Document Type

Conference Paper

Rights

This item is available under a Creative Commons License for non-commercial use only

Publication Details

This version of the paper, which was included in the IMS e-proceedings was submitted on 28th April, 2008. The subsequent oral presentaion, which was made on 20th June 2008, was based on a revised version of the paper. The significant difference between the revised 'as presented' version and this 'as submitted' version concerns the explanation of the relationship between the 4-colour rhombohedral lattice and a discrete Cartesian lattice. The 'as submitted' paper concluded that one 4-colour rhombohedral 4-tuple lattice maps onto four separate discrete cubic lattices, whereas the revised version concluded that one 4-colour rhombohedral 4-tuple lattice maps onto sixteen distinct, discrete Cartesian lattices. The revised version also included corrections to the lengths given for Cartesian lattice links in terms of rhombohedral lattice units.

Abstract

A particular discrete rhombohedral lattice consisting of four symmetrically interlaced cuboctahedral or cubic point lattices is described that is interesting because of the high degree of symmetry it exhibits. The four constituent lattices are denoted by four colours and the composite lattice is referred to as a 4-colour rhombohedral lattice. Each point of the 4-colour lattice can be referenced by an integer 4-tuple containing only the positive non-zero integers (the counting numbers). The relationship between the discrete rhombohedral lattice and a discrete Cartesian lattice is explained. Some interesting aspects of the lattice and of the counting-number 4-tuple coordinate system are pointed out.