In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, pinwheel tilings are
non-periodic tilings defined by
Charles Radin and based on a construction due to
John Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
.
They are the first known non-periodic tilings to each have the property that their tiles appear in infinitely many orientations.
Conway's tessellation
250px, Conway's triangle decomposition into smaller similar triangles.
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
.
250px, The increasing sequence of triangles which defines Conway's tiling of the plane.
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
appear in infinitely many orientations (this is due to the angles
and
of
, both non-commensurable with
).
Despite this, all the vertices have rational coordinates.
The pinwheel tilings
250px, A pinwheel tiling: tiles can be grouped in sets of five (thick lines) to form a new pinwheel tiling (up to rescaling)
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
, in which a tile may 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 in sets of 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 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 (e ...
, this means that the pinwheel tilings form a
sofic subshift.
Generalizations
Radin and Conway proposed a three-dimensional analogue which was dubbed the
quaquaversal tiling. There are other variants and generalizations of the original idea.
250px, Pinwheel fractal
One gets a fractal by iteratively dividing
in five isometric copies, following the Conway construction, and discarding the middle triangle (''ad infinitum''). This "pinwheel fractal" has
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a ...
.
Use in architecture
Federation Square
Federation Square (colloquially Fed Square) is a venue for arts, culture and public events on the edge of the Melbourne central business district. It covers an area of at the intersection of Flinders and Swanston Streets built above busy ra ...
, 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.
References
External links
Pinwheelat the Tilings Encyclopedia
Dynamic Pinwheelmade i
GeoGebra
{{DEFAULTSORT:Pinwheel Tiling
Discrete geometry
Aperiodic tilings