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Lecture 5: Triangulations & simplicial complexes (and cell complexes).

Isabel K. Darcy Mathematics Department/Applied Mathematical & Computational Sciences University of Iowa http:// www.math.uiowa.edu /~ idarcy / AppliedTopology.html. Lecture 5: Triangulations & simplicial complexes (and cell complexes).

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Lecture 5: Triangulations & simplicial complexes (and cell complexes).

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  1. Isabel K. Darcy Mathematics Department/Applied Mathematical & Computational Sciences University of Iowa http://www.math.uiowa.edu/~idarcy/AppliedTopology.html Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology Target Audience: Anyone interested in topological data analysis including graduate students, faculty, industrial researchers in bioinformatics, biology, business, computer science, cosmology, engineering, imaging, mathematics, neurology, physics, statistics, etc.

  2. Building blocks for a simplicial complex e 0-simplex = vertex = v 1-simplex = edge = {v1, v2} Note that the boundary of this edge is v2+ v1 2-simplex = triangle = {v1, v2, v3} v2 Note that the boundary of this triangle is the cycle e1 + e2 + e3 = {v1, v2} + {v2, v3} +{v1, v3} e1 e2 v1 v1 v3 v2 e3

  3. Building blocks for a simplicial complex 3-simplex = {v1, v2, v3, v4} = tetrahedron boundary of {v1, v2, v3, v4} = {v1, v2, v3} + {v1, v2, v4} + {v1, v3, v4} + {v2, v3, v4} n-simplex = {v1, v2, …, vn+1} v2 v4 v1 v3

  4. Building blocks for a simplicial complex 3-simplex = {v1, v2, v3, v4} = tetrahedron boundary of {v1, v2, v3, v4} = {v1, v2, v3} + {v1, v2, v4} + {v1, v3, v4} + {v2, v3, v4} n-simplex = {v1, v2, …, vn+1} v2 v2 Fill in v4 v4 v1 v1 v3 v3

  5. Creating a simplicial complex 0.) Start by adding 0-dimensional vertices (0-simplices)

  6. Creating a simplicial complex 1.) Next add 1-dimensional edges (1-simplices). Note: These edges must connect two vertices. I.e., the boundary of an edge is two vertices

  7. Creating a simplicial complex 1.) Next add 1-dimensional edges (1-simplices). Note: These edges must connect two vertices. I.e., the boundary of an edge is two vertices

  8. Creating a simplicial complex 1.) Next add 1-dimensional edges (1-simplices). Note: These edges must connect two vertices. I.e., the boundary of an edge is two vertices

  9. Creating a simplicial complex 2.) Add 2-dimensional triangles (2-simplices). Boundary of a triangle = a cycle consisting of 3 edges.

  10. Creating a simplicial complex 2.) Add 2-dimensional triangles (2-simplices). Boundary of a triangle = a cycle consisting of 3 edges.

  11. Creating a simplicial complex 3.) Add 3-dimensional tetrahedrons (3-simplices). Boundary of a 3-simplex = a cycle consisting of its four 2-dimensional faces.

  12. Creating a simplicial complex 3.) Add 3-dimensional tetrahedrons (3-simplices). Boundary of a 3-simplex = a cycle consisting of its four 2-dimensional faces.

  13. 4.) Add 4-dimensional 4-simplices, {v1, v2, …, v5}. Boundary of a 4-simplex = a cycle consisting of 3-simplices. = {v2, v3, v4, v5} + {v1, v3, v4, v5} + {v1, v2, v4, v5} + {v1, v2, v3, v5} + {v1, v2, v3, v4}

  14. Creating a simplicial complex n.) Add n-dimensional n-simplices, {v1, v2, …, vn+1}. Boundary of a n-simplex = a cycle consisting of (n-1)-simplices.

  15. Example: Triangulating the circle. circle = { x in R2 : ||x || = 1 }

  16. Example: Triangulating the circle. circle = { x in R2 : ||x || = 1 }

  17. Example: Triangulating the circle. circle = { x in R2 : ||x || = 1 }

  18. Example: Triangulating the circle. circle = { x in R2 : ||x || = 1 }

  19. Example: Triangulating the circle. circle = { x in R2 : ||x || = 1 }

  20. Example: Triangulating the disk. disk = { x in R2 : ||x || ≤ 1 }

  21. Example: Triangulating the disk. disk = { x in R2 : ||x || ≤ 1 }

  22. Example: Triangulating the disk. disk = { x in R2 : ||x || ≤ 1 }

  23. Example: Triangulating the disk. disk = { x in R2 : ||x || ≤ 1 }

  24. Example: Triangulating the disk. disk = { x in R2 : ||x || ≤ 1 } =

  25. Example: Triangulating the sphere. sphere = { x in R3 : ||x || = 1 }

  26. Example: Triangulating the sphere. sphere = { x in R3 : ||x || = 1 }

  27. Example: Triangulating the sphere. sphere = { x in R3 : ||x || = 1 }

  28. Example: Triangulating the sphere. sphere = { x in R3 : ||x || = 1 }

  29. Example: Triangulating the circle. disk = { x in R2 : ||x || ≤ 1 } =

  30. Example: Triangulating the circle. disk = { x in R2 : ||x || ≤ 1 }

  31. Example: Triangulating the circle. disk = { x in R2 : ||x || ≤ 1 } Fist image from http://openclipart.org/detail/1000/a-raised-fist-by-liftarn

