Tiling

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(some notes for collaborative study)

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

Penrose life.gif

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.