167 research outputs found

    Block Systems of Ranks 3 and 4 Toroidal Hypertopes

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    This dissertation deals with abstract combinatorial structure of toroidal polytopes and toroidal hypertopes. Abstract polytopes are objects satisfying the main combinatorial properties of a classical (geometric) polytope. A regular toroidal polytope is an abstract polytope which can be constructed from the string affine Coxeter groups. A hypertope is a generalization of an abstract polytope, and a regular toroidal hypertope is a hypertope which can be constructed from any affine Coxeter group. In this thesis we classify the rank 4 regular toroidal hypertopes. We also seek to find all block systems on a set of (hyper)faces of toroidal polytopes and hypertopes of ranks 3 and 4 as well as the regular and chiral toroidal polytopes of ranks 3. A block system of a set X under the action of a group G is a partition of X which is invariant under the action of G

    Regular incidence-polytopes with Euclidean or toroidal faces and vertex-figures

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    AbstractRegular incidence-polytopes are combinatorial generalizations of regular polyhedra. Certain group-theoretical constructions lead to many new regular incidence-polytopes whose faces and vertex-figures are combinatorially isomorphic to classical Euclideanly regular polytopes or regular maps on the torus

    On 4-Dimensional Point Groups and on Realization Spaces of Polytopes

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    This dissertation consists of two parts. We highlight the main results from each part. Part I. 4-Dimensional Point Groups. (based on a joint work with GĂŒnter Rote.) We propose the following classification for the finite groups of orthogonal transformations in 4-space, the so-called 4-dimensional point groups. Theorem A. The 4-dimensional point groups can be classified into * 25 polyhedral groups (Table 5.1), * 21 axial groups (7 pyramidal groups, 7 prismatic groups, and 7 hybrid groups, Table 6.3), * 22 one-parameter families of tubical groups (11 left tubical groups and 11 right tubical groups, Table 3.1), and * 25 infinite families of toroidal groups (2 three-parameter families, 19 two-parameter families, and 4 one-parameter families, Table 4.3.) In contrast to earlier classifications of these groups (notably by Du Val in 1962 and by Conway and Smith in 2003), see Section 1.7), we emphasize a geometric viewpoint, trying to visualize and understand actions of these groups. Besides, we correct some omissions, duplications, and mistakes in these classifications. The 25 polyhedral groups (Chapter 5) are related to the regular polytopes. The symmetries of the regular polytopes are well understood, because they are generated by reflections, and the classification of such groups as Coxeter groups is classic. We will deal with these groups only briefly, dwelling a little on just a few groups that come in enantiomorphic pairs (i.e., groups that are not equal to their own mirror.) The 21 axial groups (Chapter 6) are those that keep one axis fixed. Thus, they essentially operate in the three dimensions perpendicular to this axis (possibly combined with a flip of the axis), and they are easy to handle, based on the well-known classification of the three-dimensional point groups. The tubical groups (Chapter 3) are characterized as those that have (exactly) one Hopf bundle invariant. They come in left and right versions (which are mirrors of each other) depending on the Hopf bundle they keep invariant. They are so named because they arise with a decomposition of the 3-sphere into tube-like structures (discrete Hopf fibrations). The toroidal groups (Chapter 4) are characterized as having an invariant torus. This class of groups is where our main contribution in terms of the completeness of the classification lies. We propose a new, geometric, classification of these groups. Essentially, it boils down to classifying the isometry groups of the two-dimensional square flat torus. We emphasize that, regarding the completeness of the classification, in particular concerning the polyhedral and tubical groups, we rely on the classic approach (see Section 1.6). Only for the toroidal and axial groups, we supplant the classic approach by our geometric approach. We give a self-contained presentation of Hopf fibrations (Chapter 2). In many places in the literature, one particular Hopf map is introduced as “the Hopf map”, either in terms of four real coordinates or two complex coordinates, leading to “the Hopf fibration”. In some sense, this is justified, as all Hopf bundles are (mirror-)congruent. However, for our characterization, we require the full generality of Hopf bundles. As a tool for working with Hopf fibrations, we introduce a parameterization for great circles in S^3 , which might be useful elsewhere. Our main tool to understand tubical groups are polar orbit polytopes. (Chapter 1). In particular, we study the symmetries of a cell of the polar orbit polytope for different starting points. Part II. Realization Spaces of Polytopes (based on a joint work with Rainer Sinn and GĂŒnter M. Ziegler.) Robertson in 1988 suggested a model for the realization space of a d-dimensional polytope P, and an approach via the implicit function theorem to prove that the realization space is a smooth manifold of dimension NG(P) := d(f_0 + f_{d−1} ) - f{0,d-1} . We call NG(P) the natural guess for (the dimension of the realization space of) P. We build on Robertson's model and approach to study the realization spaces of higher-dimensional polytopes. We conclude combinatorial criteria (Sections 9.3.3 and 9.4.1) to decide if the realization space of the polytope in consideration is a smooth manifold of dimension given by the natural guess. As another application, we study the realization spaces of the second hypersimplices (Section 9.4.2). We apply these criteria on 4-polytopes with small number of vertices, and along the way, we analyze examples where Robertson’s approach fails, identifying the three smallest examples of 4-polytopes, for which the realization space is still a smooth manifold, but its dimension is not given by the natural guess (Section 9.5). Finally, we investigate the realization space of the 24-cell (Section 9.5.2). We construct families of realizations of the 24-cell, and using them we show that the realization space of the 24-cell has points where it is not a smooth manifold. This provides the first known example of a polytope whose realization space is not a smooth manifold. We conclude by showing that the dimension of the realization space of the 24-cell is at least 48 and at most 52.Diese Dissertation befasst sich mit zwei verschiedenen Themen, von denen jedes seinen eigenen Teil hat. Der erste Teil befasst sich mit 4-dimensionalen Punktgruppen. Wir schlagen eine neue Klassifizierung fĂŒr diese Gruppen vor (siehe Theorem A), die im Gegensatz zu frĂŒheren Klassifizierungen eine geometrische Sichtweise betont und versucht, die Aktionen dieser Gruppen zu visualisieren und zu verstehen. Im Folgenden werden diese Gruppen kurz beschrieben. Die polyedrischen Gruppen (Kapitel 5) sind mit den regelmĂ€ĂŸigen Polytopen verwandt. Die axialen Gruppen (Kapitel 6) sind diejenigen, die eine Achse festhalten. Die schlauchartigen Gruppen (Kapitel 3) werden als solche charakterisiert, die genau eine invariantes Hopf-BĂŒndel haben. Sie entstehen bei einer Zerlegung der 3-SphĂ€re in schlauchartige Strukturen (diskrete Hopf-Faserungen). Die toroidalen Gruppen (Kapitel 4) sind dadurch gekennzeichnet, dass sie einen invarianten Torus haben. Wir schlagen eine neue, geometrische Klassifizierung dieser Gruppen vor. Im Wesentlichen lĂ€uft sie darauf hinaus, die Isometriegruppen des zweidimensionalen quadratischen flachen Torus zu klassifizieren. Nebenbei geben wir eine in sich geschlossene Darstellung der Hopf-Faserungen (Kapitel 2). Als Hilfsmittel fĂŒr die Arbeit mit ihnen fĂŒhren wir eine Parametrisierung fĂŒr Großkreise in S 3 ein, die an anderer Stelle nĂŒtzlich sein könnte. Der zweite Teil befasst sich mit RealisierungsrĂ€umen von Polytopen. Wir bauen auf Robertsons Modell und Ansatz auf, um die RealisierungsrĂ€ume von Polytopen zu untersuchen. Wir stellen kombinatorische Kriterien auf (Abschnitte 9.3.3 und 9.4.1), um zu entscheiden, ob der Realisierungsraum des betrachteten Polytops eine glatte Mannigfaltigkeit der durch die “natĂŒrliche Vermutung” gegebenen Dimension ist. Als weitere Anwendung, untersuchen wir die RealisierungsrĂ€ume der zweiten Hypersimplices (Abschnitt 9.4.2). Nebenbei identifizieren wir die kleinsten Beispiele von 4-Polytopen, fĂŒr die dieser Ansatz versagt (Abschnitt 9.5). Schließlich untersuchen wir den Realisierungsraum der 24-Zelle (Abschnitt 9.5.2). Wir zeigen, dass es Punkte gibt, an denen sie keine glatte Mannigfaltigkeit ist. Zuletzt zeigen wir, dass seine Dimension mindestens 48 und höchstens 52 betrĂ€gt

    Chirality and projective linear groups

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    AbstractIn recent years the term ‘chiral’ has been used for geometric and combinatorial figures which are symmetrical by rotation but not by reflection. The correspondence of groups and polytopes is used to construct infinite series of chiral and regular polytopes whose facets or vertex-figures are chiral or regular toroidal maps. In particular, the groups PSL2(Zm) are used to construct chiral polytopes, while PSL2(Zm[i]) and PSL2(Zm[ω]) are used to construct regular polytopes

    Hexagonal extensions of toroidal maps and hypermaps

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    The rank 3 concept of a hypermap has recently been generalized to a higher rank structure in which hypermaps can be seen as “hyperfaces” but very few examples can be found in literature. We study finite rank 4 structures obtained by hexagonal extensions of toroidal hypermaps. Many new examples are produced that are regular or chiral, even when the extensions are polytopal. We also construct a new infinite family of finite nonlinear hexagonal extensions of the tetrahedron.The authors would like to thank two anonymous referees for their numerous helpful and insightful comments. This research was supported by a Marsden grant (UOA1218) of the Royal Society of New Zealand, by NSERC and by the Portuguese Foundation for Science and Technology (FCT-Fundação para a CiĂȘncia e a Tecnologia), through CIDMA - Center for Research and Development in Mathematics and Applications, within project UID/MAT/04106/2013.publishe
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