16 1. BASIC NOTIONS , , etc. Arrow MLD to chair Table 2 1 Half-hex + rotations Penrose A 1 C C A D B D D B 1 D A C C Figure 1.14. The arrow, table, half-hex, and Penrose sub- stitutions. The letter τ denotes the golden mean (1 + √ 5)/2. p = (x0, y0) ∈ R2, as in Figure 1.15. Let L be the line through p with slope α. Let S be the strip obtained by adding points in the unit square to points in L. Equivalently, it is the union of all the unit squares with southwest corner on L. Now let Λ = Z2 be the set of points in R2 with integer coordinates, let Λ1 = Λ∩S, and let Λ2 be the orthogonal projection of Λ1 onto L. This gives a point pattern on L. We then take our tiles to be the intervals between suc- cessive points. These either have length 1/ 1 + α2 if the successive points were the projections of (m, n) and (m + 1, n), or length α/ √ 1 + α2 if the

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