GEOMETRIC REALIZATIONS OF TWO-DIMENSIONAL SUBSTITUTIVE TILINGS

Two - Dimensional Random Tilings of Largecodimensionm

Random tilings of the plane by rhombi are projections into the plane of corrugated two-dimensional surfaces in a higher dimensional hypercubic crystal. We consider tilings of 2D-fold symmetry projected from D-dimensional space. Thermodynamic properties depend on the relative numbers of tiles with diierent angles and ori-entations, and also on the boundary conditions imposed on the tiling. Relat...

Geometric Realizations of Substitutions

Two-dimensional Groups, Orbifolds and Tilings

Given the triangulation of a 2-dimensional orbifold in terms of the Delaney{Dress symbol of a periodic tiling, we discuss how to compute it's orbifold symbol, as deened by J. Conway. It is shown that the number of types of equivariant tilings depends only on the \form" of the corresponding orbifold symbols. The method is applied to obtain a reened classiication of equivariant tilings for certai...

Trading spaces: building three-dimensional nets from two-dimensional tilings.

We construct some examples of finite and infinite crystalline three-dimensional nets derived from symmetric reticulations of homogeneous two-dimensional spaces: elliptic (S (2)), Euclidean (E (2)) and hyperbolic (H (2)) space. Those reticulations are edges and vertices of simple spherical, planar and hyperbolic tilings. We show that various projections of the simplest symmetric tilings of those...

Geometric Realizations of Polyhedral Complexes

We discuss two problems in combinatorial geometry. First, given a geometric polyhedral complex in R (a family of 3-polytopes attached face-to-face), can it always be realized over Q? We give a negative answer to this question, by presenting an irrational polyhedral complex with 1278 convex polyhedra. We then present a universality theorem on the space of realizations of such complexes. Second, ...