Pinwheel tiling

Pinwheel tiling

The pinwheel tiling is an aperiodic tiling proposed by John Conway and Charles Radin.It is constructed with a right triangle which appears in infinitely many orientations. This is its most remarkable feature, which was expressly sought by Radin. The first example with this property was proposed by Filipo Cesi, who used four tiles (two squares with incommensurate sides, a rectangle, and a triangle). [Radin, C., Aperiodic tilings, ergodic theory and rotations, in The Mathematics of long-range aperiodic order, ed. by V.Moody, NATO ASI series vol.489 (1997) pp.499–519] Conway proposed a solution using just one triangular prototile with dimensions 1,2, sqrt 5. If tile flipping is not allowed there should be right-handed and left-handed versions of the shape. The tiles do not match only edge-to-edge, but vertex-to-edge configurations occur. The full set of matching rules [cite journal | author = Radin, C. | title = The Pinwheel Tilings of the Plane | journal = The figure shows how a single tile is recomposed from five smaller tiles. Their type, left (L) or right (R), is indicated in subscripts.

Radin introduced the notion of statistical symmetry to describe the distribution of tile orientations. For a domino tile there are just two possible orientations, in a Penrose tiling they are ten, and in the pinwheel they are an infinite set. This happens when the basic triangle has an angle which is not a rational fraction of π, e.g. arctan(2). The tiling is not a quasicrystal and it cannot be obtained as a projection from a simple higher dimensional lattice. However, all the vertices have rational coordinates. Being obtained from substitutions, the pinwheel tiling can also be seen as a fractal. If at each iteration step the middle triangle is discarded, a fractal object with Hausdorff dimension: d = frac{ln 4}{ln sqrt 5}approx 1.7227is obtained.

Radin and Conway proposed a three dimensional analogue which was dubbed the quaquaversal tiling. [Radin, C., Conway, J., Quaquaversal tiling and rotations, preprint, Princeton University Press, 1995] There are other variants and generalizations of the original idea. [cite journal | author = Sadun, L. | title = Some Generalizations of the Pinwheel Tiling | journal = Discrete and Computational Geometry | volume = 20 | issue = 1 | pages = pp.79–110 | publisher = | location = | date = January 1998 | url = http://citeseer.ist.psu.edu/sadun96some.html | format = PDF/PostScript | accessdate = 2007-10-25 ]

References

External links

* [http://tilings.math.uni-bielefeld.de/tilings/substitution_rules/pinwheel Pinwheel] at the Tilings Encyclopedia


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Quaquaversal tiling — The quaquaveral tiling is an nonperiodic three dimensional substitution tiling of space proposed by John Conway and Charles Radin. The basic solid tiles are half prisms arranged in a pattern which relies essentially on their previous construct,… …   Wikipedia

  • Aperiodic tiling — are an aperiodic set of tiles, since they admit only non periodic tilings of the plane:] Any of the infinitely many tilings by the Penrose tiles is non periodic. More informally, many refer to the Penrose tilings as being aperiodic tilings , but… …   Wikipedia

  • Liste De Fractales Par Dimension De Hausdorff — Cet article est une liste de fractales, ordonnées par dimension de Hausdorff croissante. En mathématiques, une fractale est un ensemble dont la dimension de Hausdorff (notée δ) est strictement supérieure à la dimension topologique[1]. Sommaire 1… …   Wikipédia en Français

  • Liste de fractales — par dimension de Hausdorff Cet article est une liste de fractales, ordonnées par dimension de Hausdorff croissante. En mathématiques, une fractale est un ensemble dont la dimension de Hausdorff (notée δ) est strictement supérieure à la dimension… …   Wikipédia en Français

  • Liste de fractales par dimension de Hausdorff — Cet article est une liste de fractales, ordonnées par dimension de Hausdorff croissante. En mathématiques, une fractale est un ensemble dont la dimension de Hausdorff (notée δ) est strictement supérieure à la dimension topologique[1]. Sommaire 1… …   Wikipédia en Français

  • Liste de fractales par dimension de hausdorff — Cet article est une liste de fractales, ordonnées par dimension de Hausdorff croissante. En mathématiques, une fractale est un ensemble dont la dimension de Hausdorff (notée δ) est strictement supérieure à la dimension topologique[1]. Sommaire 1… …   Wikipédia en Français

  • Tessellation — A tessellation of pavement A honeycomb is an example of a t …   Wikipedia

  • Pavage — hexagonal d’un sol. Un pavage (ou dallage) est une partition d’un espace (généralement un espace euclidien comme le plan ou l’espace tridimensionnel) par des éléments d un ensemble fini, appelés dalles (plus précisément, ce sont des compacts… …   Wikipédia en Français

  • List of mathematics articles (P) — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… …   Wikipedia

  • John Horton Conway — Infobox Scientist name = John Horton Conway |300px image width = 300px birth date = birth date and age|1937|12|26|mf=y birth place = Liverpool, Merseyside, England residence = U.S. nationality = English death date = death place = field =… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”