David Bernier <firstname.lastname@example.org> writes: > On 10/02/2013 10:29 PM, Mike Terry wrote: > > "quasi" <email@example.com> wrote in message > > news:firstname.lastname@example.org... > >> Pubkeybreaker wrote: > >>> Susam Pal wrote: > >>>> > >>>> How can you construct a plane where every point is coloured > >>>> either black or white such that two points of the same colour > >>>> are never a unit distance apart? > >>>> > >>>> -- Originally posted at: > >>>> http://cotpi.com/p/69/ > >>>> Puzzles IRC channel: > >>>> ##puzzles on irc.freenode.net > >>>> Puzzles IRC webchat: > >>>> http://webchat.freenode.net/?channels=##puzzles > >>> > >>> It does not seem possible. Take any point. Suppose it is > >>> white. It is the center of a circle (of radius 1). All points > >>> on that circle must then be black. But given any point on > >>> that circle there is another point on the circle at distance 1. > >>> Both points are the same color. > >>> > >>> Unless, of course, I am being completely stupid. > >> > >> No, your proof is fine. > > > > 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?
2) Does Golomb's ten-vertex four-colour graph have to be rotationally symmetric?
My brief quick stabs at answers are hidden in my headers, I welcome corrections.