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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 Group Assignment 2 Due Wednesday 3/30 First project milestone

  1. 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

  2. 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

  3. 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

  4. Graphs Review COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  5. Planar Graphs COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  6. The Euler Characteristic χ = V − E + F COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  7. The Euler Characteristic χ = V − E + F Planar graphs? COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  8. The Euler Characteristic χ = V − E + F = 2 Planar graphs? COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  9. The Euler Characteristic: Proof COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  10. 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

  11. Regular Polygons COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  12. Stereographic Projection http://www.ics.uci.edu/˜eppstein/junkyard/euler/ COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  13. Regular Polyhedra (Platonic Solids) The Tetrahedron: 4 Vertices, 4 Faces, Triangle Faces COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  14. Regular Polyhedra (Platonic Solids) The Cube: 8 Vertices, 6 Faces, Square Faces COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  15. Regular Polyhedra (Platonic Solids) The Octahedron: 6 Vertices, 8 Faces, Triangle Faces COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  16. Regular Polyhedra (Platonic Solids) The Dodecahedron: 20 Vertices, 12 Faces, Pentagonal Faces COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  17. Regular Polyhedra (Platonic Solids) The Icosahedron: 12 Vertices, 20 Faces, Triangle Faces COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  18. Constructing The Tetrahedron COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  19. Constructing The Icosahedron COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. 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

  26. 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

  27. Flattening To Plane COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  28. Flattening To Plane COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  29. Flattening To Plane COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  30. 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

  31. The Torus COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  32. Constructing Torus Show Video COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  33. Torus Fundamental Polygon COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  34. Torus Fundamental Polygon ◮ What is the Euler characteristic of a torus? COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  35. Intermezzo: Rhythm And Tori / Grateful Dead COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  36. 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

  37. Duplicating Spheres What’s the euler characteristic of two spheres? COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  38. Duplicating Tori What’s the euler characteristic of two tori? COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  39. Connected Sum T 1 # T 1 = T 2 COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  40. Connected Sum T 1 # T 1 = T 2 What is the Euler characteristic? COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  41. 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

  42. 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

  43. 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

  44. Connected Sum with Spheres What is the connected sum of a sphere with a sphere? COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

  45. Connected Sum with Spheres What is the connected sum of a torus with a sphere? COMPSCI/MATH 290-04 Lecture 19: Introduction To Topology

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

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