A Guide to Penrose Tilings by Francesco D'Andrea

Author:Francesco D'Andrea
Language: eng
Format: epub
ISBN: 9783031284281
Publisher: Springer Nature Switzerland

(4.5)

Figure 4.8 shows a portion of the Cartwheel made with rhombi, and the two endless worms crossing at the center.

Fig. 4.8 Cartwheel with rhombi

4.3.1 Musical Sequences

A sequence of b’s and w’s is called a musical sequence if every b is followed by a w and there are no more than two consecutive w’s. The reason for the name is that it reminds us of the black and white keys in a piano,

where every black key is always followed by a white one, and there are at most two consecutive white keys.

To any finite or one-sided infinite Conway worm made with rhombi we can associate a sequence of b’s and w’s: one w for each long hexagon, and one b for any short hexagon, cf. (4.5).

The same can be done for two-sided infinite worms, with a little care since there is no obvious starting point for the sequence. Given an endless worm, choose any two adjacent hexagons and number them 0 and 1. This fixes the starting point and orientation for counting hexagons in the worm. Then, we can consider the function that, for each , associates to the nth hexagon his type, black or white. The function f is our doubly-infinite sequence.

Two such sequences f and g are equivalent if there exists such that either for all , or for all (i.e. they are related by a translation and possibly a reflection). For every endless Conway worm, the equivalence class of the associated sequence is uniquely defined (it does not depend on how we start numbering hexagons).

Proposition 4.13

If a Conway worm with rhombi is a patch in a tiling of the plane, then the associated binary (b’s and w’s) sequence is a musical sequence.

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