Square Multiscale

Discovered by


Nowadays (2019) there are several generalizations of a tile substitution. One was introduced already in Sad98 Even though his idea produces a welath of examples, most of these have not yet been well studied.

The idea is to give a rule how to substitute a prototile and apply this rule always to the largest current tile. The image below may give you an idea. There are several questions about this construction. A first one is: (How) does this construction yield a tiling of the entire plane? An upcoming paper by Yotam Smilansky and Yaar Solomon is dedicated to this question.

Find an example vector graphic here.

Substitution Rule

Rule Square Multiscale


Patch Square Multiscale


Smilansky, Yotam
Uniform Distribution of Kakutani Partitions Generated By Substitution Schemes
Israel Journal of Mathematics arXiv:1805.02213

Sadun, L.
Some generalizations of the pinwheel tiling
Discrete Comput. Geom. 1998, 20,1, pp. 79–110, 1626703