quasi
Posts:
12,021
Registered:
7/15/05


Re: cotpi 69  Black and white plane
Posted:
Oct 2, 2013 3:38 PM


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.
quasi

