Neighborly polytopes (original) (raw)
Arbitrarily large neighborly families of congruent symmetric convex 3-polytopes With Scott Kim
We construct, for any positive integer n, a family of n congruent convex polyhedra in IR 3 , such that every pair intersects in a common facet. Our polyhedra are Voronoi regions of evenly distributed points on the helix (t, cos t, sin t). The largest previously published example of such a family contains only eight polytopes. With a simple modification, we can ensure that each polyhedron in the family has a point, a line, and a plane of symmetry. We also generalize our construction to higher dimensions and introduce a new family of cyclic polytopes.
On sewing neighbourly polytopes
2001
In 1982, I. Shemer introduced the sewing construction for neighbourly 2m-polytopes. We extend the sewing to simplicial neighbourly d-polytopes via a verification that is not dependent on the parity of the dimension. We present also descibable classes of 4-polyopes and 5-polytopes generated by the construction.
Reports on Mathematical Physics, 2013
Regular polytopes, the generalization of the five Platonic solids in 3 space dimensions, exist in arbitrary dimension n ≥ −1; now in dim. 2, 3 and 4 there are extra polytopes, while in general dimensions only the hyper-tetrahedron, the hyper-cube and its dual hyper-octahedron exist. We attribute these peculiarites and exceptions to special properties of the orthogonal groups in these dimensions: the SO(2) = U(1) group being (abelian and) divisible, is related to the existence of arbitrarilysided plane regular polygons, and the splitting of the Lie algebra of the O(4) group will be seen responsible for the Schläfli special polytopes in 4-dim., two of which percolate down to three. In spite of dim. 8 being also special (Cartan's triality), we argue why there are no extra polytopes, while it has other consequences: in particular the existence of the three division algebras over the reals R: complex C, quaternions H and octonions O is seen also as another feature of the special properties of corresponding orthogonal groups, and of the spheres of dimension 0,1,3 and 7.
On locally spherical polytopes of type {5, 3, 5
Discrete Mathematics, 2009
There are only finitely many locally projective regular polytopes of type {5, 3, 5}. They are covered by a locally spherical polytope whose automorphism group is J 1 × J 1 × L 2 (19), where J 1 is the first Janko group, of order 175560, and L 2 (19) is the projective special linear group of order 3420. This polytope is minimal, in the sense that any other polytope that covers all locally projective polytopes of type {5, 3, 5} must in turn cover this one.
Ball polytopes and the V�zsonyi problem
Acta Math Hung, 2010
Let V be a finite set of points in Euclidean d-space (d >= 2). The intersection of all unit balls B(v,1) centered at v, where v ranges over V, henceforth denoted by B(V) is the ball polytope associated with V. Note that B(V) is non-empty iff the circumradius of V is <= 1. After some preparatory discussion on spherical convexity and spindle convexity, the paper focuses on two central themes. [a] Define the boundary complex of B(V) (assuming it is non-empty, of course), i.e., define its vertices, edges and facets in dimension 3 (in dimension 2 this complex is just a circuit), and investigate its basic properties. [b] Apply results of this investigation to characterize finite sets of diameter 1 in (Euclidean) 3-space for which the diameter is attained a maximal number of times as a segment (of length 1) with both endpoints in V. A basic result for such a characterization goes back to Grunbaum, Heppes and Straszewicz, who proved independently that the diameter of V is attained at most 2|V|-2 times, thus affirming a conjecture of Vazsonyi from circa 1935. Call V extremal if its diameter is attained this maximal number (2|V|-2) of times. We extend the aforementioned basic result by showing that V is extremal iff V coincides with the set of vertices of its ball polytope B(V) and show that in this case the boundary complex of B(V) is self-dual in some strong sense. For the sake of priority we mention that, in the present form (except for a few changes in the footnotes), the paper was submitted to a journal already in February 1, 2008.
on-2-skeleta-of-Platonic-polytopes-bhms-2018-62-94-102.pdf
Bull. Helenic Math. Society, 2018
Euler showed that every graph whose vertices have even degree can be decomposed into edge-disjoint cycles. We prove that an analogous result holds for the 2-skeleta of Platonic polytopes, where cycles are replaced by surfaces.
A Continuous d-Step Conjecture for Polytopes
Discrete & Computational Geometry, 2009
The curvature of a polytope, defined as the largest possible total curvature of the associated central path, can be regarded as the continuous analogue of its diameter. We prove the analogue of the result of Klee and Walkup. Namely, we show that if the order of the curvature is less than the dimension d for all polytope defined by 2d inequalities and for all d, then the order of the curvature is less that the number of inequalities for all polytopes.
Polytopes: Abstract, Convex and Computational
Polytopes: Abstract, Convex and Computational, 1994
Convex and Computational NATO ASI Series Advanced Science Institutes Series A Series presenting the results of activities sponsored by the NATO Science Committee, which aims at the dissemination of advanced scientific and technological knowledge, with a view to strengthening links between scientific communities.
