Limit-Periodic Tilings

tiling

half-hexagon

arrowed

Viererbaum

Socolar-Taylor

Penrose

projection

full color

Sierpinski parity

second parity

combined parity

branch permutations

I

j

I

i

I

j

This Demonstration lets you explore the similarities between the Socolar–Taylor tiling [1], the Penrose

1+ϵ+

2

ϵ

tiling [2], and the relatively new viererbaum (or quadtree) tiling [3]. 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

{I,i,j,k}

. 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

I,i,I,k,j,…

, which allow us to call tilings

n

-close whenever branch-permutation sequences agree for the first

n

terms.