177 research outputs found
Combinatorially two-orbit convex polytopes
Any convex polytope whose combinatorial automorphism group has two orbits on
the flags is isomorphic to one whose group of Euclidean symmetries has two
orbits on the flags (equivalently, to one whose automorphism group and symmetry
group coincide.) Hence, a combinatorially two-orbit convex polytope is
isomorphic to one of a known finite list, all of which are 3-dimensional: the
cuboctahedron, icosidodecahedron, rhombic dodecahedron, or rhombic
triacontahedron. The same is true of combinatorially two-orbit normal
face-to-face tilings by convex polytopes.Comment: 20 page
Combinatorial Space Tiling
The present article studies combinatorial tilings of Euclidean or spherical
spaces by polytopes, serving two main purposes: first, to survey some of the
main developments in combinatorial space tiling; and second, to highlight some
new and some old open problems in this area.Comment: 16 pages; to appear in "Symmetry: Culture and Science
Optimally Dense Packings for Fully Asymptotic Coxeter Tilings by Horoballs of Different Types
The goal of this paper to determine the optimal horoball packing arrangements
and their densities for all four fully asymptotic Coxeter tilings (Coxeter
honeycombs) in hyperbolic 3-space . Centers of horoballs are
required to lie at vertices of the regular polyhedral cells constituting the
tiling. We allow horoballs of different types at the various vertices. Our
results are derived through a generalization of the projective methodology for
hyperbolic spaces. The main result states that the known B\"or\"oczky--Florian
density upper bound for "congruent horoball" packings of remains
valid for the class of fully asymptotic Coxeter tilings, even if packing
conditions are relaxed by allowing for horoballs of different types under
prescribed symmetry groups. The consequences of this remarkable result are
discussed for various Coxeter tilings.Comment: 26 pages, 10 figure
Semi-equivelar toroidal maps and their vertex covers
If the face\mbox{-}cycles at all the vertices in a map are of same type then
the map is called semi\mbox{-}equivelar. A map is called minimal if the number
of vertices is minimal. We know the bounds of number of vertex orbits of
semi-equivelar toroidal maps. These bounds are sharp. Datta \cite{BD2020} has
proved that every semi-equivelar toroidal map has a vertex-transitive cover. In
this article, we prove that if a semi-equivelar map is orbital then it has
a finite index -orbital minimal cover for . We also show the
existence and classification of -sheeted covers of semi-equivelar toroidal
maps for each
Polygonal Complexes and Graphs for Crystallographic Groups
The paper surveys highlights of the ongoing program to classify discrete
polyhedral structures in Euclidean 3-space by distinguished transitivity
properties of their symmetry groups, focussing in particular on various aspects
of the classification of regular polygonal complexes, chiral polyhedra, and
more generally, two-orbit polyhedra.Comment: 21 pages; In: Symmetry and Rigidity, (eds. R.Connelly, A.Ivic Weiss
and W.Whiteley), Fields Institute Communications, to appea
Clusters of Cycles
A {\it cluster of cycles} (or {\it -polycycle}) is a simple planar
2--co nnected finite or countable graph of girth and maximal
vertex-degree , which admits {\it -polycyclic realization} on the
plane, denote it by , i.e. such that: (i) all interior vertices are of
degree , (ii) all interior faces (denote their number by ) are
combinatorial -gons and (implied by (i), (ii)) (iii) all vertices, edges and
interior faces form a cell-complex.
An example of -polycycle is the skeleton of , i.e. of the
-valent partition of the sphere , Euclidean plane or hyperbolic
plane by regular -gons. Call {\it spheric} pairs
; for those five pairs is
without the exterior face; otherwise .
We give here a compact survey of results on -polycycles.Comment: 21. to in appear in Journal of Geometry and Physic
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