Chaos and GraphicsA stochastic cellular automaton for three-coloring penrose tiles
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
References (6)
Penrose tiles to trapdoor ciphers
(1989)- et al.
Tilings and patterns
(1987) - et al.
Rhombic Penrose tilings can be 3-colored
The American Mathematical Monthly
(2000)
Cited by (6)
The corona limit of penrose tilings is a regular decagon
2016, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)Penrose Demosaicking
2015, IEEE Transactions on Image ProcessingUniversal von neumann neighborhood cellular automata on penrose tilings
2013, Proceedings - 2013 1st International Symposium on Computing and Networking, CANDAR 2013A 6-state universal semi-totalistic cellular automaton on kite and dart penrose tilings
2013, Fundamenta InformaticaeMulti-class blue noise sampling
2010, ACM SIGGRAPH 2010 Papers, SIGGRAPH 2010Multi-class blue noise sampling
2010, ACM Transactions on Graphics