Shape tiling.
Some Basic Properties of Packing and Covering Constants.
Some New Tilings of the Sphere with Congruent Triangles
Spaces of Domino Tilings.
Spherical F-tilings by triangles and -sided regular polygons, .
Steinhaus chessboard theorem
Symmetries of woven fabrics
Tessellations Generated by Hyperplanes.
Tessellations of random maps of arbitrary genus
We investigate Voronoi-like tessellations of bipartite quadrangulations on surfaces of arbitrary genus, by using a natural generalization of a bijection of Marcus and Schaeffer allowing one to encode such structures by labeled maps with a fixed number of faces. We investigate the scaling limits of the latter. Applications include asymptotic enumeration results for quadrangulations, and typical metric properties of randomly sampled quadrangulations. In particular, we show that scaling limits of these...
The isogonal sponges on the cubic lattice.
The dimer problem for narrow rectangular arrays: A unified method of solution, and some extensions.
The distribution of the number of nodes in the relative interior of the typical I-segment in homogeneous planar anisotropic STIT Tessellations
A result about the distribution of the number of nodes in the relative interior of the typical I-segment in homogeneous and isotropic random tessellations stable under iteration (STIT tessellations) is extended to the anisotropic case using recent findings from Schreiber/Thäle, Typical geometry, second-order properties and central limit theory for iteration stable tessellations, arXiv:1001.0990 [math.PR] (2010). Moreover a new expression for the values of this probability distribution is presented...
The geometry of non-unit Pisot substitutions
It is known that with a non-unit Pisot substitution one can associate certain fractal tiles, so-called Rauzy fractals. In our setting, these fractals are subsets of a certain open subring of the adèle ring of the associated Pisot number field. We present several approaches on how to define Rauzy fractals and discuss the relations between them. In particular, we consider Rauzy fractals as the natural geometric objects of certain numeration systems, in terms of the dual of the one-dimensional realization...
The impossibility of a tesselation of the plane into equilateral triangles whose sidelengths are mutually different, one of them being minimal.
The shape of a typical boxed plane partition.
The summation package Sigma: underlying principles and a rhombus tiling application.
There are ternary circular square-free words of length for 18.
Tiling a -board with squares and dominoes.
Tiling parity results and the holey square solution.
Tilings and isoperimetrical shapes I: Square lattice