Coboundary on colored tiling space as Rauzy fractal

Boundary of the Rauzy Fractal Sets In

We study the boundary of the 3-dimensional Rauzy fractal E ⊂ R × C generated by the polynomial P (x) = x 4 − x 3 − x 2 − x − 1. The finite automaton characterizing the boundary of E is given explicitly. As a consequence we prove that the set E has 18 neighborhoods where 6 of them intersect the central tile E in a point. Our construction shows that the boundary is generated by an iterated functi...

Parametrization of a Two-dimensional Quasiperiodic Rauzy Tiling

With the help of an affine inflation B, a two-dimensional quasiperiodic Rauzy tiling R∞ is constructed, together with a parametrization of its tiles by algebraic integers Z[ζ] ⊂ [0, 1), where ζ is a certain Pisot number (specifically, the real root of the polynomial x3 + x2 + x − 1). The coronas (clusters) of the tiling R∞ are classified by disjoint half-intervals in [0, 1) the lengths of which...

Frontière du fractal de Rauzy et système de numération complexe

A Note on Space Tiling Zonotopes

In 1908 Voronoi conjectured that every convex polytope which tiles space face-to-face by translations is affinely equivalent to the Dirichlet-Voronoi polytope of some lattice. In 1999 Erdahl proved this conjecture for the special case of zonotopes. A zonotope is a projection of a regular cube under some affine transformation. In 1975 McMullen showed several equivalent conditions for a zonotope ...

Rep-tiling Euclidean space

A rep-tiling ~is a self replicating, lattice tiling of R". Lattice tiling means a tiling by translates of a single compact tile by the points of a lattice, and self-replicating means that there is a non-singular linear map ¢: R"-o R" such that, for each T e J, the image 4~(T) is, in turn, tiled by Y-. This topic has recently come under investigation, not only because of its recreational appeal,...