Penrose tilings

Penrose tiling with a fivefold symmetry axis consisting of two kinds of rhomboid tiles... 

What is the point group of the PENROSE tiling (Fig. 3.13) if it consists of one layer of tiles What is the point group if two layers are stacked with their midpoints one on top of the other, the second layer being rotated by 180° ... 

Because a regular triacontahedron can be geometrically decomposed into ten prolate and ten oblate rhombohedra, the 1/1 and 2/1 ACs can also be viewed as two different types of periodic condensations of prolate and oblate rhombohedral building blocks. In this way, a link between AC structures and 3D Penrose tiles [93] used for i-QC modeling becomes evident. Therefore, the local atomic orders within and the linkages among triacontahedra are very useful in QC modeling. 

Penrose tiling A penrose tiling consists of rhombi (a rhombus has all four sides the same measure) that appear to have no pattern or symmetry but that, in fact, have repeated patterns within the tiling. 

In structure, the quasicrystal relates to the Penrose tile structures (polygon), originally proposed by Roger Penrose, a mathematician at Oxford University. See Crystal. 

Penrose tiling with a tile having a 36° interior angle and another with a 72° interior angle. 

An electron diffraction pattern of the aluminum-manganese alloy and a computed Fourier pattern of a three-dimensional Penrose tiling are shown in Figure 2.11. 

Electron diffraction pattern of an AIMn quasicrystal along the fivefold axis (left) and a computed Fourier pattern of a three-dimensional Penrose tiling (right). From C. Janot, Quasicrystals, A Primer, 2nd ed. (London Oxford Univ. Press, 1994), p. 3, figure 1.24. 

Figure 1.11 Portion of a Penrose tiling based on two rhombuses. Penrose tilings are nonperiodic tilings of the plane and are two-dimensional analogs of quasicrystals. (Diagram created by the free Windows application Bob s Rhombus Walker, v. 3.0.19, JKS Software, Stamford, CT.)...

Figure 1-12. Roger Penrose, 2000 (photograph by the authors) and a Penrose tiling.

Figure 1-13. AlanL. Mackay, 1982 (photograph by the authors) and his simulated electron diffraction pattern of three-dimensional Penrose tiling [31] (photograph courtesy of Alan Mackay, London).

Nevertheless, it must be pointed out that Pauling s explanation has not been generally accepted. Indeed, alternative and widely accepted rationalizations invoking such concepts as Penrose tiling and icosahedral crystals in hyperspaces have been advanced to explain the anomalous existence of the quasicrystals [95-103]. 

Using your favorite Internet search engine, type in "Escher Web Sketch." Many sites have this special (and free) online "sketching" interface that lets you create your own tessellations While online, research M.C. Escher, Penrose tiling, and the Golden Mean. 

Figure 3.17 Kites and darts (a) a rhombus with each edge equal to the golden ratio, GR (b) a kite, with arcs that force an aperiodic Penrose tiling when matched (c) a dart, with arcs that force an aperiodic Penrose tiling when matched (d) five kites make up a decagon...

Although, at first glance, it seems that a unit cell can be found, it turns out that this is never possible, Details of how to construct Figure 18 and other Penrose tilings is given in the Bibliography. 

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