On 10/09/2013 05:04 AM, Phil Carmody wrote: > David Bernier <firstname.lastname@example.org> writes: >> On 10/02/2013 10:29 PM, Mike Terry wrote: [...]
>>> The question was asked here a few years ago whether the plane can be >>> coloured along the lines above but using three colours. The answer is no, >>> and the proof is similar but less obvious... >> [...] >> >> There is a web-page called "Chromatic Number of the Plane" by >> Alexander Bogomolny that briefly discusses the question of >> the minimum number of colours needed. >> >> The relevant definition, copied from there, is: >> ``The smallest number of colors needed in a coloring of the plane to >> ensure that no monochromatic pair is at the unit distance apart is >> called the chromatic number Chi of the plane." >> >> Ref.: >> < http://www.cut-the-knot.org/proofs/ChromaticNumber.shtml > . >> >> Two or three colours won't do, from which we see that Chi >= 4. >> A 7-colouring of a regular-hexagon tiling of the plane shows >> that seven colours will do, from which we see that Chi <= 7. > > Follow-up questions: > 1) What range(s) of edge-length for said hexagon yield a valid tiling?
I don't know the answer for that 7-color tiling. There's a tiling with squares, arranged as a brick-layer lays down bricks: [o]|[oo]|[oo]|[oo]|[oo]| [[oo]|[oo]|[oo]|[oo]|[oo]|
The bricks are square, or possibly rectangular. Someone did try to optimize based upon that design through simple variation of parameters, etc. (not really really advanced stuff), trying to get away with 6 colours ...
Well, it didn't work. There was a very very small re-occuring polygon representing asymptotically a small percentage of the area for that sort of polygon in an "increasingly large circular domain" , which absolutely needed a 7th color ... As if to say: "six colours almost does it, but ... perhaps not quite ... "
> 2) Does Golomb's ten-vertex four-colour graph have to be rotationally > symmetric?
I don't know about that graph. I've been thinking that cataloguing minimal 4-colouring unit vertex graphs (one per graph isomorphism class) might be mildly interesting.
> > My brief quick stabs at answers are hidden in my headers, I welcome > corrections. > > Phil >
-- Let us all be paranoid. More so than no such agence, Bolon Yokte K'uh willing.