  32. Example: Triangulating the sphere. sphere = { x in R3 : ||x || = 1 }

  33. Example: Triangulating the sphere. sphere = { x in R3 : ||x || = 1 } =

  34. Creating a cell complex Building block: n-cells = { x in Rn : || x || ≤ 1 } Examples: 0-cell = { x in R0 : ||x || < 1 } 1-cell =open interval ={ x in R : ||x || < 1 } ( ) 2-cell = open disk = { x in R2 : ||x || < 1 } 3-cell = open ball = { x in R3 : ||x || < 1 }

  35. Building blocks for a simplicial complex e 0-simplex = vertex = v 1-simplex = edge = {v1, v2} Note that the boundary of this edge is v2+ v1 2-simplex = triangle = {v1, v2, v3} v2 Note that the boundary of this triangle is the cycle e1 + e2 + e3 = {v1, v2} + {v2, v3} +{v1, v3} e1 e2 v1 v1 v3 v2 e3

  36. Building blocks for a simplicial complex 3-simplex = {v1, v2, v3, v4} = tetrahedron boundary of {v1, v2, v3, v4} = {v1, v2, v3} + {v1, v2, v4} + {v1, v3, v4} + {v2, v3, v4} n-simplex = {v1, v2, …, vn+1} v2 v2 Fill in v4 v4 v1 v1 v3 v3

  37. Creating a cell complex Building block: n-cells = { x in Rn : || x || ≤ 1 } Examples: 0-cell = { x in R0 : ||x || < 1 } 1-cell =open interval ={ x in R : ||x || < 1 } ( ) 2-cell = open disk = { x in R2 : ||x || < 1 } 3-cell = open ball = { x in R3 : ||x || < 1 }

  38. Creating a cell complex Building block: n-cells = { x in Rn : || x || ≤ 1 } Examples: 0-cell = { x in R0 : ||x || < 1 } 1-cell =open interval ={ x in R : ||x || < 1 } ( ) 2-cell = open disk = { x in R2 : ||x || < 1 } 3-cell = open ball = { x in R3 : ||x || < 1 }

  39. Example: disk = { x in R2 : ||x || ≤ 1 } Simplicial complex = ( ) Cell complex = U U

  40. Example: disk = { x in R2 : ||x || ≤ 1 } Simplicial complex = ( ) Cell complex = U U

  41. Example: disk = { x in R2 : ||x || ≤ 1 } Simplicial complex = ( ) ( ) Cell complex = U U U

  42. Example: disk = { x in R2 : ||x || ≤ 1 } Simplicial complex = ( ) [] Cell complex = U U U

  43. Example: disk = { x in R2 : ||x || ≤ 1 } Simplicial complex = ( ) [] Cell complex = U U = U

  44. Example: disk = { x in R2 : ||x || ≤ 1 } Simplicial complex = ( ) [] Cell complex = U U = U U

  45. Example: disk = { x in R2 : ||x || ≤ 1 } Simplicial complex = ( ) [] Cell complex = U U = U = U

  46. Euler characteristic (simple form): = number of vertices – number of edges + number of faces Or in short-hand, = |V| - |E| + |F| where V = set of vertices E = set of edges F = set of 2-dimensional faces & the notation |X| = the number of elements in the set X.

  47. Example: disk = { x in R2 : ||x || ≤ 1 } Simplicial complex 3 vertices, 3 edges, 1 triangle = Cell complex 1 vertex, 1 edge, 1 disk. ( ) [] = U U = U = U

  48. Euler characteristic: Given a simplicial complex C, let Cn = the set of n-dimensional simplices in C, and let |Cn| denote the number of elements in Cn. Then = |C0| - |C1| + |C2| - |C3| + … = Σ (-1)n |Cn|

  49. Euler characteristic: Given a cell complex C, let Cn = the set of n-dimensional cells in C, and let |Cn| denote the number of elements in Cn. Then = |C0| - |C1| + |C2| - |C3| + … = Σ (-1)n |Cn|

  50. Example: sphere = { x in R3 : ||x || = 1 } = Simplicial complex 4 vertices, 6 edges, 4 triangles Cell Complex 1 vertex, 1 disk = U

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