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Wythoffian Skeletal Polyhedra Wythoffian Skeletal Polyhedra by Abigail Williams B.S. in Mathematics, Bates College M.S. in Mathematics, Northeastern University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Dedication I would like to dedicate this dissertation to my Meme. She has always been my loudest cheerleader and has supported me in all that I have done. Thank you, Meme. ii Abstract of Dissertation Wythoff's construction can be used to generate new polyhedra from the symmetry groups of the regular polyhedra. In this dissertation we examine all polyhedra that can be generated through this construction from the 48 regular polyhedra. We also examine when the construction produces uniform polyhedra and then discuss other methods for finding uniform polyhedra. iii Acknowledgements I would like to start by thanking Professor Schulte for all of the guidance he has provided me over the last few years. He has given me interesting articles to read, provided invaluable commentary on this thesis, had many helpful and insightful discussions with me about my work, and invited me to wonderful conferences. I truly cannot thank him enough for all of his help. I am also very thankful to my committee members for their time and attention. Additionally, I want to thank my family and friends who, for years, have supported me and pretended to care everytime I start talking about math. Finally, I want to thank my husband, Keith. Whenever I have doubted myself he has always been there with a hug and an encouraging word. Everyday I am thankful for his love and support. iv Table of Contents Dedication ii Abstract of Dissertation ii Acknowledgements iii Table of Contents iv 1 Background 1 1.1 Introduction . 1 1.2 Abstract polytopes . 2 1.3 Regularity . 5 1.4 Operations on regular polyhedra . 8 1.4.1 Duality - δ ..................................... 8 1.4.2 Petrie duality - π .................................. 9 1.4.3 Facetting - 'k ................................... 10 1.4.4 Halving - η ..................................... 11 1.4.5 Skewing - σ ..................................... 11 1.4.6 Mixing . 11 1.5 Realizations . 12 1.5.1 Blends of realizations . 14 1.6 Uniform polyhedra . 14 1.7 Fundamental regions . 19 1.8 Polygonal complexes . 20 2 Wythoff's construction 21 2.1 Expanding Wythoff's construction . 21 2.2 Properties of Wythoffians . 27 3 Finite regular polyhedra and their Wythoffians 33 3.1 Regular polyhedra with tetrahedral symmetry . 34 3.1.1 f3; 3g ........................................ 34 v 3.1.2 f4; 3g3 ........................................ 36 3.2 Regular polyhedra with octahedral symmetry . 38 3.2.1 f3; 4g ........................................ 38 3.2.2 f4; 3g ........................................ 39 3.2.3 f6; 4g3 ........................................ 41 3.2.4 f6; 3g4 ........................................ 42 3.3 Regular polyhedra with icosahedral symmetry . 44 3.3.1 f3; 5g ........................................ 44 3.3.2 f5; 3g ........................................ 46 3.3.3 f10; 5g3 ....................................... 47 3.3.4 f10; 3g5 ....................................... 49 5 3.3.5 f5; 2 g ........................................ 50 5 3.3.6 f 2 ; 5g ........................................ 52 5 3.3.7 f6; 2 g ........................................ 54 3.3.8 f6; 5g ........................................ 56 5 3.3.9 f3; 2 g ........................................ 57 5 3.3.10 f 2 ; 3g ........................................ 59 10 5 3.3.11 f 3 ; 2 g ........................................ 61 10 3.3.12 f 3 ; 3g ........................................ 63 4 Planar regular apeirohedra and their Wythoffians 65 4.1 Square tiling . 65 4.1.1 f4; 4g ........................................ 65 4.1.2 f1; 4g4 ....................................... 67 4.2 Triangular tiling . 68 4.2.1 f3; 6g ........................................ 68 4.2.2 f1; 6g3 ....................................... 70 4.3 Hexagonal tiling . 71 4.3.1 f6; 3g ........................................ 71 4.3.2 f1; 3g6 ....................................... 72 5 Blended regular apeirohedra and their Wythoffians 75 5.1 Blended apeirohedra related to the square tiling of the plane . 76 5.1.1 f4; 4g#fg ...................................... 76 5.1.2 f1; 4g4#fg ..................................... 78 5.1.3 f4; 4g#f1g ..................................... 80 5.1.4 f1; 4g4#f1g .................................... 83 5.2 Blended apeirohedra related to the hexagonal tiling of the plane . 86 5.2.1 f6; 3g#fg ...................................... 86 5.2.2 f1; 3g6#fg ..................................... 88 vi 5.2.3 f6; 3g#f1g ..................................... 