Regular Polytopes of Nearly Full Rank

An abstract regular polytope P\mathcal{P} of rank n can only be realized faithfully in Euclidean space \mathbbE^d\mathbb{E}^{d} of dimension d if d≥n when P\mathcal{P} is finite, or d≥n−1 when P\mathcal{P} is infinite (that is, P\mathcal{P} is an apeirotope). In case of equality, the realization P of P\mathcal{P} is said to be of full rank. If there is a faithful realization P of P\mathcal{P} of dimension d=n+1 or d=n (as P\mathcal {P} is finite or not), then P is said to be of nearly full rank. In previous papers, all the at most four-dimensional regular polytopes and apeirotopes of nearly full rank have been classified. This paper classifies the regular polytopes and apeirotopes of nearly full rank in all higher dimensions.     

Regular Polytopes of Full Rank

The dimension of a faithful realization of a finite abstract regular polytope in some euclidean space is no smaller than its rank. Similarly, that of a discrete faithful realization of a regular apeirotope is at least one fewer than the rank. Realizations which attain the minimum are said to be of full rank. The regular polytopes and apeirotopes of full rank in two and three dimensions were classified in an earlier paper. In this paper these polytopes and apeirotopes are classified in all dimensions. Moreover, it is also shown that there are no chiral polytopes of full rank.     

Radii of Regular Polytopes

For the first time complete lists of two pairs of inner and outer radii classes of the three types of regular polytopes which exist in all dimensions are presented. A new approach using isotropic polytopes provides better understanding of the underlying geometry and helps to unify the results.     

Reciprocating the Regular Polytopes

For reciprocation with respect to a sphere x²=c in Euclideann-space, there is a unitary analogue: Hermitian reciprocationwith respect to an antisphere u=c. This is now applied, forthe first time, to complex polytopes. When a regular polytope has a palindromic Schläfli symbol,it is self-reciprocal in the sense that its reciprocal ', withrespect to a suitable concentric sphere or antisphere, is congruentto . The present article reveals that and ' usually have togetherthe same vertices as a third polytope ⁺ and the same facet-hyperplanesas a fourth polytope ^– (where ⁺ and ^– are againregular), so as to form a ‘compound’, ⁺[2]^–.When the geometry is real, ⁺ is the convex hull of and ', while^– is their common content or ‘core’. For instance,when is a regular p-gon {p}, the compound is The exceptions are of two kinds. In one, ⁺ and ^– are notregular. The actual cases are when is an n-simplex {3, 3, ...,3} with n4 or the real 4-dimensional 24-cell {3, 4, 3}=2{3}2{4}2{3}2or the complex 4-dimensional Witting polytope 3{3}3{3}3{3}3.The other kind of exception arises when the vertices of arethe poles of its own facet-hyperplanes, so that , ', ⁺ and ^–all coincide. Then is said to be strongly self-reciprocal.     

Hamiltonian Submanifolds of Regular Polytopes

We investigate polyhedral 2k-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex k -Hamiltonian if it contains the full k-skeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so-called super-neighborly triangulations), we focus on the case of the cross polytope and the sporadic regular 4-polytopes. By our results the existence of 1-Hamiltonian surfaces is now decided for all regular polytopes. Furthermore we investigate 2-Hamiltonian 4-manifolds in the d-dimensional cross polytope. These are the “regular cases” satisfying equality in Sparla’s inequality. In particular, we present a new example with 16 vertices which is highly symmetric with an automorphism group of order 128. Topologically it is homeomorphic to a connected sum of seven copies of S ²×S ². By this example all regular cases of n vertices with n<20 or, equivalently, all cases of regular d-polytopes with d≤9 are now decided.     

Polytopes of Minimum Positive Semidefinite Rank

The positive semidefinite (psd) rank of a polytope is the smallest $k$ k for which the cone of $k \times k$ k × k real symmetric psd matrices admits an affine slice that projects onto the polytope. In this paper we show that the psd rank of a polytope is at least the dimension of the polytope plus one, and we characterize those polytopes whose psd rank equals this lower bound. We give several classes of polytopes that achieve the minimum possible psd rank including a complete characterization in dimensions two and three.     

Regular Polytopes in Ordinary Space

The three aims of this paper are to obtain the proof by Dress of the completeness of the enumeration of the Grünbaum—Dress polyhedra (that is, the regular apeirohedra, or apeirotopes of rank 3) in ordinary space E ³ in a quicker and more perspicuous way, to give presentations of those of their symmetry groups which are affinely irreducible, and to describe all the discrete regular apeirotopes of rank 4 in E ³ . The paper gives a complete classification of the discrete regular polytopes in ordinary space. Received June 6, 1996, and in revised form October 28, 1996.     

Symmetric Graphs from Polytopes of High Rank

Given a polytope ${{\mathcal{P}}}$ of rank 2n, the faces of middle ranks n ? 1 and n constitute the vertices of a bipartite graph, the medial layer graph ${{M(\mathcal{P})}}$ of ${{\mathcal{P}}}$ . The group ${{D(\mathcal{P})}}$ of automorphisms and dualities of ${{\mathcal{P}}}$ has a natural action on this graph. We prove algebraic and combinatorial conditions on ${{\mathcal{P}}}$ that ensure this action is transitive on k-arcs in ${{M(\mathcal{P})}}$ for some small k (in particular focussing on k?=?3), and provide examples of families of polytopes that satisfy these conditions. We also examine how ${{D(\mathcal{P})}}$ acts on the k-stars based at vertices of ${{M(\mathcal{P})},}$ and describe self-dual regular polytopes (in particular those of rank 6) for which this action is transitive on the k-stars for small k.     

