Penrose tiling

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A Penrose tiling is a non-periodic tiling generated by an aperiodic set of prototiles named after Sir Roger Penrose, who investigated these sets in the 1970s. Because all tilings obtained with the Penrose tiles are non-periodic, Penrose tiles are considered aperiodic tiles. A Penrose tiling may be constructed so as to exhibit both reflection symmetry and fivefold rotational symmetry, as in the diagram at the right.

A Penrose tiling has many remarkable properties, most notably:

  • It is non-periodic, which means that it lacks any translational symmetry. More informally, a shifted copy will never match the original.
  • It is self-similar, so the same patterns occur at larger and larger scales. Thus, the tiling can be obtained through "inflation" (or "deflation") and any finite patch from the tiling occurs infinitely many times.
  • It is a quasicrystal: implemented as a physical structure a Penrose tiling will produce Bragg diffraction and its diffractogram reveals both the fivefold symmetry and the underlying long range order.

Various methods to construct Penrose tilings have been discovered, including matching rules, substitutions, cut and project schemes and coverings.

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