Higman has questioned which discrete hyperbolic groups [p, q] have representations onto almost all symmetric and alternating groups. We call this property 3tf and show that, except perhaps for finitely many values of/? and q, [p,q] has property JC. It is well known that the modular group F = (x,y\ x = y = 1> has the property that every alternating and symmetric group is a homomorphic image of F...

László Fejes Tóth’s fascinating book [2] demonstrates in many ways the phenomenon that figures of discrete or convex geometry that are very economical, namely solving an extremal problem of isoperimetric type, often show a high degree of symmetry. Among the examples are also planar mosaics where, for instance, an extremal property leads to the hexagonal pattern. Mosaics, or tessellations, have ...

It is well known that there are 17 crystallographic groups that determine the possible tessellations of the Euclidean plane. We approach them from an unusual point of view. Corresponding to each crystallographic group there is an orbifold. We show how to think of the orbifolds as artifacts that serve to create tessellations. 1. TESSELLATIONS. The history of human civilization gives an enormous ...

Sébastien Truchet was a pioneer in applying combinatorics to the study of regular patterns. He enumerated the patterns that could be formed from square tiles that were divided by a diagonal into a black and a white triangle Following Truchet, others have created Truchet-like tilings composed of circular arcs and other motifs. These patterns are all based on Euclidean tessellations, usually the ...

The so-called STIT tessellations form the class of homogeneous (spatially stationary) tessellations of R which are stable under the nesting/iteration operation. In this paper, we establish the strong mixing property for these tessellations and give the optimal form of the rate of decay for the quantity |P(A ∩ Y = ∅, ThB ∩ Y = ∅) − P(A ∩ Y = ∅)P(B ∩ Y = ∅)| when A and B are two compact sets, h a...

We find that 48/64 hard spheres per unit cell on the gyroid minimal surface are entropically self-organized. Striking evidence is obtained in terms of the acceptance ratio of Monte Carlo moves and order parameters. The regular tessellations of the spheres can be viewed as hyperbolic tilings on the Poincaré disc with a negative Gaussian curvature, one of which is, equivalently, the arrangement o...

A regular map is a family of equivalent polygons, glued together to form a closed surface without boundaries which is vertex, edge and face transitive. The commonly known regular maps are derived from the Platonic solids and some tessellations of the torus. There are also regular maps of genus greater than 1 which are traditionally viewed as finitely generated groups. RMS (Regular Map Smoothing...

Classical low-density parity-check (LDPC) codes were first introduced by Robert Gallager in the 1960's and have reemerged as one of the most influential coding schemes. We present new families of quantum low-density parity-check error-correcting codes derived from regular tessellations of Platonic 2-manifolds and from embeddings of the Lubotzky-Phillips-Sarnak Ramanujan graphs. These families o...