885 research outputs found
Symmetries of Monocoronal Tilings
The vertex corona of a vertex of some tiling is the vertex together with the
adjacent tiles. A tiling where all vertex coronae are congruent is called
monocoronal. We provide a classification of monocoronal tilings in the
Euclidean plane and derive a list of all possible symmetry groups of
monocoronal tilings. In particular, any monocoronal tiling with respect to
direct congruence is crystallographic, whereas any monocoronal tiling with
respect to congruence (reflections allowed) is either crystallographic or it
has a one-dimensional translation group. Furthermore, bounds on the number of
the dimensions of the translation group of monocoronal tilings in higher
dimensional Euclidean space are obtained.Comment: 26 pages, 66 figure
Quasi Regular Polyhedra and Their Duals with Coxeter Symmetries Represented by Quaternions I
In two series of papers we construct quasi regular polyhedra and their duals
which are similar to the Catalan solids. The group elements as well as the
vertices of the polyhedra are represented in terms of quaternions. In the
present paper we discuss the quasi regular polygons (isogonal and isotoxal
polygons) using 2D Coxeter diagrams. In particular, we discuss the isogonal
hexagons, octagons and decagons derived from 2D Coxeter diagrams and obtain
aperiodic tilings of the plane with the isogonal polygons along with the
regular polygons. We point out that one type of aperiodic tiling of the plane
with regular and isogonal hexagons may represent a state of graphene where one
carbon atom is bound to three neighboring carbons with two single bonds and one
double bond. We also show how the plane can be tiled with two tiles; one of
them is the isotoxal polygon, dual of the isogonal polygon. A general method is
employed for the constructions of the quasi regular prisms and their duals in
3D dimensions with the use of 3D Coxeter diagrams.Comment: 22 pages, 16 figure
Tilings of an Isosceles Triangle
An N-tiling of triangle ABC by triangle T is a way of writing ABC as a union
of N trianglescongruent to T, overlapping only at their boundaries. The
triangle T is the "tile". The tile may or may not be similar to ABC. In this
paper we study the case of isosceles (but not equilateral) ABC. We study three
possible forms of the tile: right-angled, or with one angle double another, or
with a 120 degree angle. In the case of a right-angled tile, we give a complete
characterization of the tilings, for N even, but leave open whether N can be
odd. In the latter two cases we prove the ratios of the sides of the tile are
rational, and give a necessary condition for the existence of an N-tiling. For
the case when the tile has one angle double another, we prove N cannot be prime
or twice a prime.Comment: 34 pages, 18 figures. This version supplies corrections and
simplification
Proof of two conjectures of Zuber on fully packed loop configurations
Two conjectures of Zuber [``On the counting of fully packed loops
configurations. Some new conjectures,'' preprint] on the enumeration of
configurations in the fully packed loop model on the square grid with periodic
boundary conditions, which have a prescribed linkage pattern, are proved.
Following an idea of de Gier [``Loops, matchings and alternating-sign
matrices,'' Discrete Math., to appear], the proofs are based on bijections
between such fully packed loop configurations and rhombus tilings, and the
hook-content formula for semistandard tableaux.Comment: 20 pages; AmS-LaTe
Enumeration of Matchings: Problems and Progress
This document is built around a list of thirty-two problems in enumeration of
matchings, the first twenty of which were presented in a lecture at MSRI in the
fall of 1996. I begin with a capsule history of the topic of enumeration of
matchings. The twenty original problems, with commentary, comprise the bulk of
the article. I give an account of the progress that has been made on these
problems as of this writing, and include pointers to both the printed and
on-line literature; roughly half of the original twenty problems were solved by
participants in the MSRI Workshop on Combinatorics, their students, and others,
between 1996 and 1999. The article concludes with a dozen new open problems.
(Note: This article supersedes math.CO/9801060 and math.CO/9801061.)Comment: 1+37 pages; to appear in "New Perspectives in Geometric
Combinatorics" (ed. by Billera, Bjorner, Green, Simeon, and Stanley),
Mathematical Science Research Institute publication #37, Cambridge University
Press, 199
Periodic Planar Disk Packings
Several conditions are given when a packing of equal disks in a torus is
locally maximally dense, where the torus is defined as the quotient of the
plane by a two-dimensional lattice. Conjectures are presented that claim that
the density of any strictly jammed packings, whose graph does not consist of
all triangles and the torus lattice is the standard triangular lattice, is at
most , where is the number of packing
disks. Several classes of collectively jammed packings are presented where the
conjecture holds.Comment: 26 pages, 13 figure
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