
Re: cotpi 69  Black and white plane
Posted:
Oct 9, 2013 5:04 AM


David Bernier <david250@videotron.ca> writes: > On 10/02/2013 10:29 PM, Mike Terry wrote: > > "quasi" <quasi@null.set> wrote in message > > news:v9vo49lvr7j4ns7i82s869ou749lgdmrup@4ax.com... > >> 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 webpage 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.cuttheknot.org/proofs/ChromaticNumber.shtml > . > > Two or three colours won't do, from which we see that Chi >= 4. > A 7colouring of a regularhexagon tiling of the plane shows > that seven colours will do, from which we see that Chi <= 7.
Followup questions: 1) What range(s) of edgelength for said hexagon yield a valid tiling?
2) Does Golomb's tenvertex fourcolour graph have to be rotationally symmetric?
My brief quick stabs at answers are hidden in my headers, I welcome corrections.
Phil  The list of trusted root authorities in your browser included the United Arab Emiratesbased Etisalat, which was caught secretly uploading spyware onto 100,000 customers' BlackBerries. http://www.wired.com/threatlevel/2009/07/blackberryspies/

