On 12/12/06, Michael Reid <reid@math.ucf.edu> wrote:
some time ago, i wrote:
Before people go too far looking for such charts, I have a question: is it possible to have 4 points in a plane, no 3 colinear, so that all 6 distances are integers? And with the additional constraint that no 2 distances are equal?
it is possible to have arbitrarily many points in the plane, no three collinear, such that all distances between them are integers, and no two such distances are equal.
there hasn't been any response, except for dan's query "why is this true?". since no one has answered, here is my solution.
I developed a geometric construction, which boils down to the following: given a circle, assume that we can find a triangle with rational sides inscribed within it. Permuting two sides creates a congruent triangle, also inscribed in the circle. By Ptolemy, all distances between the four points are rational. Now continue, permuting the new triangles. True, I haven't shown that this process does not terminate. But it does apparently produce results for circles of radius sqrt(1/m) for many natural m; m = 17 is the smallest so far not observed. So I enquire Does this construction find all charts with given radius, assuming we can find an initial triangle? Can Mike Reid's algebraic construction be modified to deal with radius the reciprocal square root of arbitrary natural m? Example: for radius 1/sqrt(3) we find charts [3,5,7,7,8,3]/7, [3,5,7,7,5,8]/7, [3,5,7,8,7,7]/7, [3,7,8,8,7,5]/7, [7,8,13,13,15,7]/13, [7,8,13,13,8,15]/13, [7,8,13,15,13,13]/13, [7,13,15,15,13,8]/13, ...
i also do not know about dan's follow-up question:
| And what about the higher-dimensional analogues: | | Q3: What is the maximum number of points in R^n, no hyperplane | (i.e., n-1 dimensional plane) containing n+1 of them, such that all | interpoint distances are integers?
but i note that even with the weaker condition that no hyperplane contains all of the points, the question is still non-trivial!
Making everything concyclic no longer helps here. The condition for n+2 points to lie on a (n-1)-sphere is that the matrix with elements (s_ij)^2 be singular, where s_ij is the distance between points i,j. For n = 1 this happens to decompose into four bilinear factors, each a version of Ptolemy's theorem with various signs; but for larger n it is irreducible. Fred Lunnon