Lattice Octahedra
- 1 January 1960
- journal article
- Published by Canadian Mathematical Society in Canadian Journal of Mathematics
- Vol. 12, 297-302
- https://doi.org/10.4153/cjm-1960-025-4
Abstract
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).Keywords
This publication has 1 reference indexed in Scilit:
- Über kritische Gitter im vierdimensionalen Raum (R 4)Monatshefte für Mathematik, 1954