An Atlas of Small Regular Abstract Polytopes

Summary This article announces the creation of an atlas of small regular abstract polytopes. The atlas contains information about all regular abstract polytopes whose automorphism group has order 2000 or less, except those of order <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>512k$ where $k\geq1$. The article explains also the techniques used to create the atlas, and gives some summary tables. At the time of printing, the url for the atlas is http://<a href = "http://www.abstract-polytopes.com/atlas">www.abstract-polytopes.com/atlas</a>.     

Twisted Groups and Locally Toroidal Regular Polytopes

In recent years, much work has been done on the classification of abstract regular polytopes by their local and global topological type. Abstract regular polytopes are combinatorial structures which generalize the well-known classical geometric regular polytopes and tessellations. In this context, the classical theory is concerned with those which are of globally or locally spherical type. In a sequence of papers, the authors have studied the corresponding classification of abstract regular polytopes which are globally or locally toroidal. Here, this investigation of locally toroidal regular polytopes is continued, with a particular emphasis on polytopes of ranks and . For large classes of such polytopes, their groups are explicitly identified using twisting operations on quotients of Coxeter groups. In particular, this leads to new classification results which complement those obtained elsewhere. The method is also applied to describe certain regular polytopes with small facets and vertex-figures.

     

Size Analysis of Nearly Regular Delaunay Triangulations

Consider a nearly regular point pattern in which a Delaunay triangulation is comprised of nearly equilateral triangles of the same size. We propose to model this set of points with Gaussian perturbations about a regular mean configuration. By investigating triangle subsets in detail we obtain various distributions of statistics based on size, or squared size of the triangles which is closely related to the mean (squared) distance to the six nearest neighbors. A scaleless test statistic, corresponding to a coefficient of variation for squared sizes, is proposed and its asymptotic properties described. The methodology is applied to an investigation of regularity in human muscle fiber cross-sections. We compare the approach with an alternative technique in a power study.     

Regular Apeirotopes of Dimension and Rank 4

In previous papers, all the four-dimensional (finite) regular polytopes have been classified, as well as the regular apeirotopes of full rank (that is, of rank 5). Of the two problems in \(\mathbb{E}^{4}\) thus left open (namely, regular apeirotopes of ranks 3 and 4), this paper describes the regular apeirotopes of rank 4. The methods employed here are somewhat different from those in earlier work; while knowledge of the possible dimension vectors (dim?R ₀,…,dim?R ₃) of the mirrors R ₀,…,R ₃ of the generating reflexions of the symmetry groups plays a rôle, the crystallographic restriction leads to a considerable emphasis being placed on the vertex-figures.     

矩阵的满秩正交分解

旨在给出矩阵一种新分解(满秩正交分解).分解简单易求,且与矩阵的奇异值分解有类似的性质和应用.     

全正则子半群构成链的正则半群(英文)

本文考虑全正则子半群构成链的正则半群,得到了正则半群具有全正则子半群构成链的一个充分必要条件,这推广了Jones关于具有全正则子半群构成链的逆半群的结果.特别地,建立了具有全正则子半群构成链的完全0-单半群的结构.     

满秩Hankel矩阵的拟直分解

本文研究行满秩Hankel矩阵分解为一个真正的(proper)Hankel矩阵与一个退化的(de- generate)Hankel矩阵之拟直和的存在性及唯一性问题.     

有限秩的可解群的正则自同构

设G是有限秩的剩余有限可解群或是有限秩的剩余有限可解群的有限扩张,α是G的一个索数p阶正则自同构且φ:G→G(g→[g,α])是满射,则G是幂零类不超过h(p)的幂零群,其中h(p)是只与p有关的函数.     

A Full Rank Perfect Code of Length 31

A full rank perfect 1-error correcting binary code of length 31 with a kernel of dimension 21 is described. This was the last open case of the rank-kernel problem of Etzion and Vardy. AMS Classification: 94B25     

On Some Families of Smooth Affine Spherical Varieties of Full Rank

Let G be a complex connected reductive group. Losev has shown that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. In this paper we use a combinatorial characterization of the weight monoids of smooth affine spherical varieties to classify:(a) all such varieties for G = SL(2) × C~×and(b) all such varieties for G simple which have a G-saturated weight monoid of full rank. We also use the characterization and Knop's classification theorem for multiplicity free Hamiltonian manifolds to give a new proof of Woodward's result that every reflective Delzant polytope is the moment polytope of such a manifold.     

Finite Groups of Bounded Rank with an Almost Regular Automorphism of Prime Order

We prove that if a finite group G of rank r admits an automorphism of prime order having exactly m fixed points, then G has a -invariant subgroup of (r,m)-bounded index which is nilpotent of r-bounded class (Theorem 1). Thus, for automorphisms of prime order the previous results of Shalev, Khukhro, and Jaikin-Zapirain are strengthened. The proof rests, in particular, on a result about regular automorphisms of Lie rings (Theorem 3). The general case reduces modulo available results to the case of finite p-groups. For reduction to Lie rings powerful p-groups are also used. For them a useful fact is proved which allows us to glue together nilpotency classes of factors of certain normal series (Theorem 2).     

Law of Sums of the Squares of Areas, Volumes and Hyper-Volumes of Regular Polytopes from Clifford Algebras

Inspired by the recent sums of the squares law obtained by [1] we derive the law of the sums of the squares of the areas, volumes and hyper-volumes associated with the faces, cells and hyper-cells of regular polytopes in diverse dimensions by using Clifford algebraic methods [5].