Multiplication cubes and multiplication automata

TL;DR

This paper generalizes two-dimensional multiplication tiling systems to higher dimensions, establishing connections between different systems through macrotile operations that create conjugacies and factor maps in cellular automata.

Contribution

It introduces higher-dimensional multiplication tessellation systems and develops a theory linking them via macrotile operations, revealing new topological conjugacies.

Findings

Extended multiplication tiling systems to higher dimensions.

Established macrotile operations linking different tessellations.

Demonstrated conjugacies and factor maps between cellular automata.

Abstract

We extend previously known two-dimensional multiplication tiling systems that simulate multiplication by two natural numbers $p$ and $q$ in base $pq$ to higher dimensional multiplication tessellation systems. We develop the theory of these systems and link different multiplication tessellation systems with each other via macrotile operations that glue cubes in one tessellation system into larger cubes of another tessellation system. The macrotile operations yield topological conjugacies and factor maps between cellular automata performing multiplication by positive numbers in various bases.