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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 3:38 PM
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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

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
10/2/13 Mark Brader
10/3/13 William Elliot

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