We will also introduce below the concept of tilings and tiles.
Many of the common features of Penrose tilings follow from a hierarchical pentagonal structure given by substitution rules: this is often referred to as inflation and deflation, or composition and decomposition, of tilings or (collections of) tiles.
The durability of tilings is affected by the size and type of ceramic tiles.
The construction of tilings from an IFS can be generalized to the construction of tilings from a graph IFS.
In the case of tilings of Euclidean space, the basin is the entire Euclidean space.
The use of the inverses of the IFS functions in the study of tilings is well-established, in particular in many of the references cited above.
Subdivision rules give a sequence of tilings of a surface, and tilings give an idea of distance, length, and area (by letting each tile have length and area 1).
The aesthetic value of tilings has long been appreciated, and remains a source of interest in them; here the visual appearance (rather than the formal defining properties) of Penrose tilings has attracted attention.
The Combinatorial Riemann Mapping Theorem implies that a group G acts geometrically on \mathbb{H}^3 if and only if it is Gromov hyperbolic, it has a sphere at infinity, and the natural subdivision rule on the sphere gives rise to a sequence of tilings that is conformal in the sense above.
Given any subdivision complex for a subdivision rule, we can subdivide it over and over again to get a sequence of tilings.
