Lecture 5 Triangulations simplicial complexes and cell complexes
- Slides: 55
Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH: 7450 (22 M: 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. Isabel K. Darcy Mathematics Department/Applied Mathematical & Computational Sciences University of Iowa http: //www. math. uiowa. edu/~idarcy/Applied. Topology. html
Building blocks for oriented simplicial complex 0 -simplex = vertex = v 1 -simplex = oriented edge = (v 1, v 2) v 1 v 2 e 2 -simplex = oriented face = (v 1, v 2, v 3) v 2 e 1 v 1 e 2 e 3 v 3
Building blocks for oriented simplicial complex 2 -simplex = oriented face = v 2 f = (v 1, v 2, v 3) = (v 2, v 3, v 1) = (v 3, v 1, v 2) e 1 v 1 e 2 e 3 – f = (v 2, v 1, v 3) = (v 3, v 2, v 1) = (v 1, v 3, v 2) v 2 e 1 v 3 e 2 e 3 v 3
Building blocks for oriented simplicial complex 3 -simplex = σ = (v 1, v 2, v 3, v 4) = (v 2, v 3, v 1, v 4) = (v 3, v 1, v 2, v 4) = (v 2, v 1, v 4, v 3) = (v 3, v 2, v 4, v 1) = (v 1, v 3, v 4, v 2) = (v 4, v 2, v 1, v 3) = (v 4, v 3, v 2, v 1) = (v 4, v 1, v 3, v 2) = (v 1, v 4, v 2, v 3) = (v 2, v 4, v 3, v 1) = (v 3, v 4, v 1, v 2) – σ = (v 2, v 1, v 3, v 4) = (v 3, v 2, v 1, v 4) = (v 1, v 3, v 2, v 4) = (v 2, v 4, v 1, v 3) = (v 3, v 4, v 2, v 1) = (v 1, v 4, v 3, v 2) v 2 = (v 1, v 2, v 4, v 3) = (v 2, v 3, v 4, v 1) = (v 3, v 1, v 4, v 2) = (v 4, v 1, v 2, v 3) = (v 4, v 2, v 3, v 1) = (v 4, v 3, v 1, v 2) v 1 v 3 v 4
Building blocks for oriented simplicial complex 0 -simplex = vertex = v 1 -simplex = oriented edge = (v 1, v 2) Note that the boundary v 1 v 2 of this edge is v – v 2 1 e 2 -simplex = oriented face = (v 1, v 2, v 3) v 2 Note that the boundary of this face is the cycle e 1 e 2 e 1 + e 2 + e 3 v 1 v 3 = (v , v ) + (v , v ) – (v , v ) e 3 1 2 2 3 1 3 = (v 1, v 2) – (v 1, v 3) + (v 2, v 3)
Building blocks for oriented simplicial complex 3 -simplex = (v 1, v 2, v 3, v 4) = solid tetrahedron v 2 v 4 v 1 v 3 boundary of (v 1, v 2, v 3, v 4) = – (v 1, v 2, v 3) + (v 1, v 2, v 4) – (v 1, v 3, v 4) + (v 2, v 3, v 4) n-simplex = (v 1, v 2, …, vn+1)
Building blocks for an unoriented simplicial complex using Z 2 coefficients 0 -simplex = vertex = v 1 -simplex = edge = {v 1, v 2} v 1 e v 2 Note that the boundary of this edge is v 2 + v 1 2 -simplex = face = {v 1, v 2, v 3} v 2 Note that the boundary of this face is the cycle e 1 e 2 e 1 + e 2 + e 3 v 1 v 3 e 3 = {v 1, v 2} + {v 2, v 3} + {v 1, v 3}
Creating a simplicial complex 0. ) Start by adding 0 -dimensional vertices (0 -simplices)
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
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
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
Creating a simplicial complex 2. ) Add 2 -dimensional triangles (2 -simplices). Boundary of a triangle = a cycle consisting of 3 edges.
Creating a simplicial complex 2. ) Add 2 -dimensional triangles (2 -simplices). Boundary of a triangle = a cycle consisting of 3 edges.
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.
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.
4. ) Add 4 -dimensional 4 -simplices, {v 1, v 2, …, v 5}. Boundary of a 4 -simplex = a cycle consisting of 3 -simplices. = {v 2, v 3, v 4, v 5} + {v 1, v 2, v 3, v 4}
Creating a simplicial complex n. ) Add n-dimensional n-simplices, {v 1, v 2, …, vn+1}. Boundary of a n-simplex = a cycle consisting of (n-1)-simplices.
