Summary: 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 \(m\)th aggregate, for each \(m\) in exactly the same way. This induces a discrete hierarchical addressing system 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 replication 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.