University of California, Berkeley Bridges 1999 CS 39 (2017) Key Concepts Carlo H. Séquin University of California, Berkeley

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!

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

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

Cn Dn Symmetry (2D) Conway: n *n Bridges 1999 Symmetry (2D) An exact definition of ‘What is symmetry?’ A finite set of symmetry classes. For 2D finite patterns, there are just 2 classes: The different ways in which a pattern can be mapped back onto itself with a rigid-body transformation form the elements of a mathematical group. Cn has n elements; Dn has 2*n elements. Cn Dn Conway: n *n

Symmetry of 3D Objects Seven highly regular “spherical” symmetries based on the Platonic solids:

Symmetry of 3D Objects (cont.) Bridges 1999 Symmetry of 3D Objects (cont.) Seven “cylindrical” symmetries based on the seven linear Frieze patterns: Here are the 7 frieze patterns.

2D Infinite “Wallpaper” Symmetry Bridges 1999 2D Infinite “Wallpaper” Symmetry 17 symmetry types of plane patterns: The circle symbol indicates only pure translational symmetry in two different directions – nothing else.

Topology of 2-Manifolds Granada 2003 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 Three parameters are needed to topologically classify a 2-manifold: sidedness, number of borders, connectivity (expressed as EC or genus).

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

Manifold Connectivity 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. Disk EC = 1 - #cuts genus = #cuts genus = 4

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.

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

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

Graph Embeddings (1D-M. in 2D-M.) 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: Open Problem: What is the surface of minimal genus that allows crossing-free embedding of a graph or knot.

Intriguing Open Problems . . . Minhyong Kim (Oxford University) “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/