Multiplication cubes and multiplication automata
Abstract
We extend previously known two-dimensional multiplication tiling systems that simulate multiplication by two natural numbers and in base 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.
Cite
@article{arxiv.2211.15293,
title = {Multiplication cubes and multiplication automata},
author = {Johan Kopra},
journal= {arXiv preprint arXiv:2211.15293},
year = {2025}
}
Comments
40 pages, 6 figures. The part from the end of Example 4.21 to the start of Subsection 4.4 is new. Additionally, added Propositions 4.30, 4.37, 5.8, Corollary 4.39, Theorem 5.14. Augmented statements for Lemmas 4.29, 4.36, Theorems 5.9, 5.12, Corollaries 5.10, 5.13. A different, shorter proof given for Theorem 4.40. Other minor improvements