Robinson Triangle Decompositions
The substitution method for both P2 and P3 tilings can be described using Robinson triangles of different sizes. The Robinson triangles arising in P2 tilings (by bisecting kites and darts) are called A-tiles, while those arising in the P3 tilings (by bisecting rhombs) are called B-tiles. The smaller A-tile, denoted AS, is an obtuse Robinson triangle, while the larger A-tile, AL, is acute; in contrast, a smaller B-tile, denoted BS, is an acute Robinson triangle, while the larger B-tile, BL, is obtuse.
Concretely, if AS has side lengths (1, 1, φ), then AL has side lengths (φ, φ, 1). B-tiles can be related to such A-tiles in two ways:
- If BS has the same size as AL then BL is an enlarged version φAS of AS, with side lengths (φ, φ, φ2=1+φ) – this decomposes into an AL tile and AS tile joined along a common side of length 1.
- If instead BL is identified with AS, then BS is a reduced version (1/φ)AL of AL with side lengths (1/φ,1/φ,1) – joining a BS tile and a BL tile along a common side of length 1 then yields (a decomposition of) an AL tile.
In these decompositions, there appears to be an ambiguity: Robinson triangles may be decomposed in two ways, which are mirror images of each other in the (isosceles) axis of symmetry of the triangle. In a Penrose tiling, this choice is fixed by the matching rules – furthermore, the matching rules also determine how the smaller triangles in the tiling compose to give larger ones.Partial inflation of star to yield rhombs, and of a collection of rhombs to yield an ace.
It follows that the P2 and P3 tilings are mutually locally derivable: a tiling by one set of tiles can be used to generate a tiling by another – for example a tiling by kites and darts may be subdivided into A-tiles, and these can be composed in a canonical way to form B-tiles and hence rhombs. The P2 and P3 tilings are also both mutually locally derivable with the P1 tiling (see figure 2 above).
The decomposition of B-tiles into A-tiles may be written
- BS = AL, BL = AL + AS
(assuming the larger size convention for the B-tiles), which can be summarized in a substitution matrix equation:
Combining this with the decomposition of enlarged φA-tiles into B-tiles yields the substitution
so that the enlarged tile φAL decomposes into two AL tiles and one AS tiles. The matching rules force a particular substitution: the two AL tiles in a φAL tile must form a kite – thus a kite decomposes into two kites and a two half-darts, and a dart decomposes into a kite and two half-darts. Enlarged φB-tiles decompose into B-tiles in a similar way (via φA-tiles).
Composition and decomposition can be iterated, so that, for example
The number of kites and darts in the nth iteration of the construction is determined by the nth power of the substitution matrix:
where Fn is the nth Fibonacci number. The ratio of numbers of kites to darts in any sufficiently large P2 Penrose tiling pattern therefore approximates to the golden ratio φ. A similar result holds for the ratio of the number of thick rhombs to thin rhombs in the P3 Penrose tiling.
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