These pages have moved to http://www.dr-mikes-maths.com/
See:
What I
live for
- What I
laugh at
- What I
like
- What I
look like
- Who I
love
-
Contact me
Contents
- Title Page......i
- Abstract......iii
- Contents......iv
- Acknowledgments......vii
- CHAPTER 1: How To Read This Thesis......1
- 1.1: Aim of the Work......1
- 1.2: Thesis Outline......3
- 1.3: Knowledge Required and Other Warnings......5
- CHAPTER 2: Geometric Polytopes......6
- 2.1: Introductory Concepts......6
- 2.2: Definitions of a Polytope......8
- 2.3: Properties of Polytopes......9
- 2.3.1: Faces......9
- 2.3.2: Combinatorial Equivalence......11
- 2.3.3: Duals......13
- 2.3.4: Vertex Figures......14
- 2.4: Regular Polytopes......15
- 2.4.1: The Definition......15
- 2.4.2: Schlafli Symbols......17
- 2.5: The Classification of Regular Polytopes......18
- 2.5.1: The 1- and 2-polytopes......18
- 2.5.2: The 3-polytopes......18
- 2.5.3: The 4-polytopes......20
- 2.5.4: From Then On......21
- CHAPTER 3: Combinatorial Polytopes......23
- 3.1: Incidence Complexes......23
- 3.2: Euler Polytopes......30
- 3.3: Properties of Euler Polytopes......32
- 3.3.1: Isomorphism......32
- 3.3.2: Duals......37
- 3.3.3: (i,j)-Connectivity......39
- 3.3.4: Subpolytopes......42
- 3.3.5: Decomposable Polytopes......44
- 3.4: Regular Incidence Polytopes......50
- 3.4.1: Automorphism Groups......50
- 3.4.2: Regular Incidence Polytopes......52
- 3.4.3: Schlafli Symbols......55
- 3.4.4: Number Crunching......59
- 3.5: The Geometric Link......60
- CHAPTER 4: Some Examples......64
- 4.1: Combinatorial Simplices......65
- 4.2: Combinatorial Cubes......69
- 4.3: Halfcubes......76
- 4.4: The Crosses and Halfcrosses......85
- 4.5: The Lattices......86
- 4.6: Sporadic Examples......97
- 4.6.1: The 1-polytopes......97
- 4.6.2: The 2-polytopes......98
- 4.6.3: 3- and 4-polytopes......98
- CHAPTER 5: Groups And Polytopes......100
- 5.1: Coxeter Groups......100
- 5.1.1: Introduction......100
- 5.1.2: Other Notes......105
- 5.1.3: Coxeter Groups and the Flag Action......108
- 5.2: Complexes Constructed from Groups......115
- 5.2.1: Universal Complexes......116
- 5.2.2: Quotient Polytopes......124
- 5.2.3: Groups Acting on Quotients......134
- 5.3: Applications of the Constructions......138
- CHAPTER 6: Some Classifications......143
- 6.1: General Classes......143
- 6.1.1: The Simplices......143
- 6.1.2: Cubes and Halfcubes......149
- 6.1.3: Crosses and Halfcrosses......154
- 6.2: The Lattices and Related Polytopes......154
- 6.2.1: Preliminaries......154
- 6.2.2: The Coxeter Group......157
- 6.2.3: Sparse Subgroups......163
- 6.2.4: The Link With the Lattices......172
- 6.3: Particular Dimensions......178
- 6.3.1: The 1- and 2-polytopes......178
- 6.3.2: The 3-polytopes......178
- 6.3.3: The 4-polytopes......181
- 6.3.4: The 5-polytopes......189
- 6.3.5: From Then On......193
- CHAPTER 7: Conclusion......195
- 7.1: Summary......195
- 7.2: What Follows......198
- APPENDIX A: Information On Selected Coxeter Groups......200
- APPENDIX B: Non-Lattices......205
- B.1: The Example......206
- B.2: The Non-Existence......207
- APPENDIX C: CAYLEY Code......211
- C.1: The Naive Algorithm......211
- C.2: Refining the Search......213
- C.3: An Infinite Search......215
- Bibliography......217
- Index......222
List of Tables
- Table 2.5.1: The Regular Polyhedra......18
- Table 2.5.2: Vertex Sets for the Regular Polyhedra......19
- Table 2.5.3: The Regular Geometric 4-Polytopes......21
- Table 2.5.4: The Regular Geometric d-Polytopes, d>=5 (i)......22
- Table 2.5.5: The Regular Geometric d-Polytopes, d>=5 (ii)......22
- Table 6.3.1: Overview of Combinatorially Regular 4-Polytopes......183
- Table 6.3.2: Some 4-Polytopes from Section 6.1......183
- Table 6.3.3: Combinatorially Regular 4-Polytopes with... ......184
- Table 6.3.4: Combinatorially Regular 4-Polytopes with... ......185
- Table 6.3.5: Some Combinatorially Regular Polytopes Corresp... ......187
- Table 6.3.6: Some Restrictions on Sparse Subgroups A......189
- Table 6.3.7: Analyzing Euler's Condition for 5-Polytopes......192
- Table 6.3.8: Further Analysis of Euler's Condition......192
- Table 6.3.9: Combinatorially Regular Euler d-Polytopes for d>=5......194
- Table 7.1.1: Overview of Combinatorially Regular Euler Polytopes......199
- Table A.1: Finite Coxeter Groups Satisfying C1......200
- Table A.2: Infinite Coxeter Groups Satisfying C1 and C2......201
- Table A.3: Quotient Polytopes for d>=3......202
- Table A.4: Quotient Polytopes for d=4......203
- Table A.5: Quotient Polytopes for d=5......203
- Table A.6: Quotient Polytopes for d>=6......204
- Table A.7: Finite Universal Polytopes......204