89 5.2.4 f1; 3g6#f1g .................................... 93 5.3 Blended apeirohedra related to the triangular tiling of the plane . 95 5.3.1 f3; 6g#fg ...................................... 95 5.3.2 f1; 6g3#fg ..................................... 98 5.3.3 f3; 6g#f1g ..................................... 99 5.3.4 f1; 6g3#f1g ....................................102 5.4 Remarks on blended apeirohedra . 104 6 Pure regular apeirohedra and their Wythoffians 105 6.1 f4; 6j4g ...........................................106 6.2 f6; 4j4g ...........................................108 6.3 f1; 6g4;4 ..........................................110 6.4 f1; 4g6;4 ..........................................112 6.5 f1; 4g·;∗3 ..........................................114 6.6 f6; 6g4 ............................................116 6.7 f4; 6g6 ............................................118 6.8 f6; 4g6 ............................................120 6.9 f6; 6j3g ...........................................122 6.10 f1; 6g6;3 ..........................................123 6.11 f1; 3g(a) ..........................................125 6.12 f1; 3g(b) ...........................................127 7 Uniform polyhedra 130 7.1 New uniform polyhedra resulting from Wythoff's construction . 131 7.1.1 Uniform polyhedra derived from finite regular polyhedra . 132 7.1.2 Uniform apeirohedra derived from planar regular apeirohedra . 133 7.1.3 Uniform apeirohedra derived from blended regular apeirohedra . 134 7.1.4 Uniform apeirohedra derived from pure regular apeirohedra . 139 7.2 New construction for uniform polyhedra . 146 7.3 Uniform polyhedra generated through alternate construction . 148 7.3.1 Tetrahedral family . 148 7.3.2 Octahedral family . 148 7.3.3 Icosahedral family . 151 7.4 Summary of results . 157 Appendix 159 Bibliography 208 vii Chapter 1 Background 1.1 Introduction Since ancient times, mathematicians have been studying polyhedra. Beginning with the convex regular Platonic solids, the subject then progressed to the non-regular but still convex Archimedean solids, [41]. By the 1800s, the non-convex Kepler-Poinsot star polyhedra had been discovered. In the last century, mathematicians began taking a more combinatorial approach to the study. This opened the door for the Petrie-Coxeter infinite polyhedra which have convex faces and skew vertex figures. Then came Coxeter's Regular Polytopes, which, in addition to examing the geometry of polyhedra, examined the underlying symmetries and combinatorial properties of polyhedra. By the 1970s Gr¨unbaum had extended these ideas about symmetry to allow for new types of polygons. These polygons can be skew or even infinite, [19]. To make polyhedra with these new polygons as faces, Gr¨unbaum introduced the idea of skeletal polyhedra. This allowed for the regular Gr¨unbaum- Dress apeirohedra (see [16], [17]) with skew and infinite faces. At this point, the list of regular polyhedra was complete. Then McMullen and Schulte examined and fully classified the regular polyhedra by their combinatorial properties (see [30], [31]). This study gave life to the field of abstract regular polytopes, [32]. The study of abstract regular polyhedra naturally gave rise to the study of non-regular abstract polyhedra, in particular, the study of chiral polyhedra. The chiral polyhedra have been fully classified by Schulte (see [37], [39]) and continue to be studied (see, for example, [34], [33], [25], [11]). The primary aim of this thesis is to take one of the techniques used to examine chiral polyhedra and apply it to skeletal polyhedra to find analogues of the Archimedean solids. More specifically, we take abstract regular polyhedra and realize them in Euclidean space, then based on those real- izations and their symmetry groups we generate more polyhedra. To do this, we find a symmetry group in ordinary Euclidean space corresponding to the automorphism group of an abstract poly- hedron. We then use Wythoff's construction on the symmetry group to generate a realization of the abstract regular polyhedron and to generate other polyhedra with the same symmetries. These realizations are based on abstract objects, so their geometric properties only need fulfill the most basic combinatorial properties of an abstract polyhedron. As such, we need to refine our 1 definitions of polygon and polyhedron to account for the realizations of the abstract polyhedra. Based on Gr¨unbaum's work, polygons are viewed as sequences of connected edges, and polyhedra are viewed as skeletons comprised of vertices and edges, [19]. This allows us to look at polyhedra with skew faces and infinite faces. While the regular polyhedra with finite, planar faces have been studied extensively, there
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