Gluing and cutting cube tiling codes in dimension six

TL;DR

This paper demonstrates that in six-dimensional space, any two cube tiling codes can be transformed into each other through a sequence of gluing and cutting operations, revealing a fundamental connectivity in their structure.

Contribution

The paper introduces a method to connect any two six-dimensional cube tiling codes via gluing and cutting operations, establishing a new understanding of their structural relationships.

Findings

Any two six-dimensional cube tiling codes are interconnected through gluing and cutting sequences.

The operations of gluing and cutting can transform one cube tiling code into another in dimension six.

The result provides insights into the combinatorial structure of cube tilings in higher dimensions.

Abstract

Let $S$ be a set of arbitrary objects, and let $s\mapsto s'$ be a permutation of $S$ such that $s"=(s')'=s$ and $s'\neq s$. Let $S^d=\{v_1...v_d\colon v_i\in S\}$. Two words $v,w\in S^d$ are dichotomous if $v_i=w'_i$ for some $i\in [d]$, and they form a twin pair if $v_i'=w_i$ and $v_j=w_j$ for every $j\in [d]\setminus \{i\}$. A polybox code is a set $V\subset S^d$ in which every two words are dichotomous. A polybox code $V$ is a cube tiling code if $|V|=2^d$. A $2$-periodic cube tiling of $\mathbb{R}^d$ and a cube tiling of flat torus $\mathbb{T}^d$ can be encoded in a form of a cube tiling code. A twin pair $v,w$ in which $v_i=w_i'$ is glue (at the $i$th position) if the pair $v,w$ is replaced by one word $u$ such that $u_j=v_j=w_j$ for every $j\in [d]\setminus \{i\}$ and $u_i=*$, where $*\not\in S$ is some extra fixed symbol. A word $u$ with $u_i=*$ is cut (at the $i$th position) if $u$ is replaced by a twin pair $q,t$ such that $q_i=t_i'$ and $u_j=q_j=t_j$ for every $j\in [d]\setminus \{i\}$. If $V,W\subset S^d$ are two cube tiling codes and there is a sequence of twin pairs which can be interchangeably gluing and cutting in a way which allows us to pass from $V$ to $W$, then we say that $W$ is obtained from $V$ by gluing and cutting. In the paper it is shown that for every two cube tiling codes in dimension six one can be obtained from the other by gluing and cutting.