Replicating Tessellations
Andrew Vince · SIAM Journal on Discrete Mathematics · 1993
A theory of replicating tessellation of $\mathbb{R}^n $ is developed that simultaneously generalizes radix representation of integers and hexagonal addressing in computer science. The tiling aggregates tesselate Euclidean space so that the $( m + 1 )$st aggregate is, in turn, tiled by translates of the mth aggregate, for each m in exactly the same way. This induces a discrete hierarchical addressing systsem on $\mathbb{R}^n $. Necessary and sufficient conditions for the existence of replicating tessellations are given, and an efficient algorithm is provided to determine whether or not a replicating tessellation is induced. It is shown that the generalized balanced ternary is replicating in all dimensions. Each replicating tessellation yields an associated self-replicating tiling with the following properties: (1) a single tile T tesselates $\mathbb{R}^n $ periodically and (2) there is a linear map A, such that $A( T )$ is tiled by translates of T. The boundary of T is often a fractal curve.