quasi
Posts:
12,023
Registered:
7/15/05


Re: cotpi 69  Black and white plane
Posted:
Oct 2, 2013 8:14 PM


quasi wrote: >Michael F. Stemper wrote: >>Eric Lafontaine wrote: >>>cotpi 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? >>> >>> I must have missed something? >>> Place an equilateral triangle anywhere on the plane >>> Set one of its corners white, another one black >>> What do you do with the third one? >> >>That was my reaction, as well. >> >>> Next question: what kind of plane makes equilateral >>> triangles impossible? >> >>How about the Gaussian integers? > >Geometrically, we can cast the underlying set of the ring Z[i] >as the subset Z^2 of R^2. > >Just color the point (x,y) white if x + y is even, black if >x + y is odd. > >>Is there something similar to the Gaussian integers, but for >>rationals? In other words, {a+bi  a,b in Q} > >The field Q(i). > >Geometrically, we can cast the underlying set of the field Q(i) >as the subset Q^2 of R^2. > >>You can't have equilateral triangles in whatever this would >>be called, > >Right. > >>but would it have the property initially specified? > >Good question.
Actually, I think it's a great question.
I'll state it in full ...
Question:
Does there exist a 2coloring of Q^2 such that no two points of Q^2 which are 1 unit apart have the same color?
quasi

