This Demonstration lets you explore the similarities between the Socolar–Taylor tiling , the Penrose
tiling , and the relatively new viererbaum (or quadtree) tiling . Each of these tilings is an areal inflation-factor four substitution tiling and also a limit-periodic tiling. Here we construct each uniquely displaced tiling as a sequence of periodic approximants. Each periodic approximant is determined by a sequence of branch permutations taken from the set
. The set elements are also actions of the Klein four-group.
The limit periodic construction has a number of advantages over the more common construction using substitution rules. It lends itself to a graphical proof that the various matching rules enforce aperiodicity. It explicitly shows the limit periodic structure. Finally, it introduces sequences of branch permutations such as
, which allow us to call tilings
-close whenever branch-permutation sequences agree for the first