Maximal Sets of Unit-distant Points
by
Christian Elsholtz
Department of Mathematics, Royal Holloway, University of London
Coauthors: W. Klotz (Clausthal)

We study maximal sets of points mutually distance 1 apart. Let F denote a field, and f(Fⁿ) be the cardinality of a maximal set in dimension n. The regular simplex shows that f(Rⁿ)=n+1. For which n can this simplex be rotated such that all coordinates are rational? A full evaluation of f(Qⁿ) is given, depending only on the prime factorizations of n and n+1. Our results imply that for almost all even n one has f(Qⁿ)=n and for almost all odd n one has f(Qⁿ)=n-1.

This apparently geometric or algebraic question is solved by methods from number theory and design theory. We also study the case of general fields.

Date received: April 30, 2004