Cubes of integral vectors in dimension four

TL;DR

This paper characterizes and counts m-icubes in four-dimensional integer space using Hurwitz quaternions, demonstrating the possibility of infinite extension of such structures.

Contribution

It provides a comprehensive description and enumeration of m-icubes in b4^4, and proves that they can be extended infinitely.

Findings

Describes m-icubes in b4^4 for 2 d7 m d7 4 using Hurwitz quaternions

Counts the number of m-icubes with a given edge length

Proves that m-icubes can be extended infinitely in b4^4

Abstract

A system of $m$ nonzero vectors in $\mathbb{Z}^n$ is called an $m$-icube if they are pairwise orthogonal and have the same length. The paper describes $m$-icubes in $\mathbb{Z}^4$ for $2\le m\le 4$ using Hurwitz integral quaternions, counts the number of them with given edge length, and proves that unlimited extension is possible in $\mathbb{Z}^4$.