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Tilings for Pisot beta numeration

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Figure 1. The patch β − 2 R (0) of the aperiodic tiling C aper and the corre- corre-sponding integral beta-tiles, β 2 = 2β + 2.
Figure 4. Periodic tiling ˜ C ext ⊆ C ext , β 2 = 2β +2. On the left, the following tiles are represented: X (yellow), X + δ(1) (purple), X + δ(β − 2) (brown), X +δ(β − 1) (light green)
Figure 5. Patch of the periodic tiling R (0) + δ ′ ( h β − 2, β 2 − 2β i Z ) induced by β 3 = 2β 2 − β + 1.
Figure 6. The pruned boundary graph (after merging the states with same middle component) for β 2 = aβ + b, a ≥ b ≥ 1.
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