Canadian Journal of Mathematics, 1969
Our aim in this paper is to describe a new class of convex polytopes, which will be called linearly stable. These have properties analogous to those of projectively stable polytopes (called projectively unique by Grunbaum (1, exercise 4.8.30)), which were first investigated early in 1966 by Grunbaum and Pedes. Although many particular examples of projectively stable polytopes have been found, at present no general criteria for projective stability are known. The main result of this paper is a theorem which enables us to classify linearly stable polytopes completely. I wish to thank Professor G. C. Shephard for his many helpful suggestions for improvements to this paper. During the period of this research I held a research studentship, for which I would like to thank the Science Research Council. 2. The main theorem. The definitions of Grunbaum's recent book (1) will be followed throughout the paper. All {/-polytopes (^-dimensional convex polytopes (1, § 3.1)) will be assumed to lie in ^-dimensional Euclidean space E d , unless the context demands otherwise. The polytopes which are the subject of the paper are mainly centrally symmetric (1, § 6.4) ; the centre of symmetry will always be taken to be the origin o. Throughout the paper we write c.s. for "centrally symmetric". Two c.s. polytopes P\ and Pi are said to be linearly equivalent if there is a non-singular linear transformation A such that P\A = P 2. Clearly Pi and Pi are combinatorially equivalent (1, § 3.2), that is, there is a one-to-one inclusionpreserving correspondence between the faces of P\ and those of Pi. A c.s. polytope is called linearly stable if every c.s. polytope which is combinatorially equivalent to P is linearly equivalent to P. The regular d-cube C d is C d = {x = (Si,. .. , y G E d | |É,| ^ 1, i = 1,. .. , d). Clearly C d is centrally symmetric, and any polytope linearly equivalent to C d will be called a d-cube. (Strictly we should call it a d-parallelotope.) The symbol C d will, however, be reserved for the regular d-cube. C d has 2 d vertices, whose coordinates are (±1,. .. , ±1), with all possible changes of sign. The theorem which enables us to classify the linearly stable polytopes can now be stated.
Archiv der Mathematik, 2007
Cyclic polytopes are characterized as simplicial polytopes satisfying Gale's evenness condition (a combinatorial condition on facets relative to a fixed ordering of the vertices). Periodically-cyclic polytopes are polytopes for which certain subpolytopes are cyclic. Bisztriczky discovered a class of periodically-cyclic polytopes that also satisfy Gale's evenness condition. The faces of these polytopes are braxtopes, a certain class of nonsimplicial polytopes studied by the authors. In this paper we prove that the periodicallycyclic Gale polytopes of Bisztriczky are exactly the polytopes that satisfy Gale's evenness condition and are braxial (all faces are braxtopes). The existence of other periodically-cyclic Gale polytopes is open.
On the Size of Equifacetted Semi-Regular Polytopes
2011
Unlike the situation in the classical theory of convex polytopes, there is a wealth of semi-regular abstract polytopes, including interesting examples exhibiting some unexpected phenomena. We prove that even an equifacetted semi-regular abstract polytope can have an arbitrary large number of flag orbits or face orbits under its combinatorial automorphism group.
On face numbers of neighborly cubical polytopes
2013
Neighborly cubical polytopes are known as the cubical analogues of the cyclic polytopes. Using the short cubical hhh-vectors of cubical polytopes (introduced by Adin), we derive an explicit formula for the face numbers of the neighborly cubical polytopes. These face numbers form a unimodal sequence.
Regular Polytopes in Ordinary Space
Discrete & Computational Geometry, 1997
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.
Vertex embeddings of regular polytopes
Expositiones Mathematicae, 2003
The question of when one regular polytope (finite, convex) embedds in the vertices of another, of the same dimension, leads to a fascinating interplay of geometry, combinatorics, and matrix theory, with farther relations to number theory and algebraic topology. This mainly expository paper is an account of this subject, its history, and the principal results together with an outline of their proofs. The relationships with other branches of mathematics are also explained.
2008
Some basic mathematical tools such as convex sets, polytopes and combinatorial topology, are used quite heavily in applied fields such as geometric modeling, meshing, computer vision, medical imaging and robotics. This report may be viewed as a tutorial and a set of notes on convex sets, polytopes, polyhedra, combinatorial topology, Voronoi Diagrams and Delaunay Triangulations. It is intended for a broad audience of mathematically inclined readers. I have included a rather thorough treatment of the equivalence of V-polytopes and H-polytopes and also of the equivalence of V-polyhedra and H-polyhedra, which is a bit harder. In particular, the Fourier-Motzkin elimination method (a version of Gaussian elimination for inequalities) is discussed in some detail. I also included some material on projective spaces, projective maps and polar duality w.r.t. a nondegenerate quadric in order to define a suitable notion of ``projective polyhedron'' based on cones. To the best of our knowledge, this notion of projective polyhedron is new. We also believe that some of our proofs establishing the equivalence of V-polyhedra and H-polyhedra are new.
An Introduction to Polytope Theory through Ehrhart's Theorem
2019
A classic introduction to polytope theory is presented, serving as the foundation to develop more advanced theoretical tools, namely the algebra of polyhedra and the use of valuations. The main theoretical objective is the construction of the so called Berline-Vergne valuation. Most of the theoretical development is aimed towards this goal. A little survey on Ehrhart positivity is presented, as well as some calculations that lead to conjecture that generalized permutohedra have positive coefficients in their Ehrhart polynomials. Throughout the thesis three different proofs of Ehrhart’s theorem are presented, as an application of the new techniques developed.