1.4. CONTRUCTING INTERESTING TILINGS 19
Kite Dart
Figure 1.16. Penrose kite and dart tiles, with bumps to
enforce matching rules
A remarkable theorem of Shahar Mozes [Moz] says that for any substi-
tution σ involving square tiles of equal size, there exists a set S and a factor
map f : ΩS → Ωσ that is onto, measure-preserving and 1:1 away from a
negligible set (meaning a set of measure zero with respect to all translation-
invariant probability measures). Mozes’ argument was generalized by Radin
[Rad] to apply to the pinwheel tiling, and Mozes’ theorem was generalized
by Goodman-Strauss [GS] to apply to any substitution tiling (as defined in
this book).
Despite their historical significance, and despite Goodman-Strauss’ the-
orem, we will not be talking much about matching rules tiling spaces. While
powerful techniques have been developed to explore the topology of substi-
tution and cut-and-project tiling spaces, at this time the topology of most
local matching rules tiling spaces remains a mystery.
