1. University of California, Berkeley
Bridges 1999
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 D D D
This series extents to arbitrary dimensions!

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

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

5. 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

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

7. 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.

8. 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.

9. 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).

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

11. 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

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)
(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)

14. 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.

15. 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.

16. Intriguing Open Problems . . .
Minhyong Kim (Oxford University)
“Secret Link Uncovered Between Pure Math and Physics”
A graph embedded in a 3-hole torus