That is a great problem. Is it possibly related to percolation, where you have an infinite grid of "cells" each of which is "open" or "closed", independently with prob. = p. Instead of open or closed, color the cells black and white. Then they try to answer the question: What is the inf of the p for which there exists, with probability one, an infinite connected component that is all the same color ??? (A component being an equivalence class under the relation: cell A ~ cell B when there is a monochromatic path P starting with cell A and ending with cell B, such that for each cell of P, the next cell is one sharing an edge with it.) E.g., for cells being the tiles of the hexagonal tessellation of the plane, the critical probability is p = 1/2. The field of percolation originated with models for liquid percolating through various types of rock, so maybe there is a connection with networks of electrical resistors. —Dan
On May 29, 2017, at 8:18 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Time to exhume that glorious problem about the infinite chessboard grid of resistors, perhaps?
http://www.mathpages.com/home/kmath668/kmath668.htm <http://www.mathpages.com/home/kmath668/kmath668.htm>
And get a new bathroom ... WFL
On 5/29/17, James Propp <jamespropp@gmail.com <mailto:jamespropp@gmail.com>> wrote:
Can a square-tiling of the plane, like a square-tiling of a square, be converted into an electrical network (a la Brooks, Smith, Stone, and Tutte)? That might give a picture that would be suggestive of a different name than "bathroom floor tiling".
Incidentally, I've always thought that the tiling of the plane by squares and octagons was the "bathroom floor tiling":
Me, I grew up in a house with a bathroom floor tiled by regular hexagons. Many of the bathrooms of that day (1950s) had this pattern (legend has it that this is one reason for the name of the game Hex).