Penrose Tiling

"Sir Roger Penrose is one of the world's most widely known mathematicians. His popular books describe his insights and speculations about the workings of the human mind and the relationships between mathematics and physics. His interests range from astrophysics and quantum mechanics to mathematical puzzles and games."

A problem from recreational mathematics to which Penrose has made a significant contribution is the tiling problem.The tiling problem is this: given a collection of polygonal shapes, is it possible to cover the whole plane using just these shapes, with no overlaps? Such an arrangement of shapes is called a tiling. A Penrose tiling has many remarkable properties, most notably:
It is

  • nonperiodic, which means that it lacks any translational symmetry. More informally, a shifted copy will never match the original exactly.
  • Any finite region in a tiling appears infinitely many times in that tiling and, in fact, in any other tiling. This property would be trivially true of a tiling with translational symmetry but is non-trivial when applied to the non-periodic Penrose tilings.
  • It is a quasicrystalQuasicrystal. Quasicrystals are structure that are both ordered and nonperiodic. They form patterns that fill all the space but lack translational symmetry. Crystallographic restriction theorem allows only 2, 3, 4, and 6-fold rotational symmetries, but quasicrystals display symmetry of other orders ….
  • implemented as a physical structure a Penrose tiling will produce Bragg diffraction; the diffractogram reveals both the underlying fivefold symmetry and the long range order. This order reflects the fact that the tilings are organized, not through translational symmetry, but rather through a process sometimes called "deflation" or "inflation."


Penrose tiling, the Fibonacci sequence and the Golden ratio

are intricately related and perhaps they should be considered as different aspects of the same phenomenon.

  • The ratio of thick to thin rhombuses in the infinite tile is the golden ratio
  • The Conway worms, sequences of neighbouring rhombuses with parallel sides, are Fibonacci ordered appearances of and and thus the Ammann bars also form Fibonacci ordered grids around each star a segmented Fibonacci spiral is formed by the sides of rhombuses.
  • The distances between repeated finite motifs in the tiling grow as Fibonacci numbers when the size of the motif increases
  • the distribution of oscillation frequencies in a Penrose tiling shows bands and gaps whose widths are in proportions expressed by f.
  • The substitution scheme introduces f as a scaling factor; its matrix is the square of the Fibonacci substitution matrix; implemented as a symbol sequence (e.g. 1?101, 0?10) this substitution produces a series of words with lengths which are the Fibonacci numbers with odd index, for , the limit being the infinite Fibonacci binary sequence.
  • The eigenvalues of the substitution matrix are f+1 (=f²) and 2-f (=1/f²).

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