# Contents

• Title Page......i
• Abstract......iii
• Contents......iv
• Acknowledgements......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
• 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
• 4.6.1: 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: Analysing 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