# The Mathematical Tourist

4/30/07

## Integral Heptagons

It isn't hard to find three points such that the distance between each pair of points is an integer. Three points defining a right triangle with sides 3, 4, and 5 represent one such example. Triangles characterized by Pythagorean triples and many other triangles exhibit such integral relationships.

It's not so clear that there are sets of four points in which the distance between each pair is an integer, but there are. It's even less clear for five, six, seven, or more points.

The problem of finding sets of points with all mutual distances integers has intrigued many mathematicians, including Abram Besicovitch (1891–1970) and Paul Erdos (1913–1996). Erdős originally asked for five points in the plane, no three on a line, no four on a circle with the distance between each pair of points an integer.

When that problem was solved, six points became the target. There proved to be infinite families of such point sets.

Now, the seven-point case has been solved. Using an exhaustive computer search, Tobias Kreisel and Sascha Kurz of the University of Bayreuth found a integral heptagon, in which no three points lie on a line and no four points lie on a circle. In fact, they came up with two examples.

The following table gives the distances between the pairs of points in the smallest possible integral heptagon.

 0 22270 22098 16637 9248 8908 8636 22270 0 21488 11397 15138 20698 13746 22098 21488 0 10795 14450 13430 20066 16637 11397 10795 0 7395 11135 11049 9248 15138 14450 7395 0 5780 5916 8908 20698 13430 11135 5780 0 10744 8636 13746 20066 11049 5916 10744 0

For a diagram, see Ed Pegg's current Math Games column.

In each case, you can also look the smallest possible diameter, d, where the diameter is the largest occurring distance in a point set. For four points, d = 8; for five points, d = 73, and for six points, d = 174. The new results show that, for seven points, d = 22,270.

The new target? Are there eight points in the plane, no three on a line, no four on a circle with pairwise integral distances?

Comments are welcome. You can reach Ivars Peterson at ipeterson@maa.org.

References:

Brass, P., W. Moser, and J. Pach. 2005. Research Problems in Discrete Geometry. New York: Springer.

Guy, R.K. 1994. Unsolved Problems in Number Theory, 2nd. ed. New York: Springer.

Kreisel, T., and S. Kurz. Preprint. There are integral heptagons, no three points on a line, no four on a circle.

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