3-colourability of Penrose kite-and-dart tilings

A Universal Semi-totalistic Cellular Automaton on Kite and Dart Penrose Tilings

In this paper we investigate certain properties of semi-totalistic cellular automata (CA) on the well known quasi-periodic kite and dart two dimensional tiling of the plane presented by Roger Penrose. We show that, despite the irregularity of the underlying grid, it is possible to devise a semitotalistic CA capable of simulating any boolean circuit and any Turing machine on this aperiodic tiling.

The Symplectic Penrose Kite

The purpose of this article is to view the Penrose kite from the perspective of symplectic geometry. Mathematics Subject Classification 2000. Primary: 53D20 Secondary: 52C23 Introduction The kite in a Penrose aperiodic tiling by kite and darts [8, 9] is an example of a simple convex polytope. By the Atiyah, Guillemin–Sternberg convexity theorem [1, 6], convex polytopes that are rational can be ...

Rhombic Penrose Tilings Can be 3-Colored

Homogenization of Penrose tilings

where Ω is an open subset of R, and f depends on x also through the shape and the orientation of the cell contaning x in an a-periodic tiling of the space R. As an example we may consider mixtures of two (linear) conducting materials with different dielectric constants depending on the type of the tile, and f(y,Du) = a(y)|Du|, where a takes two values α, β depending whether y is in one type of ...

Subdivision Surfaces and Penrose Tilings