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CS 39 (2017)

Explore the fascinating world of higher-dimensional spaces, including measure polytopes and regular polytopes. Learn about symmetries, topologies, regular homotopies, mathematical knots, and graph embeddings in different dimensions. Discover the intriguing link between pure mathematics and physics.

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CS 39 (2017)

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  1. CS 39 (2017) Key Concepts Carlo H. Séquin University of California, Berkeley

  2. Higher-Dimensional Spaces • Extrapolating by analogy from 1-, 2-, 3-D spaces,with which we are intuitively familiar,we can think of spaces of higher dimensions. • Consecutive perpendicular extrusions leads to the“measure polytopes”, the “units” for these spaces. 1D 2D 3D 4D This series extents to arbitrary dimensions!

  3. Komplete Graphs  Simplex Series • To draw Kn with all edges of the same lengthwe need to use (n-1)-dimensional space. 1D 2D 3D This series also goes on indefinitely!

  4. Regular Polytopes All dimensions enable some regular polytopes: 2D 3D 4D 5D 6D 7D 8D 9D …5 6 3 3 3 3 3 3 There are always 3 polytopes that result from the: • Simplex series • Measure polytope series • Cross polytope series

  5. Symmetry (2D) Cn Dn Conway: n *n An exact definition of ‘What is symmetry?’ A finite set of symmetry classes. For 2D finite patterns, there are just 2 classes:

  6. Symmetry of 3D Objects Sevenhighly regular “spherical” symmetries based on the Platonic solids:

  7. Symmetry of 3D Objects (cont.) Seven “cylindrical” symmetries based onthe seven linear Frieze patterns:

  8. 2D Infinite “Wallpaper” Symmetry 17 symmetry types of plane patterns:

  9. Topology of 2-Manifolds Defining Characteristics: Double-sided (orientable) Number of borders b = 3 Euler characteristic χ = –5 Genus g = (2 – χ – b)/2 = 2 Independent cutting lines: 2

  10. Some Prototypical Surfaces Möbius bands Boy surface Klein bottles Two-sided handle bodies:

  11. Manifold Connectivity Disk EC = 1 - #cuts genus = #cutsgenus = 4 Determining the Euler characteristic:Cut “ribbons” until shape is a topological disk. Determining the genus of a handle-body:Cut tubes until there are no more loops.

  12. Regular Homotopies(1D-M. in 2D) Smooth deformations, without any cuts or sharp kinks!  Curves in 2D can only transform into one another if they have the same turning number.

  13. Regular Homotopies(2D-M. in 3D) NOT possible in 2D Turning a 3D-sphere inside out (B. Morin) (Hyper-) Spheres can only be turned inside out only in spaces with an odd number of dimensions.

  14. Mathematical Knots (1D-M. in 3D) Open Problem: To determine unambiguously whether two (complicated) knots are the same. Closed Loops in 3D space.Strand is not allowed to pass through itself!

  15. Graph Embeddings(1D-M. in 2D-M.) Open Problem: What is the surface of minimal genus that allows crossing-free embedding of a graph or knot. Planar versus non-planar Graphs: Utility Graph (K3,3) is non-planar;can be embedded in a torus (g = 1).“Torus knots” are embedded in a torus surface:

  16. Intriguing Open Problems . . . “Secret Link Uncovered Between Pure Math and Physics” A graph embedded in a 3-hole torus https://www.quantamagazine.org/secret-link-uncovered-between-pure-math-and-physics-20171201/ Minhyong Kim (Oxford University)

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