A theory of replicating tessellation of Rⁿ 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 Rⁿ. 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 Rⁿ 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.
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Andrew Vince (1993) studied this question.