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Topic: cotpi 69 - Black and white plane
Replies: 26   Last Post: Oct 9, 2013 10:23 AM

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 quasi Posts: 12,067 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?

>>

>
>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 2-coloring of Q^2 such that no two points
of Q^2 which are 1 unit apart have the same color?

quasi

Date Subject Author
10/2/13 cotpi
10/2/13 Pubkeybreaker
10/2/13 quasi
10/2/13 Mike Terry
10/6/13 David Bernier
10/6/13 David Bernier
10/6/13 David Bernier
10/6/13 David Bernier
10/6/13 David Bernier
10/6/13 quasi
10/6/13 quasi
10/6/13 David Bernier
10/6/13 David Bernier
10/6/13 quasi
10/6/13 David Bernier
10/9/13 Phil Carmody
10/9/13 David Bernier
10/2/13 Eric Lafontaine
10/2/13 Michael F. Stemper
10/2/13 quasi
10/2/13 quasi
10/2/13 Haran Pilpel
10/2/13 quasi
10/2/13 quasi
10/2/13 Ted Schuerzinger
10/3/13 William Elliot