Pinwheel tiling
Encyclopedia
Pinwheel tilings are non-periodic tilings
defined by Charles Radin and based on a construction due to John Conway.
They are the first known non-periodic tilings to each have the property that their tiles appear in infinitely many orientations.
be the right triangle with side length 
, 
and 
.
Conway noticed that
can be divided in five isometric copies of its image by the dilation of factor 
.
By suitably rescaling and translating/rotating, this operation can be iterated to obtain an infinite increasing sequence of growing triangles all made of isometric copies of
.
The union of all these triangles yields a tiling of the whole plane by isometric copies of
.
In this tiling, isometric copies of
appears in infinitely many orientations (this is due to the angles 
and 
of 
, both non-commensurable with 
).
Despite of this, all the vertices have rational coordinates.
Formally, the Pinwheel tilings are the tilings whose tiles are isometric copies of
, with a tile maying intersect another tile only either on a whole side or on half the length 
side, and such that the following property holds.
Given any Pinwheel tiling
, there is a Pinwheel tiling 
which, once each tile is divided in five following the Conway construction and the result is dilated by a factor 
, is equal to 
.
In other words, the tiles of any Pinwheel tilings can be grouped five-by-five into homothetic tiles, so that these homothetic tiles form (up to rescaling) a new Pinwheel tiling.
The tiling constructed by Conway is a Pinwheel tiling, but there are uncountably many other different Pinwheel tiling.
They are all locally undistinguishable (i.e., they have the same finite patches).
They all share with the Conway tiling the property that tiles appear in infinitely many orientations (and vertices have rational coordinates).
The main result proven by Radin is that there is a finite (though very large) set of so-called prototiles, with each being obtained by coloring the sides of
, so that the Pinwheel tilings are exactly the tilings of the plane by isometric copies of these prototiles, with the condition that whenever two copies intersect in a point, they have the same color in this point.
In terms of symbolic dynamics
, this means that the Pinwheel tilings form a sofic subshift
. There are other variants and generalizations of the original idea.
One get a fractal by iteratively dividing
in five isometrics copies, following the Conway construction, and discarding the middle triangle (ad infinitum). This Pinwheel fractal has Hausdorff dimension
.
, a building complex in Melbourne, Australia features the pinwheel tiling. In the project, the tiling pattern is used to create the structural sub-framing for the facades, allowing for the facades to be fabricated off-site, in a factory and later erected to form the facades. The pinwheel tiling system was based on the single triangular element, composed of zinc, perforated zinc, sandstone or glass (known as a tile), which was joined to 4 other similar tiles on an aluminum frame, to form a "panel". Five panels were affixed to a galvanized steel frame, forming a "mega-panel", which were then hoisted onto support frames for the facade. The rotational positioning of the tiles gives the facades a more random, uncertain compositional quality, even though the process of its construction is based on pre-fabrication and repetition. The same pinwheel tiling system is used in the development of the structural frame and glazing for the "Atrium" at Federation Square, although in this instance, the pin-wheel grid has been made "3-dimensional" to form a portal frame structure.
Aperiodic tiling
An aperiodic tiling is a tiling obtained from an aperiodic set of tiles. Properly speaking, aperiodicity is a property of particular sets of tiles; any given finite tiling is either periodic or non-periodic...
defined by Charles Radin and based on a construction due to John Conway.
They are the first known non-periodic tilings to each have the property that their tiles appear in infinitely many orientations.
The Conway tesselation
Let be the right triangle with side length , and .Conway noticed that
can be divided in five isometric copies of its image by the dilation of factor .By suitably rescaling and translating/rotating, this operation can be iterated to obtain an infinite increasing sequence of growing triangles all made of isometric copies of
.The union of all these triangles yields a tiling of the whole plane by isometric copies of
.In this tiling, isometric copies of
appears in infinitely many orientations (this is due to the angles and of , both non-commensurable with ).Despite of this, all the vertices have rational coordinates.
The Pinwheel tilings
Radin relied on the above construction of Conway to define Pinwheel tilings.Formally, the Pinwheel tilings are the tilings whose tiles are isometric copies of
, with a tile maying intersect another tile only either on a whole side or on half the length side, and such that the following property holds.Given any Pinwheel tiling
, there is a Pinwheel tiling which, once each tile is divided in five following the Conway construction and the result is dilated by a factor , is equal to .In other words, the tiles of any Pinwheel tilings can be grouped five-by-five into homothetic tiles, so that these homothetic tiles form (up to rescaling) a new Pinwheel tiling.
The tiling constructed by Conway is a Pinwheel tiling, but there are uncountably many other different Pinwheel tiling.
They are all locally undistinguishable (i.e., they have the same finite patches).
They all share with the Conway tiling the property that tiles appear in infinitely many orientations (and vertices have rational coordinates).
The main result proven by Radin is that there is a finite (though very large) set of so-called prototiles, with each being obtained by coloring the sides of
, so that the Pinwheel tilings are exactly the tilings of the plane by isometric copies of these prototiles, with the condition that whenever two copies intersect in a point, they have the same color in this point.In terms of symbolic dynamics
Symbolic dynamics
In mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics given by the shift operator...
, this means that the Pinwheel tilings form a sofic subshift
Generalizations
Radin and Conway proposed a three dimensional analogue which was dubbed the quaquaversal tilingQuaquaversal tiling
The quaquaversal tiling is a nonperiodic tiling of the euclidean 3-space introduced by John Conway and Charles Radin. The basic solid tiles are half prisms arranged in a pattern which relies essentially on their previous construct, the pinwheel tiling...
. There are other variants and generalizations of the original idea.
One get a fractal by iteratively dividing
in five isometrics copies, following the Conway construction, and discarding the middle triangle (ad infinitum). This Pinwheel fractal has Hausdorff dimensionHausdorff dimension
thumb|450px|Estimating the Hausdorff dimension of the coast of Great BritainIn mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space...
.Curiosity
Federation SquareFederation Square
Federation Square is a civic centre and cultural precinct in the city of Melbourne, Victoria, Australia....
, a building complex in Melbourne, Australia features the pinwheel tiling. In the project, the tiling pattern is used to create the structural sub-framing for the facades, allowing for the facades to be fabricated off-site, in a factory and later erected to form the facades. The pinwheel tiling system was based on the single triangular element, composed of zinc, perforated zinc, sandstone or glass (known as a tile), which was joined to 4 other similar tiles on an aluminum frame, to form a "panel". Five panels were affixed to a galvanized steel frame, forming a "mega-panel", which were then hoisted onto support frames for the facade. The rotational positioning of the tiles gives the facades a more random, uncertain compositional quality, even though the process of its construction is based on pre-fabrication and repetition. The same pinwheel tiling system is used in the development of the structural frame and glazing for the "Atrium" at Federation Square, although in this instance, the pin-wheel grid has been made "3-dimensional" to form a portal frame structure.
External links
- Pinwheel at the Tilings Encyclopedia