Paving

Periodic pavings

Periodic pavings of the plan or space are known since antiquity and were often used like decorative reasons in architecture. In Crystallography, these pavings model periodic arrangements of atoms (crystals). In 1891, the crystallographer and Russian mathematician Fedorov (University of Saint-Petersbourg) showed that there existed only 17 types of periodic pavings of the plan (two pavings are in the same way standard if they are invariant by same the Groupe of Isométrie, i.e. by axial rotations, symmetries and translations). All these types, except two, can be realized by pavings whose tiles all are of the regular polygons. The Alhambra de Grenade is considered to contain mosaics illustrating all these types of pavings.

Aperiodic pavings

It was thought a long time that only pavings by translations of the plan were necessarily periodic. In particular, Hao Wang conjectured into 1961 that it was the case, and in deduced that one could conceive a Computer program which would decide if a set of tiles given made it possible to pave or not the plan. However, in 1964, Robert Berger (a pupil of Wang) found a whole of 20426 tiles being able to pave only apériodiquement the plan. The conjecture is thus false: to know if a set of tiles can pave or not the plan is Indécidable.

Plays increasingly smaller of tiles paving only apériodiquement for summer have found:

in 1976, Raphael Robinson simplifies the set of tiles of Robert Berger in a set of 24 tiles (6 rotating near);
in 1974, Roger Penrose, following an order to create a puzzle, finds a set of 20 tiles (2 rotating near);
in 1996, Karel Culik and Jarkko Kari found (by a completely different method) a set of 13 tiles.

One can note that in 1994 John Horton Conway and Charles Radin found a play comprising an infinity of tiles but which, rotating close, is reduced to a single tile: a right-angled triangle on sides 1,2 and $\ sqrt 5$ . Paving obtained is known under the name of Pinwheel.

Pavings quasiperiodic

Among aperiodic pavings, some are it less than others… in other words, one can quantify the degree of aperiodicity. In this way, one can quote for example the concepts of recurrence and uniform recurrence (or quasiperiodicity ). A paving is known as recurring if, when a reason (finished tiles together) appears once, it appears in any sufficiently large zone. If, moreover, one can fix the size of this zone according to the size of the reason, then paving is known as uniformly recurring (or quasiperiodic).

Thus, a paving uniformly recurring of the plan is such as if one considers an any reason appearing in a circle of radius R traced on paving, then it exists a number R such as one can be sure that this reason reappears in any circle of radius R traced on paving.

In particular, periodic pavings are uniformly recurring ( a fortiori recurring). It is also the case of the Pavage of Penrose. In fact, one can show that if a play of tile paves the plan, then it can also pave it in a way uniformly recurring (the proof rests on a diagonal Argument).

See too

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