Example: Triangulating the circle = { x in R 2 : ||x || = 1 }
Example: Triangulating the circle = { x in R 2 : ||x || = 1 }
Example: Triangulating the circle = { x in R 2 : ||x || = 1 } Not a triagulation Two vertices define a single edge
Example: Triangulating the circle = { x in R 2 : ||x || = 1 }
Example: Triangulating the circle = { x in R 2 : ||x || = 1 }
Example: Triangulating the disk = { x in R 2 : ||x || ≤ 1 }
Example: Triangulating the disk = { x in R 2 : ||x || ≤ 1 }
Example: Triangulating the disk = { x in R 2 : ||x || ≤ 1 }
Example: Triangulating the disk = { x in R 2 : ||x || ≤ 1 }
Example: Triangulating the disk = { x in R 2 : ||x || ≤ 1 } =
Example: Triangulating the sphere = { x in R 3 : ||x || = 1 }
Example: Triangulating the sphere = { x in R 3 : ||x || = 1 }
Example: Triangulating the sphere = { x in R 3 : ||x || = 1 }
Example: Triangulating the sphere = { x in R 3 : ||x || = 1 }
Example: Triangulating the circle. disk = { x in R 2 : ||x || ≤ 1 } =
Example: Triangulating the circle. disk = { x in R 2 : ||x || ≤ 1 }
Example: Triangulating the circle. disk = { x in R 2 : ||x || ≤ 1 } Fist image from http: //openclipart. org/detail/1000/a-raised-fist-by-liftarn
Example: Triangulating the sphere = { x in R 3 : ||x || = 1 }
Example: Triangulating the sphere = { x in R 3 : ||x || = 1 } =
Creating a cell complex Building block: n-cells = { x in Rn : || x || ≤ 1 } Examples: 0 -cell = { x in R 0 : ||x || < 1 } 1 -cell =open interval ={ x in R : ||x || < 1 } ( 2 -cell = open disk = { x in R 2 : ||x || < 1 } 3 -cell = open ball = { x in R 3 : ||x || < 1 } )
Building blocks for a simplicial complex 0 -simplex = vertex = v 1 -simplex = edge = {v 1, v 2} v 1 e v 2 Note that the boundary of this edge is v 2 + v 1 2 -simplex = triangle = {v 1, v 2, v 3} v 2 Note that the boundary of this triangle is the cycle e 1 e 2 e 1 + e 2 + e 3 v 1 v 3 e 3 = {v 1, v 2} + {v 2, v 3} + {v 1, v 3}
Creating a cell complex Building block: n-cells = { x in Rn : || x || ≤ 1 } Examples: 0 -cell = { x in R 0 : ||x || < 1 } 1 -cell =open interval ={ x in R : ||x || < 1 } ( 2 -cell = open disk = { x in R 2 : ||x || < 1 } 3 -cell = open ball = { x in R 3 : ||x || < 1 } )
Creating a cell complex Building block: n-cells = { x in Rn : || x || ≤ 1 } Examples: 0 -cell = { x in R 0 : ||x || < 1 } 1 -cell =open interval ={ x in R : ||x || < 1 } ( 2 -cell = open disk = { x in R 2 : ||x || < 1 } 3 -cell = open ball = { x in R 3 : ||x || < 1 } )
Example: disk = { x in R 2 : ||x || ≤ 1 } Simplicial complex Cell complex = = U ( ) U
Example: disk = { x in R 2 : ||x || ≤ 1 } Simplicial complex Cell complex = = U ( ) U
Example: disk = { x in R 2 : ||x || ≤ 1 } Simplicial complex = Cell complex = U ( ) U
Example: disk = { x in R 2 : ||x || ≤ 1 } Simplicial complex = Cell complex = U [ ] U ( ) U
Example: disk = { x in R 2 : ||x || ≤ 1 } Simplicial complex = Cell complex = U [ ] U ( ) U =
Example: disk = { x in R 2 : ||x || ≤ 1 } Simplicial complex = Cell complex = U [ U ( ] = U ) U
Example: disk = { x in R 2 : ||x || ≤ 1 } Simplicial complex = 3 vertices, 3 edges, 1 triangle Cell complex = 1 vertex, 1 edge, 1 disk. U [ U ( ) U ] = U =
Example: sphere = { x in R 3 : ||x || = 1 } = Simplicial complex 4 vertices, 6 edges, 4 triangles Cell Complex 1 vertex, 1 disk = U
Example: sphere = { x in R 3 : ||x || = 1 } = Simplicial complex Cell complex = U
Example: sphere = { x in R 3 : ||x || = 1 } = Simplicial complex Cell complex = U U
Example: sphere = { x in R 3 : ||x || = 1 } = Simplicial complex Cell complex = U U =
Example: sphere = { x in R 3 : ||x || = 1 } = Simplicial complex Cell complex = U U = Fist image from http: //openclipart. org/detail/1000/a-raised-fist-by-liftarn
Example: sphere = { x in R 3 : ||x || = 1 } = Cell complex = U U =
Example: sphere = { x in R 3 : ||x || = 1 } = Cell complex = U U =
Example: sphere = { x in R 3 : ||x || = 1 } Simplicial complex Cell complex = = U U =
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