Lecture 19: Introduction To Topology COMPSCI/MATH 290-04 Chris - - PowerPoint PPT Presentation

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Lecture 19: Introduction To Topology COMPSCI/MATH 290-04 Chris - - PowerPoint PPT Presentation

Feb 26, 2023 430 likes •913 views

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

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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

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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

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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

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Graphs Review

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Planar Graphs

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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The Euler Characteristic

χ = V − E + F

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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The Euler Characteristic

χ = V − E + F Planar graphs?

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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The Euler Characteristic

χ = V − E + F = 2 Planar graphs?

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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The Euler Characteristic: Proof

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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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

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Regular Polygons

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Stereographic Projection

http://www.ics.uci.edu/˜eppstein/junkyard/euler/

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Regular Polyhedra (Platonic Solids)

The Tetrahedron: 4 Vertices, 4 Faces, Triangle Faces

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Regular Polyhedra (Platonic Solids)

The Cube: 8 Vertices, 6 Faces, Square Faces

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Regular Polyhedra (Platonic Solids)

The Octahedron: 6 Vertices, 8 Faces, Triangle Faces

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Regular Polyhedra (Platonic Solids)

The Dodecahedron: 20 Vertices, 12 Faces, Pentagonal Faces

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Regular Polyhedra (Platonic Solids)

The Icosahedron: 12 Vertices, 20 Faces, Triangle Faces

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Constructing The Tetrahedron

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Constructing The Icosahedron

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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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

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Platonic Solids: Is This it??

Let p be the number of sides per face, q be the degree of each vertex pF = 2E = qV

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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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

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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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 = 1 2 + 1 E

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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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 = 1 2 + 1 E = ⇒ 1 q + 1 p > 1 2

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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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

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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

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Flattening To Plane

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Flattening To Plane

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Flattening To Plane

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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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

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The Torus

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Constructing Torus

Show Video

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Torus Fundamental Polygon

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Torus Fundamental Polygon

◮ What is the Euler characteristic of a torus?

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Intermezzo: Rhythm And Tori / Grateful Dead

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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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

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Duplicating Spheres

What’s the euler characteristic of two spheres?

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Duplicating Tori

What’s the euler characteristic of two tori?

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Connected Sum

T1#T1 = T2

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Connected Sum

T1#T1 = T2 What is the Euler characteristic?

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Connected Sum: g Tori

What is the Euler characteristic of TN = T1#T1# . . . #T1 g times?

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Connected Sum: g Tori

What is the Euler characteristic of TN = T1#T1# . . . #T1 g times? χ = 2 − 2g

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Connected Sum: g Tori

What is the Euler characteristic of TN = T1#T1# . . . #T1 g times? χ = 2 − 2g ◮ g is known as the “genus”

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Connected Sum with Spheres

What is the connected sum of a sphere with a sphere?

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Connected Sum with Spheres

What is the connected sum of a torus with a sphere?

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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Euler Characteristic: Homology

χ = β0 − β1 + β2 ◮ β0: Number of connected components ◮ β1: Number of independent loops/cycles ◮ β2 Number of independent voids

COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology