Part II. Mathematics Background

Here scientific background information is provided. Be aware that the following chapters have not yet been proofread as thoroughly as they should. They are for courageous readers only.

Well yes, it seems our dear project has to do with about all other branches of mathematics one can think of, but a little bit of hands waving is necessary as well. We aren't mathematical crackpots, are we?

Perhaps some things here might turn out as mere mess. Since we are non-experts in the fields addressed, some inaccuracies are of course unavoidable. Anyway it is crucial to be able to explore the unknown terrain by using computation (that's how the saying goes).

But there is no reason for a too submissive behavior either. On one hand one can try to use information from other parts of science for our concrete strategy games. And on the other side in return, one might even ask if by the help of computational trial / error and learning by doing we address some interesting mathematics valuable in its own right. General theorems emerge starting from shared properties of a wealth of examples.

Table of Contents

7. Case Studies
Tetrahedron Stellation I (Deltoid)
Truncated Deltoid
Tetrahedron Iterated Stellation 3_1
Tetrahedron Stellation II
Tetrahedron Reticulated
Tetrahedron Reticulated (big)
Tetrahedron Reticulated (tall)
Tetrahedron Reticulated (huge)
Octahedron stellation II
Icosahedron
Standard (regular)
Variant 1
Icosahedron Stellation I
Icosahedron Stellation II
Cube Triang II
Yabi
Bipyramid Octa
L2(7)
Cartographic
Regular
M(24)
Torus 3x4
Trapezohedron
S(5)
A(6)
Quad Star
Reticulated 2x2 Cube
Reticulated 3x3 Cube Variants
standard Rubik's Cube
rotated corner
rotated 2x2 area
rotated 3x3 area
a follow up to variant 3
Rhombic Dodecahedron
Zonotope 5
Monkgau
Costa-Hoffman minimal surface
Quad Hex tesselation on a torus
Pentagonal Quasicrystals
Initial decagon
Iteration one
Iteration two
Iteration three
Iteration four
Heptagonal Quasicrystals
Initial 14-gon
DoubleDodecahedron
Icositetra
Dode Tor 6
Torus Hexa 3x5
Hexa Tor
Hexa UnOrient
Fancy Heptas
ToDo
8. Graphs and Combinatorics
Adjacency matrix
Coloring
Combinatorical aspects
ToDo
9. Algebraic, Difference and Geometric Topology
Euler formula
Homology and Cohomology
Homotopy
Cohomology with Coefficients
Exact Sequences
Homology Spheres
ToDo
10. Knots and Links
Generalized Torus Knots
Invariant Polynomials
ToDo
11. Group Theory
General Theory
Holonomy Groups
Group Cohomology
Introduction
Cohomology using Sylow subgroups
What _is_ a Schur cover?
Schur covers and lifting a quotient
Results
Morphing Group
Galois Groups
ToDo
12. Representation Theory
Group Rings and Algebras
Ordinary Characters
Modular Characters
13. Lie Groups and Algebras
General Theory
SL(2,5) and sl(2,5)
McKay Correspondence
14. Algebra
Commutative Algebra and Algebraic Geometry
Ideal Theory, CoInvariant Graded Algebras
Gröbner Bases, Buchberger's Algorithm, Syzygies
Homological Algebra
Schemes
Geometric Invariant Theory
Geometric Algebra
Geometric Geometry and Algebraic Algebra
15. Lattices
Division Rings and Finite Fields
Icosians
Designs, Codes
Packings, Coverings and Combinatorics
Quasicrystals
Modular functions
ToDo
16. Geometry
Calculus of Variations
Riemannian Surfaces
Difference Geometry and Metric Spaces
Difference Equations and Cellular Automata
17. Mathematical Physics
Discrete Mechanics
Discrete Chaos Theory
Relativity
Quantum Mechanics
Static and Dynamic Field Theory
Thermodynamics and Statistical Physics
18. Chess Pieces Groups
2-Body Systems
King A versus King A
King C versus King A
King F versus King A
King B versus King A
King C versus King C
King F versus King C
King B versus King C
King F versus King F
King B versus King F
King B versus King B
3-Body 2-Party Systems
King A, Std A versus King A
General Remarks
19. Complexity Theory
NP != P and beyond
Introduction
Games
20. Resume
What's next?