Convex Polytopes and Tilings with Few Flag Orbits by Nicholas Matteo B.A. in Mathematics, Miami University M.A. in Mathematics, Miami 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 Abstract of Dissertation The amount of symmetry possessed by a convex polytope, or a tiling by convex polytopes, is reflected by the number of orbits of its flags under the action of the Euclidean isometries preserving the polytope. The convex polytopes with only one flag orbit have been classified since the work of Schläfli in the 19th century. In this dissertation, convex polytopes with up to three flag orbits are classified. Two-orbit convex polytopes exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron, and the rhombic triacontahedron. Two-orbit face-to-face tilings by convex polytopes exist on E1, E2, and E3; the only ones which are also combinatorially two-orbit are the trihexagonal plane tiling, the rhombille plane tiling, the tetrahedral-octahedral honeycomb, and the rhombic dodecahedral honeycomb. Moreover, any combinatorially two-orbit convex polytope or tiling is isomorphic to one on the above list. Three-orbit convex polytopes exist in two through eight dimensions. There are infinitely many in three dimensions, including prisms over regular polygons, truncated Platonic solids, and their dual bipyramids and Kleetopes. There are infinitely many in four dimensions, comprising the rectified regular 4-polytopes, the p; p-duoprisms, the bitruncated 4-simplex, the bitruncated 24-cell, and their duals. There are only finitely many other examples, comprising the Gosset polytopes k21 and their duals in five through eight dimensions, and the polytopes k22 and their duals in five or six dimensions. The polygons in the plane with any number k of flag orbits are described. These are pk certain pk-gons, and also certain 2 -gons if k ≥ 4 is even, for any p so that the number of sides is at least three. The resulting flag orbits from various standard operations used to construct new polytopes from others are also discussed. Finally, it is proven that for any k, the polytopes with k flag orbits occur in only finitely many dimensions, and that a certain class of d-dimensional polytopes (including most uniform polytopes) is either regular or has at least d − 1 flag orbits. ii Table of Contents Abstract of Dissertation ii Table of Contents iii List of Figures v List of Tables vii Chapter I. Introduction 1 Chapter II. Orbit Graphs 21 Chapter III. Standard Operations 31 Chapter IV. k-orbit Polygons 59 Chapter V. Two-orbit Polytopes and Tilings 67 Chapter VI. Combinatorially Two-orbit Polytopes and Tilings 87 Chapter VII. Three-orbit Polytopes and Tilings 111 Chapter VIII. Restrictions on k-orbit Polytopes 155 Bibliography 171 iii List of Figures IIa Possible walks on i-edges and j-edges with ji − jj ≥ 2 25 IIb Possible quotients of an (i; i + 1)-walk of 3 steps 27 IIc The lattice of orbit graphs for a quadrilateral 29 IId Example quadrilaterals corresponding to the symmetry types in Figure IIc 30 IIIa The orbit graph of a pyramid over a regular polygon fpg 38 3 IIIb The orbit graph of a pyramid over a cuboctahedron 4 38 IIIc The orbit graph of the product of a regular p-gon P with an isogonal two-orbit q-gon Q, under G(P ) × G(Q) 44 IIId The orbit graph of a prism over a three-orbit polygon 44 IIIe A polyhedron not admitting an extended Kleetope 48 IVa Constructing new line segments above an edge of the regular p-gon Q 62 IVb Constructing new line segments above a point, producing k-gons with k flag orbits 64 IVc Example polygons of the cyclic type 65 Va The first few two-orbit convex polygons, in pairs of duals 68 Vb The two-orbit convex polyhedra 69 Vc A rectangle, overlaid with its midpoint dual and polar dual 73 Vd The fully-transitive rhombus tiling 78 Ve The trihexagonal tiling 79 Vf The 1-intransitive rectangle tiling 81 v Vg The rhombille tiling 81 VIa The Coxeter diagram of a facet-intransitive two-orbit polytope 93 VIb The region R composed of (d − 1)-adjacent chambers, in the case d = 3 (left) or d = 4 (right) 95 VIc Potential Coxeter diagrams for the automorphism group of a two-orbit polytope 96 VId Tiles of An+1 n An. Vertices and edges on the boundary of An+1 are labeled by the number of tiles of An+1 containing them. Elements belonging to An are darkened.106 VIIa Three-orbit polygons with flags colored by orbit 112 VIIb Some three-orbit hexagons 113 VIIc Some polyhedra of type 31;2, with flags colored to match the colored orbit graph 115 VIId The tilings of type 31;2 116 VIIe Some polyhedra of type 30;1, with flags colored to match the colored orbit graph 117 VIIf Three-orbit tilings of type 31 119 VIIIa A partial Kleetope of the cube with an orthodivision but no flag triangulation by orthoschemes. 