XiStrat Manual

December 2006


To our customers. They encourage us to go on.

Table of Contents

Preface
I. User Guide
1. Introduction
Architecture and Idea behind it
Focus
Feedback
Thanx
Epilog
Formal Stuff
ToDo
2. Graph Construction
Introduction
Creating 3D data
Infiles
Property Files
Faces and Orientations
Colorings
Procedural 3D Textures
Autogenerated Files
ToDo
3. Playing Games
Introduction to our Internet Games
General Procedure
Chess Variant
Go Variant
Still more features
Single Agent Games
ToDo
4. Problems, Compositions and Studies
Chess problems
Go
ToDo
5. Retrograde Analysis
Chess
Go
6. Engines
Non-cooperative Games
General Programming, Brute Force and others
GCJ and JNI
Chess
Go
Reversi, Othello
Puzzles Autosolving
ToDo
II. Mathematics Background
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?
III. Appendix
A. Installation and Usage
Compiling
Gaming
Server
Client
Creating 3D data
Knot Viewer
Export ASCII data
XML persistence
target GAP™ export and autogenerating infiles
B. Hardware Platforms, Third Party Software
C. Bibliography
Books, Articles, URLs
Usenet
D. GNU Free Documentation License
PREAMBLE
APPLICABILITY AND DEFINITIONS
VERBATIM COPYING
COPYING IN QUANTITY
MODIFICATIONS
COMBINING DOCUMENTS
COLLECTIONS OF DOCUMENTS
AGGREGATION WITH INDEPENDENT WORKS
TRANSLATION
TERMINATION
FUTURE REVISIONS OF THIS LICENSE
ADDENDUM: How to use this License for your documents
E. Glossary
What's up?

List of Figures

2.1. 3D Texture example
3.1. The Overall Architecture
3.2. New Game
3.3. Chess Piece Type C
3.4. Chess Piece Type F
3.5. After a Checkmate
3.6. Go on Hexa Tor
3.7. Ko on a boundary
3.8. 2-stage Ko 0
3.9. 2-stage Ko I
3.10. 2-stage Ko II
3.11. 2-stage Ko III
3.12. 2-stage Ko IV
3.13. 2-stage Ko V
3.14. Go 3 players
3.15. Random Go
3.16. Load Game
3.17. Client Mirror View
3.18. XiStrat Client in Applet Mode
7.1. Stellated Tetrahedron
7.2. Truncated Deltoid
7.3. Tetrahedron Stellation I(iterated)
7.4. Tetrahedron Stellation II
7.5. Tetrahedron Reticulated 0
7.6. Tetrahedron Reticulated I
7.7. Tetrahedron Reticulated II
7.8. Tetrahedron Reticulated III
7.9. Octahedron stellation II
7.10. Icosahedron
7.11. Icosahedron Variant 1
7.12. Icosahedron Stellation I
7.13. Icosahedron Stellation II
7.14. Cube Triang II
7.15. Yabi
7.16. Bipyramid Octa
7.17. L2(7) Regular
7.18. M(24)
7.19. Torus 3x4
7.20. Trapezohedron
7.21. S(5)
7.22. A(6)
7.23. Quad Star
7.24. Cube 2x2
7.25. Cube 3x3
7.26. 3x3 rotated corner
7.27. 3x3 rotated 2x2
7.28. 3x3 rotated 3x3 area
7.29. 3x3 follow up
7.30. Rhombic Dodecahedron
7.31. Zonotope 5
7.32. Monkgau
7.33. Costa-Hoffman
7.34. Quad Hex
7.35. Pentagonal Quasicrystal 0
7.36. Pentagonal Quasicrystal I
7.37. Pentagonal Quasicrystal II
7.38. Pentagonal Quasicrystal III
7.39. Pentagonal Quasicrystal IV
7.40. Heptagonal Quasicrystal 0
7.41. DoubleDodecahedron
7.42. Icositetra
7.43. Dode Tor 6
7.44. Torus Hexa 3x5
7.45. Hexa Tor
7.46. Hexa UnOrient
7.47. Fancy Heptas
10.1. KnotViewer Trefoil
10.2. Dode Tor 6 knot
10.3. Borromean rings
10.4. Dode Tor 6 Link
10.5. Hexa Tor Knot
11.1. Holonomy
15.1. Penrose Quasicrystal in Progress

List of Tables

3.1. Piece Moving Rules
8.1. Polyhedra Cycle Index
11.1. Holonomy Groups
11.2. c and rot as words
11.3. Relations of (Binary) Polyhedral Groups
11.4. Schur multiplier
11.5. Holonomy Group Schur Covers
11.6. Cross Groups on Polyhedra
11.7. Galois Groups
18.1. 2-Body Groups
B.1. Hardware
B.2. Software

List of Examples

11.1. Dodecahedron Holonomy
11.2. Cube and Schur Cover
11.3. DoubleTetrahedron and Schur Cover
11.4. Tetrahedron Reticulated and Schur Cover
11.5. DoubleCube with two holes and Schur Cover
11.6. Tetrahedron Stellated and Schur Cover

List of Equations

8.1. Polyhedra Cycle Index
9.1. Euler formula
9.2. Euler Characteristic and Betti Numbers
11.1. Riemann-Hurwitz identity
11.2. Frobenius formula
11.3. Short Sequence
11.4. Commutative Diagram
11.5. A(4) Short Exact Sequence
13.1. Baker-Campbell-Hausdorff formula
14.1. Hilbert's Nullstellensatz
15.1. Dirichlet Series Generating Function F2
17.1. Simple Dynamical 2-Body System