From: Robin Chapman
Newsgroups: rec.puzzles,sci.math
Subject: Re: Integer coordinates
Date: Thu, 11 Jun 1998 14:04:18 GMT
In article <357DCB82.66C9@pratique.fr>,
JeanSeB wrote:
>
> It is not possible to draw an equilateral triangle in 2D space whose
> vertices have integer coordinates.
>
> This is possible in 3D space: ((0,0,1),(0,1,0),(1,0,0))
>
> It is also possible to draw such a tetrahedron in 3D space, if you add
> vertex (1,1,1)
>
> Is it possible to draw a regular simplex in 4D space, such that all
> coordinates are integers?
>
> A simplex ia an hyper-tetrahedron with 5 equidistant vertices.
>
> In other words, is it possible to find 5 points (Ai) with integer
> coordinates in 4D space, such that
> dist(Ai,Aj)=K
>
> Or to prove that such a simplex does not exist.
>
> Is this problem still open?
This is a good problem. It reduces to a question in the theory of rational
quadratic forms. Let's ask for which n an n-simplex can be embedded n-space
with integral coordinates. The answer is if and only if
(i) n + 1 is an odd square, or
(ii) n + 1 is a sum of 2 odd squares, or
(iii) n + 1 = 0 (mod 4).
So for the first few values we have
YES 1, 3, 7, 8, 9, 11, 12, 15, 16, 19, 23, 24, 25, ...
NO 2, 4, 5, 6, 10, 13, 14, 17, 18, 20, 21, 22, ...
It's equivalent to consider rational coordinates, as we can scale, and we can
also translate to put one of the vertices at the origin. Let v_1, ..., v_n be
the other vertices. Then for some rational number m we have v_i.v_i = 2m and
v_i.v_j = m for i <> j. This means that the quadratic forms Q_1 = x_1^2 +
x_2^2 + ... + x_n^2 and 2m Q_2 where Q_2 = x_1^2 + x_1 x_2 + x_1 x_3 + ... +
x_1 x_n + x_2^2 + x_2 x_3 + ... + x_n^2 are equivalent over the rationals.
Indeed this is a necessary and sufficient condition. One can use the
Hasse-Minkowski theory of rational quadratic forms to determine when an m
exists so that this is the case, and doing so leads, after some effort, to
the stated condition.
For the Hasse-Minkowski theory, see e.g., Serre's A Course in Arithmetic.
Robin Chapman + "They did not have proper
Department of Mathematics - palms at home in Exeter."
University of Exeter, EX4 4QE, UK +
rjc@maths.exeter.ac.uk - Peter Carey,
http://www.maths.ex.ac.uk/~rjc/rjc.html + Oscar and Lucinda
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