Extended Substitution Tilings
In this post I preset a novel class of quasiperiodic tilings which I call Extended Substitution Tilings (EST). EST may be used to solve certain problems which are unsolvable using a standard substitution tiling.
In a standard substitution tiling, inflation proceeds by applying the substitution rules n times. As n tends toward infinity, so the tiling expands to fill the entire plane. In an EST there are two sets of substitution rules, termed recursive(R) and terminal(T). Inflation proceeds in two corresponding phases: recursive and terminal. In the recursive phase R is applied n times. Then in the terminal phase T is applied just once. As n tends toward infinity, the tiling expands to fill the entire plane.
Both R and T have the same set of prototiles, but the set of tiles used within the prototiles may be different. Provided some basic preconditions are met, an EST generates a tiling that is quasiperiodic.
Example
The example I’ve chosen to illustrate this concept is a tiling I call Hexagon Boat. For more detail on how I found this tiling see my upcoming blog post.
Recursive Rule
[pic recursive rule]
Terminal Rule
[pic terminal rule]
Patch
[pic patch]
TODO:
- discuss pre-conditions