Gluing and Cutting Cube Tiling Codes in Dimension Six

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 $$s'\ne s$$ . Let $$S^d=\{v_1\dots v_d:v_i\in S\}$$ . Two words $$v,w\in S^d$$ are dichotomous if $$v_i=w'_i$$ for some $$i\in [d]=\{1,\dots ,d\}$$ , 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 distinct 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 ith 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 $$*\notin S$$ is some extra fixed symbol. A word u with $$u_i=*$$ is cut (at the ith position) if u is replaced by a twin pair q, t such that $$q_i=t_i'\ne *$$ 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.