duplicating Spheres What’s the euler characteristic of two spheres? COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Duplicating Tori What’s the euler characteristic of two tori? COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Connected Sum T 1 # T 1 = T 2 COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Connected Sum T 1 # T 1 = T 2 What is the Euler characteristic? COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Connected Sum: g Tori What is the Euler characteristic of T N = T 1 # T 1 # . . . # T 1 g times? COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Connected Sum: g Tori What is the Euler characteristic of T N = T 1 # T 1 # . . . # T 1 g times? χ = 2 − 2 g COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Connected Sum: g Tori What is the Euler characteristic of T N = T 1 # T 1 # . . . # T 1 g times? χ = 2 − 2 g ◮ g is known as the “genus” COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Connected Sum with Spheres What is the connected sum of a sphere with a sphere? COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Connected Sum with Spheres What is the connected sum of a torus with a sphere? COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
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 COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
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