An Exploration of a Discrete Rhombohedral Lattice; of Possible Engineering or Physical Relevance (Revised Version)

TL;DR

This paper describes a highly symmetric discrete rhombohedral lattice composed of interlaced cuboctahedral point lattices, exploring its structure, coordinate system, and potential physical or engineering relevance.

Contribution

It introduces a novel 4-colour rhombohedral lattice with a unique coordinate system and analyzes its relationship to Cartesian lattices.

Findings

High symmetry of the lattice is demonstrated.

Coordinate system based on positive integers is established.

Relationship to Cartesian lattices is clarified.

Abstract

A particular discrete rhombohedral lattice consisting of four symmetrically interlaced cuboctahedral point lattices is described that is interesting because of the high degree of symmetry it exhibits. The four constituent cuboctahedral 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.