Let Ai, A2, … , An be n linearly independent points in n-dimensional Euclidean space of a lattice Λ. The points ± A1, ±A2, . . , ±An define a closed n-dimensional octahedron (or “cross poly tope“) K with centre at the origin O. Our problem is to find a basis for the lattices Λ which have no points in K except ±A1, ±A2, … , ±An.

Let the position of a point P in space be defined vectorially by

1

where the p are real numbers. We have the following results.

When n = 2, it is well known that a basis is

2

When n = 3, Minkowski (1) proved that there are two types of lattices, with respective bases

3

When n = 4, there are six essentially different bases typified by A1, A2, A3 and one of

4

In all expressions of this kind, the signs are independent of each other and of any other signs. This result is a restatement of a result by Brunngraber (2) and a proof is given by Wolff (3).