  The main goal of this paper is to define a 1-1 correspondence between between
substitution tilings constructed by inflation and the arithmetic of positional
representation in the underlying real vector space.
  It introduces a generalization of inflationary tessellations to equivalence
classes of tiles. Two tiles belong to the same class if they share a defined
geometric property, such as equivalence under a group of isometries, having the
same measure, or having the same `decoration'. Some properties of ordinary
tessellations for which the equivalence relation is congruence with respect to
the full group of isometries are already determined by the weaker relation of
equivalence with respect to equal measure. In particular, the multiplier for an
inflationary tiling (such as a Penrose aperiodic tiling) is an algebraic
number.
  Equivalence of tiles under measure facilitates the investigation of
properties of tilings that are independent of dimension, and provides a method
for transferring tilings from one dimension to another.
  Three well-known aperiodic tilings illustrate aspects of the correspondence:
a tiling of Ammann, a Penrose tiling, and the monotiling of Taylor and
Socolar-Taylor.
