Enzymind Labs

A 2-D Thue-Morse (TM) pattern has rows and columns that are 1-D Thue-Morse sequences. In TM tilings, each symbol of a pattern is replaced by an image, and these images define the tiling motif.

There is only a single two symbol 2-D (and higher dimension) TM pattern. But with three and more symbols more than one array may be TM. These can be specified by recursive string rewriting system, such as an L-System used here.

2 symbol, 2-D TM L-System:
symbols : A, B
start : A
rules : A → [AB;BA], B → [BA;AB]

where ";" indicates a new row of a matrix.

In general, a 2-D Thue-Morse L-System has these restrictions:

1) Pairs of symbols, either horizontally or diagonally, don't appear in the rules arrays.

2) Each symbol is replaced by a pattern that has all symbols with equal frequency (NOTE: I've got to check whether this is strictly true, or is a consequence of (3) below.)

3) The rule patterns are orthogonal, in that no pair of patterns has the same symbol in the same location.

An example of a four symbol 2-D TM pattern:

symbols : A, B, C, D
start : A
rules : A → [AB;CD], B → [BC;DA], C → [CD;AB], D → [DA;BC]