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 the vector graphic here.

[YS20]

Smilansky, Yotam

**Uniform Distribution of Kakutani Partitions Generated By Substitution Schemes**

*Israel Journal of Mathematics*
arXiv:1805.02213

[Sad98]

Sadun, L.

**Some generalizations of the pinwheel tiling**

*Discrete Comput. Geom.*
1998,
20,1,
pp. 79–110,
1626703