Multiplication cubes and multiplication automata

TL;DR

This work generalizes multiplication tilings from two dimensions to higher-dimensional tessellations using multiplication cubes, establishing a unified framework that links tilings with multiplication automata. Central to the approach are macrotile and microtile constructions that realize topological conjugacies and factor maps between different base representations and CA performing multiplication by $oldsymbol{eta}$ in base $N$. The paper proves that macrotile operations respect the encoded real numbers and compose according to matrix multiplication, enabling a hierarchical connection between tessellations and automata. It also delivers a complete classification of the regularity of multiplication automata, showing non-regularity in broad cases and identifying regular instances, thereby clarifying the dynamical complexity of these multiplication systems.

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.

Figures (6)

• Figure 1: The multiplication cube set $T_{(2,5)}$ and a part of a tiling from $X_{(2,5)}=X_{T_{(2,5)}}$ (with the upper right corner of the cube at the origin marked by a black dot). Consecutive powers of two (starting with $4,8,16,\dots$) can be found in the tiling along diagonals that go from bottom left to top right: for an explanation of this, see Example \ref{['powersof2']} or Proposition \ref{['infShiftMul']}.
• Figure 2: The multiplication cube $\mathop{\mathrm{cube}}\nolimits_{(2,3,5)}(10)$ with the faces $\mathop{\mathrm{\tau}}\nolimits_1$ (right), $\mathop{\mathrm{\beta}}\nolimits_2$ (front) and $\mathop{\mathrm{\tau}}\nolimits_3$ (top) visible. One can also verify that these faces and their adjacent edges yield the lower-dimensional multiplication cubes $\mathop{\mathrm{cube}}\nolimits_{(3,5)}(10)$, $\mathop{\mathrm{cube}}\nolimits_{(2,5)}(3)$ and $\mathop{\mathrm{cube}}\nolimits_{(2,3)}(4)$.
• Figure 3: Left: A tessellation with $\mathop{\mathrm{cube}}\nolimits_{(2,3,5)}(10)$ positioned at the origin in $\mathbb{Z}^3$. Middle: $\mathbb{Z}^3$ as a directed graph together with labels given by $\mathop{\mathrm{cube}}\nolimits_{(2,3,5)}(10)$. Right: The weights $\mathop{\mathrm{wgt}}\nolimits_{(2,3,5)}(v)$ of the points $v\in\mathbb{Z}^3$ have been added to the grid, the point ${\mathbf{0}}$ is at the upper right with $\mathop{\mathrm{wgt}}\nolimits_{(2,3,5)}({\mathbf{0}})=1$.
• Figure 4: Let $P$, $P'$ be the two paths between two opposite corners of a square. The path integrals of a tessellation $f$ over $P$ and $P'$ are equal.
• Figure 5: A partial valid tiling $f$ using $T_{(2,5)}$. There is no way to complete the tiling, because the path integral around the non-tiled part is not zero.

Theorems & Definitions (123)

• Definition 2.1
• Definition 2.2
• Remark 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5