New perspectives on forbidden symmetries, quasicrystals, and Penrose tilings

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   Figure 1

   Quasiperiodic tiling can be forced using a single tile, the marked decagons shown in a. Matching rules demand that two decagons may overlap, as shown in b, only if shaded regions overlap and the overlap area is greater than or equal to the hexagonal overlap region indicated as A. This permits two possible types of overlap between neighbors: either small (A type) or large (B type), as shown in b. If each decagon is inscribed with an fat rhombus, as shown in c, a tiling of overlapping decagons (d Left) can be transformed into a Penrose tiling (d Right), where space for the skinny rhombi incorporated.

   
   
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   Figure 2

   Cluster C consists of five fat and two skinny rhombi with two side hexagons composed of two fat and one skinny rhombus each. There are two possible configurations for filling each side hexagon; the two possibilities are shown with dashed lines on either side in a. Under deflation, each C cluster can be replaced by a single “deflated” fat rhombus, as shown in b. There is a configuration of nine C clusters shown in c (thin lines) that, under deflation, forms a scaled-up C configuration (medium lines), called a DC cluster. Under double-deflation, each DC cluster is replaced by “doubly deflated” fat rhombus (thick lines).

   
   
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   Figure 3

   Associated with each C cluster is core area (with area 3τ + 2) consisting of a kite-shaped region, shown as shaded in a. In a Penrose tiling, core areas of neighboring tiles either join edge-to-edge (A overlap) or overlap by a fixed amount (B overlap), as shown with dark shading in b. c is a DC cluster that illustrates the core areas of the nine C clusters that compose it. An isolated DC cluster contains one B overlap (see dark shading) and the rest A overlaps.