Gluing and cutting cube tiling codes in dimension six

Abstract

Let S be a set of arbitrary objects, and let s → s ′ be a permutation of S such that s ′′ = (s ′ ) ′ = s and sand they form a twin pair if v ′ i = w i and v j = w j for every j ∈periodic cube tiling of d and a cube tiling of flat torus 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 ∈ [d ] \ {i } and u i = * , where * ∈ 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 ∈ [d ] \ {i }. If V , W ⊂ 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.