# Difference between revisions of "Tiling"

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(some notes for collaborative study) | (some notes for collaborative study) | ||

− | [https://noisebridge.net/images/c/c2/Gardner_PenroseTilings1-1977.pdf | + | * http://preshing.com/20110831/penrose-tiling-explained/ |

+ | * [https://noisebridge.net/images/c/c2/Gardner_PenroseTilings1-1977.pdf 2 chapters from Martin Gardner's book "Penrose Tiles to Trapdoor Ciphers"] | ||

+ | |||

+ | ''1977'' -> <pre> | ||

+ | A Penrose tiling can, of course, always be colored with four colors so | ||

+ | that no two tiles of the same color share a common edge. Can it always be | ||

+ | colored with three? It can be shown, Conway said, from the local isomorphism | ||

+ | theorem, that if any Penrose tiling is three-colorable, all are, but so far | ||

+ | no one has proved that any infinite Penrose tiling is three-colorable.</pre> | ||

+ | |||

+ | ''2002'' -< | ||

+ | * [http://www.ual.es/~jcaceres/ptbpcb3c.ps Penrose Tilings by Pentacles can be 3-Colored] | ||

+ | <pre>by J. Caceres, M.E. Gegundez, A. Marquez , 2002 | ||

+ | "In this paper the problem is solved for the last | ||

+ | Penrose tiling for which the problem remain ..."</pre> | ||

+ | |||

+ | ''note: stochastic cellular automata don't care about proof'' | ||

+ | * [http://www.cs.unca.edu/~mcclure/professional/Penrose3Color.pdf A Stochastic Cellular Automata for Three-Coloring Penrose Tiles] | ||

+ | <pre>First, assign one of three possible colors to each tile randomly. | ||

+ | Then, allow the cellular automata to evolve accoring to the following | ||

+ | set of rules: | ||

+ | * If the value fo a cell (or tile) equals the value of a bordering | ||

+ | cell which is closer to the origin (as measured by some arbitrary | ||

+ | point chosen within each tile), then with 90% probability, the | ||

+ | cell changes value randomly to one of the other two colors. | ||

+ | * If the value of a cell does not equal the value of a bordering | ||

+ | cell which is closer to the origin, but does equal the value | ||

+ | of a cell farther away from the origin, then with 10% probability | ||

+ | the cell changes value | ||

+ | * If the value of a cell does not equal the value of any bordering | ||

+ | cell, the cell does not change value</pre> | ||

+ | |||

+ | [[Image:Penrose_life.gif]] | ||

+ | |||

+ | ''Conway's game of life on Penrose tiles'' | ||

+ | * [http://www-users.cs.york.ac.uk/~susan/bib/ss/nonstd/ecal05.pdf Penrose Life: ash and oscillators] | ||

+ | <pre>Margaret Hill, Susan Stepney, and Francis Wan | ||

+ | We compare the long term behaviour of Conway’s Game | ||

+ | of Life cellular automaton, from initial random configurations, | ||

+ | on a bounded rectangular grid and a bounded Penrose tiling grid. | ||

+ | We investigate the lifetime to stability, the final ‘ash’ density, | ||

+ | and the number and period of final oscillators. Penrose grids have | ||

+ | similar qualitative behaviour but different quantitative behaviour, | ||

+ | with shorter lifetimes, lower ash densities, and higher ocurrence | ||

+ | of long-period oscillators.</pre> | ||

+ | |||

+ | |||

[[Category:Math]] | [[Category:Math]] |

## Latest revision as of 02:38, 3 January 2015

(some notes for collaborative study)

- http://preshing.com/20110831/penrose-tiling-explained/
- 2 chapters from Martin Gardner's book "Penrose Tiles to Trapdoor Ciphers"

*1977* ->

A Penrose tiling can, of course, always be colored with four colors so that no two tiles of the same color share a common edge. Can it always be colored with three? It can be shown, Conway said, from the local isomorphism theorem, that if any Penrose tiling is three-colorable, all are, but so far no one has proved that any infinite Penrose tiling is three-colorable.

*2002* -<

by J. Caceres, M.E. Gegundez, A. Marquez , 2002 "In this paper the problem is solved for the last Penrose tiling for which the problem remain ..."

*note: stochastic cellular automata don't care about proof*

First, assign one of three possible colors to each tile randomly. Then, allow the cellular automata to evolve accoring to the following set of rules: * If the value fo a cell (or tile) equals the value of a bordering cell which is closer to the origin (as measured by some arbitrary point chosen within each tile), then with 90% probability, the cell changes value randomly to one of the other two colors. * If the value of a cell does not equal the value of a bordering cell which is closer to the origin, but does equal the value of a cell farther away from the origin, then with 10% probability the cell changes value * If the value of a cell does not equal the value of any bordering cell, the cell does not change value

*Conway's game of life on Penrose tiles*

Margaret Hill, Susan Stepney, and Francis Wan We compare the long term behaviour of Conway’s Game of Life cellular automaton, from initial random configurations, on a bounded rectangular grid and a bounded Penrose tiling grid. We investigate the lifetime to stability, the final ‘ash’ density, and the number and period of final oscillators. Penrose grids have similar qualitative behaviour but different quantitative behaviour, with shorter lifetimes, lower ash densities, and higher ocurrence of long-period oscillators.