**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 A_{S}, is an obtuse Robinson triangle, while the larger A-tile, A_{L}, is acute; in contrast, a smaller B-tile, denoted B_{S}, is an acute Robinson triangle, while the larger B-tile, B_{L}, is obtuse.

Concretely, if A_{S} has side lengths (1, 1, *φ*), then A_{L} has side lengths (*φ*, *φ*, 1). B-tiles can be related to such A-tiles in two ways:

- If B
_{S}has the same size as A_{L}then B_{L}is an enlarged version*φ*A_{S}of A_{S}, with side lengths (*φ*,*φ*,*φ*2=1+*φ*) – this decomposes into an A_{L}tile and A_{S}tile joined along a common side of length 1. - If instead B
_{L}is identified with A_{S}, then B_{S}is a reduced version (1/*φ*)A_{L}of A_{L}with side lengths (1/*φ*,1/*φ*,1) – joining a B_{S}tile and a B_{L}tile along a common side of length 1 then yields (a decomposition of) an A_{L}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.

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

- B
_{S}= A_{L}, B_{L}= A_{L}+ A_{S}

(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 *φ*A-tiles into B-tiles yields the substitution

so that the enlarged tile *φ*A_{L} decomposes into two A_{L} tiles and one A_{S} tiles. The matching rules force a particular substitution: the two A_{L} tiles in a *φ*A_{L} 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 *φ*B-tiles decompose into B-tiles in a similar way (via *φ*A-tiles).

Composition and decomposition can be iterated, so that, for example

The number of kites and darts in the *n*th iteration of the construction is determined by the *n*th power of the substitution matrix:

where *F*_{n} is the *n*th 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.

Read more about this topic: Penrose Tiling, Features and Constructions, Inflation and Deflation

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