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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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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
B Group Assignment 2 Due Wednesday 3/30B First project milestone Friday 4/8/2016B Welcome to unit 3!
COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Table of Contents
I The Euler Characteristic
B Spherical Polytopes / Platonic Solids
B Fundamental Polygons, Tori
B 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
Planar graphs?
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
B The Euler Characteristic
I Spherical Polytopes / Platonic Solids
B Fundamental Polygons, Tori
B 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 eachvertex
pF = 2E = qV
Combine with V − E + F = 2
2Eq− E +
2Ep
= 2
1q+
1p=
12+
1E
=⇒ 1q+
1p>
12
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 eachvertex
pF = 2E = qV
Combine with V − E + F = 2
2Eq− E +
2Ep
= 2
1q+
1p=
12+
1E
=⇒ 1q+
1p>
12
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 eachvertex
pF = 2E = qV
Combine with V − E + F = 2
2Eq− E +
2Ep
= 2
1q+
1p=
12+
1E
=⇒ 1q+
1p>
12
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 eachvertex
pF = 2E = qV
Combine with V − E + F = 2
2Eq− E +
2Ep
= 2
1q+
1p=
12+
1E
=⇒ 1q+
1p>
12
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 eachvertex
pF = 2E = qV
Combine with V − E + F = 2
2Eq− E +
2Ep
= 2
1q+
1p=
12+
1E
=⇒ 1q+
1p>
12
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
B The Euler Characteristic
B Spherical Polytopes / Platonic Solids
I Fundamental Polygons, Tori
B 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
I What is the Euler characteristic of a torus?
COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Torus Fundamental Polygon
I 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
B The Euler Characteristic
B Spherical Polytopes / Platonic Solids
B Fundamental Polygons, Tori
I 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
T1#T1 = T2
COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Connected Sum
T1#T1 = T2What is the Euler characteristic?
COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Connected Sum: g Tori
What is the Euler characteristic of TN = T1#T1# . . .#T1 gtimes?
χ = 2− 2g
I g is known as the “genus”
COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Connected Sum: g Tori
What is the Euler characteristic of TN = T1#T1# . . .#T1 gtimes?
χ = 2− 2g
I g is known as the “genus”
COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
Connected Sum: g Tori
What is the Euler characteristic of TN = T1#T1# . . .#T1 gtimes?
χ = 2− 2g
I 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 = β0 − β1 + β2
I β0: Number of connected componentsI β1: Number of independent loops/cyclesI β2 Number of independent voids
COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology
