On regular polyhedra with hidden symmetries

We consider polyhedral realizations of oriented regular maps with or without self-intersections in E³ whose symmetry group is a subsgroup of small index in their. automorphism group. The four classical kepler-poinsot polyhedra are the only ones of index 1. There are exactly five of Index 2, all with icosahedral symmetry group [W2] as the Kepler-poinsot polyhedra. In this paper we show that there are no such polyhedral realizations with octahedral (tetrahedral) symmetry group and index 2 or 3 (2,3,4,5), which is best possible in the octahedral case.     

A combinatorial theory of Grünbaum's new regular polyhedra,Part II: Complete enumeration

The new regular polyhedra as defined by Branko Grünbaum in 1977 (cf. [5]) are completely enumerated. By means of a theorem of Bieberbach, concerning the existence of invariant affine subspaces for discrete affine isometry groups (cf. [3], [2] or [1]) the standard crystallographic restrictions are established for the isometry groups of the non finite (Grünbaum-)polyhedra. Then, using an appropriate classification scheme which—compared with the similar, geometrically motivated scheme, used originally by Grünbaum—is suggested rather by the group theoretical investigations in [4], it turns out that the list of examples given in [5] is essentially complete except for one additional polyhedron.So altogether—up to similarity—there are two classes of planar polyhedra, each consisting of 3 individuals and each class consisting of the Petrie duals of the other class, and there are ten classes of non planar polyhedra: two mutually Petrie dual classes of finite polyhedra, each consisting of 9 individuals, two mutually Petrie dual classes of infinite polyhedra which are contained between two parallel planes with each of those two classes consisting of three one-parameter families of polyhedra, two further mutually Petrie dual classes each of which consists of three one parameter families of polyhedra whose convex span is the whole 3-space, two further mutually Petrie dual classes consisting of three individuals each of which spanE ³ and two further classes which are closed with respect to Petrie duality, each containing 3 individuals, all spanningE ³, two of which are Petrie dual to each other, the remaining one being Petrie dual to itself.In addition, a new classification scheme for regular polygons inE ⁿ is worked out in §9.     

Translative integral formulae for convex bodies

Translative versions of the principal kinematic formula for quermassintegrals of convex bodies are studied. The translation integral is shown to be a sum of Crofton type integrals of mixed volumes. As corollaries new integral formulas for mixed volumes are obtained. For smooth centrally symmetric bodies the functionals occurring in the principal translative formula are expressed by measures on Grassmannians which are related to the generating measures of the bodies.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

Sur le tétraèdre régulier dans un espace projectif complexe

Résumé On montre l'existence, dans un espace projectif complexe, d'un tétraèdre régulier ayant un groupe de symétrie isomorphe à celui du tétraèdre régulier euclidien. On précise la classe d'homologie de dimension 2, qui contient le complexe simplicial qui lui est associé.

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We show the existence, in a complex projective space, of a regular tetrahedron which has its symmetry group isomorphic to that of the euclidean regular tetrahedron. We give precisely the two-dimensional homology class, containing the simplicial complexe associated to it.

     

Realizations of regular apeirotopes

Summary In an earlier paper, a theory of realizations of (finite) regular polytopes in euclidean spaces was developed. Here, the analogous problem of realizing regular apeirotopes (infinite polytopes) is investigated. While no complete theory is expounded, several basic results are established. Among these is the curious fact that, if a regular apeirotope has a discrete realization, then it has one with no translations in its symmetry group.     

On the construction of highly symmetric tight frames and complex polytopes

Many “highly symmetric” configurations of vectors in C^d${\mathbb{C}}^d$, such as the vertices of the platonic solids and the regular complex polytopes, are equal-norm tight frames by virtue of being the orbit of the irreducible unitary action of their symmetry group. For nonabelian groups there are uncountably many such tight frames up to unitary equivalence. The aim of this paper is to single out those orbits which are particularly nice, such as those which are the vertices of a complex polytope. This is done by defining a finite class of tight frames of n   vectors for C^d${\mathbb{C}}^d$ (n and d fixed) which we call the highly symmetric tight frames. We outline how these frames can be calculated from the representations of abstract groups using a computer algebra package. We give numerous examples, with a special emphasis on those obtained from the (Shephard–Todd) finite reflection groups. The interrelationships between these frames with complex polytopes, harmonic frames, equiangular tight frames, and Heisenberg frames (maximal sets of equiangular lines) are explored in detail.     

Gated sets in metric spaces

The concept of a gated subset in a metric space is studied, and it is shown that properties of disjoint pairs of gated subsets can be used to investigate projections in Tits buildings.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

Infinite series of combinatorially regular polyhedra in three-space

The paper discusses polyhedral realizations in ordinary Euclidean 3-space of Coxeter's regular skew polyhedra {4, p|4 ^p/2]–1} and their duals on an orientable surface of genus 2 ^p–3(p–4)+1. Our considerations are based on work of Coxeter, Ringel and McMullen et al., revealing that certain polyhedral manifolds discovered by the last three authors are in fact the polyhedra in question. We also describe Kepler-Poinsot-type polyhedra in 3-space obtained by projections from Coxeter's regular skew star-polyhedra in 4 dimensions.     

Puzzles and polytope isomorphisms

We show that an isomorphism between the graphs of two simple polytopes of arbitrary dimension can always be extended to an isomorphism between the polytopes themselves. It has been convenient to study the dual situation, involving what we like to call the puzzle of a simplicial polytope.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.     

A combinatorial theory of Grünbaum's new regular polyhedra,part I: Grünbaum's new regular polyhedra and their automorphism group

The new regular polyhedra, defined and investigated by Branko Grünbaum in [4], and theirn-dimensional generalizations are classified in terms of their symmetry group.     

Constructing infinite families of regular incidence (quasi-)polytopes

Given a regular incidence (quasi-)polytopeP of type {a ₁,a ₂, ...,a _n–1} and a function on its directed edges satisfying certain conditions, we construct for everym 2 a regular incidence (quasi-)polytope of type {ma ₁,a ₂, ...,a _n–1} with the same vertex figure asP.     

Geometric realizations for Dyck's regular map on a surface of genus 3

Klein's and Dyck's regular maps on Riemann surfaces of genus 3 were one impetus for the theory of regular maps, automorphic functions, and algebraic curves. Recently a polyhedral realization inE ³ of Klein's map was discovered [18], thereby underlining the strong analogy to the icosahedron. In this paper we show that Dyck's map can be realized inE ³ as a polyhedron of Kepler-Poinsot-type, i.e., with maximal symmetry and minimal self-intersections. Furthermore some closely related polyhedra and a Kepler-Poinsot-type realization of Sherk's regular map of genus 5 are discussed.     

How many atoms can be defined by boxes?

We study the functionb(n, d), the maximal number of atoms defined byn d-dimensional boxes, i.e. parallelopipeds in thed-dimensional Euclidean space with sides parallel to the coordinate axes. We characterize extremal interval families definingb(n, 1)=2n-1 atoms and we show thatb(n, 2)=2n ²-6n+7. We prove that for everyd, exists and . Moreover, we obtainb*(3)=8/9.     

Sur les m-uples F-réguliers dans les espaces projectifs complexes

Résumé On étudie dans P ⁿ les m-uples de points, appelés F-réguliers, dont les sous-triplets ordonnés sont deux à deux isométriques. On montre qu'il existe au plus deux classes d'isométrie de quintuplets F-réguliers contenant un triangle équilatère T donné. On étudie aussi les m-uples F-réguliers, dont les sous k-uples (k<m) non ordonnés sont deux à deux isométriques. Ces m-uples sont appelés k-réguliers. On montre que la 4-régularité implique la k-régularité pour tous les k5.

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We investigate in P ⁿ m-tuples of points in which all ordered triples are pairwise isometric. Such m-tuples are called F-regular. We show that for a given triangle T there exist at most two isometry classes of F-regular quintuples containing T. We also investigate F-regular m-tuples in which all (unordered) k-tuples (k<m) are pairwise isometric. Such m-tuples are called k-regular. We show that 4-regularity implies k-regularity for all k5.

     

Eine differentialgeometrische Kennzeichnung der allgemeinen Dupinschen Zykliden

Zusammenfassung Es werden die Flächen des projektiven dreidimensionalen Raumes bestimmt, die ein konjugiertes Netz aus Kegelschnitten, die zugleich Kegelschattengrenzen sind, tragen. Es kommen nur algebraische Flächen in Frage. Abgesehen von einem Sonderfall, handelt es sich um allgemeine Dupin'sche Zykliden. Die Untersuchung ist so angelegt, daß Fallunterscheidungen möglichst vermieden werden.

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Herrn Professor Dr. Dr. h.c.mult. Otto Haupt mit den besten Wünschen zum 100. Geburtstag gewidmet     

Diameters of 3-sphere quotients

Let G⊂O(4) act isometrically on S³. In this article we calculate a lower bound for the diameter of the quotient spaces S³/G. We find it to be , which is exactly the value of the lower bound for diameters of the spherical space forms. In the process, we are also able to find a lower bound for diameters for the spherical Aleksandrov spaces, Sn/G, of cohomogeneities 1 and 2, as well as for cohomogeneity 3 (with some restrictions on the group type). This leads us to conjecture that the diameter of Sn/G is increasing as the cohomogeneity of the group G increases.     

Semigroup of matrices acting on the max-plus projective space

We investigate the action of semigroups of d×d matrices with entries in the max-plus semifield on the max-plus projective space. Recall that semigroups generated by one element with projectively bounded image are projectively finite and thus contain idempotent elements.In terms of orbits, our main result states that the image of a minimal orbit by an idempotent element of the semigroup with minimal rank has at most d! elements. Moreover, each idempotent element with minimal rank maps at least one orbit onto a singleton.This allows us to deduce the central limit theorem for stochastic recurrent sequences driven by independent random matrices that take countably many values, as soon as the semigroup generated by the values contains an element with projectively bounded image.     

On quadratic forms

The present paper is related to the Jordan—von Neumann characterization of inner product spaces, to the Halperin problem concerning quadratic forms, to some results of the present author on quadratic and sesquilinear forms and to recently obtained results of C. T. Ng and of J. Vukman.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.     

Regular polytopes of type {4,4,3} and {4,4,4}

Abstract regular polytopes generalize the classical concept of a regular polytope and regular tessellation to more complicated combinatorial structures with a distinctive geometrical and topological flavour. In this paper the authors give an almost complete classification of the (universal) locally toroidal regular 4-polytopes of Schläfli types {4,4,3} and {4,4,4}.