Elsevier

Computers & Graphics

Volume 26, Issue 3, June 2002, Pages 519-524
Computers & Graphics

Chaos and Graphics
A stochastic cellular automaton for three-coloring penrose tiles

https://doi.org/10.1016/S0097-8493(02)00094-8Get rights and content

Abstract

We present a three-state, stochastic cellular automaton that runs on Penrose tilings and seems to evolve to a three-colored equilibrium.

Introduction

In 1973 and 1974, Roger Penrose discovered three sets of polygons each of which tiles the plane aperiodically and (if certain matching conditions are enforced) only aperiodically. Later, John H. Conway asked if such tilings can be three-colored, where adjacent tiles are to receive different colors. This question has been answered affirmatively for two types of Penrose tilings, but appears to be open for the remaining type. In this paper, we present an algorithm that seems to three-color finite parts of Penrose tilings of all types. The algorithm works by running a particular three-state, stochastic cellular automaton on a given Penrose tiling. The cellular automaton is chosen so that three-colorings are stable and it seems to generally evolve to such an equilibrium.

Section snippets

Penrose tilings

There are three types of Penrose tilings: tilings by kites and darts, tilings by rhombs, and tilings by pentacles. We describe them briefly here. More details are in [1], [2].

Coloring the tiles

A tiling is called three-colorable if we may assign one of three distinct colors to each tile such that adjacent tiles have different colors. Tiles are said to be adjacent if their intersection is a line segment. Fig. 7, Fig. 8, Fig. 9 show three-colored tilings by kites and darts, rhombs, and pentacles respectively. Sibley and Wagon [3] proved that tilings by rhombs are three-colorable and Babilon [4] proved that tilings by kites and darts are three-colorable. The equivalent question for the

Related coloring schemes

Clearly the basic idea of this paper is applicable in other situations. A change in the number of states yields a class of algorithms for n-coloring planar maps or graphs in general. For example, taking the number of states to be two, we can use the algorithm to two-color a checker board. While two-coloring a checker board is very simple, this gives us a rudimentary way of measuring the efficiency of the algorithm. Experiments indicate that a 8×8 checkerboard is two-colored in about 32

Implementation notes

All the images for this paper were generated with Mathematica. The tilings were generated using the DigraphFractals Mathematica package by the author as described in [6]. These images were then converted to PlanarMap and PlanarGraph objects as defined in the GraphColoring Mathematica package by Stan Wagon [5]. Code to run the cellular automaton on the PlanarGraph objects was written by the author. Finally three-colored images were rendered by the ShowMap function defined in the GraphColoring

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There are more references available in the full text version of this article.

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