162 VIIIb The situation of Lemma VIII.7 165 VIIIc The situation of Lemma VIII.9. The shaded triangles remain intact. 166 vi List of Tables Va Two-orbit convex d-polytopes 85 Vb Two-orbit tilings of Ed 85 VIIa Three-orbit convex d-polytopes 152 VIIb Three-orbit tilings of Ed 154 vii CHAPTER I Introduction The amount of symmetry possessed by a convex polytope, or a tiling by convex polytopes, is reflected by the number of orbits of its flags under the action of the Euclidean isometries preserving the polytope. The polytopes with the fewest such orbits are the “most symmetric”. The convex polytopes with only one flag orbit have been classified since the work of Schläfli in the 19th century. In this dissertation, a classification is given for convex polytopes with up to three flag orbits. A description of the polygons in the plane with any number k of flag orbits is also given. The resulting flag orbits from various standard operations used to construct new polytopes from others are also discussed. Finally, it is proven that for any k, the polytopes with k flag orbits occur in only finitely many dimensions, and that a certain class of d-dimensional polytopes (including most uniform polytopes) is either regular or has at least d − 1 flag orbits. A reference to a section in the current chapter appears as §1; a reference to a section in another chapter always includes the chapter number in Roman numerals, such as §II.1. A reference to an item in the current chapter, such as the first lemma in §4, has the form “Lemma 1”. A reference to another chapter, for instance the first proposition in §II.4, has the form “Proposition II.1”. 1. Convex polytopes Convex polytopes and their symmetries are some of the oldest objects of mathematical study. The oldest extant work of axiomatic mathematics, Euclid’s Elements, is concerned with the construction of the Platonic solids (or as we will call them, the one-orbit 3-polytopes). Through the ensuing millennia, highly symmetric figures in the plane or 3-space, and highly symmetric tilings using these as building blocks, have been of enduring interest to artists 1 and mathematicians alike. In the past two centuries, mathematicians have extended the investigation into higher dimensions. Over this time, polytopes have been given many definitions. Here, a convex polytope is the convex hull of finitely many points in d-dimensional Euclidean space Ed (Grünbaum 2003). These are the objects which (in 2 or 3 dimensions) would have been familiar to the ancient Greeks. Throughout this dissertation, the word “polytope” without qualification means “convex polytope.” A d-polytope is one whose affine hull is d-dimensional. Some authors define a (convex) polyhedron as the intersection of finitely many closed half-spaces in Ed, or equivalently, as the solution set to a finite system of linear inequalities: d f x 2 E j Ax ≤ b g; m T where A is an m × d matrix, b 2 E , and for two vectors a = (a1; : : : ; am) and b = T (b1; : : : ; bm) , a ≤ b if and only if ai ≤ bi for each i. This is a convex object, but is not necessarily bounded. These authors define “(con- vex) polytope” to mean a bounded convex polyhedron, or equivalently, a compact convex polyhedron. This is equivalent to our earlier definition. However, we will not use this definition of polyhedron: throughout the sequel, polyhedron means 3-polytope, and polygon means 2-polytope. A (proper) face of a polytope P is the intersection of P with a supporting hyperplane H, i.e. a hyperplane such that H \ P is not empty, and P is contained in one of the closed half-spaces determined by H. In addition to these proper faces, we admit two improper faces: the empty face, ;, and P itself. With the inclusion of these improper faces, the faces of P , ordered by containment, form a lattice: the face lattice of P , denoted L(P ).A j-face is a face whose affine hull is j-dimensional; we call 0-faces vertices, 1-faces edges, (d − 3)-faces subridges, (d − 2)-faces ridges, and (d − 1)-faces facets. The empty face is considered a (−1)-face. Two faces are said to be incident if one contains the other.