Pinwheel tiling

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In geometry, 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.

Contents

Conway's tessellation

Conway's triangle decomposition into smaller similar triangles. Pinwheel 1.svg
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 .

The increasing sequence of triangles which defines Conway's tiling of the plane. Pinwheel 2.gif
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

A pinwheel tiling: tiles can be grouped in sets of five (thick lines) to form a new pinwheel tiling (up to rescaling) Pinwheel 3.jpg
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. [1] In terms of symbolic dynamics, 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. [2] There are other variants and generalizations of the original idea. [3]

Pinwheel fractal Pinwheel fractal.png
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 .

Use in architecture

Federation Square's sandstone facade Federation-square-sandstone-facade.jpg
Federation Square's sandstone façade

Federation Square, 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.

Related Research Articles

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Koch snowflake Fractal and mathematical curve

The Koch snowflake is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curve Without Tangents, Constructible from Elementary Geometry" by the Swedish mathematician Helge von Koch.

Tessellation Tiling of a plane in mathematics

A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries.

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Aperiodic tiling Specific form of plane tiling in mathematics

An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types is aperiodic if copies of these tiles can form only non-periodic tilings. The Penrose tilings are the best-known examples of aperiodic tilings.

In geometry, the gyrobifastigium is the 26th Johnson solid (J26). It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile three-dimensional space.

In mathematics, the T-square is a two-dimensional fractal. It has a boundary of infinite length bounding a finite area. Its name comes from the drawing instrument known as a T-square.

Gosper curve Space-filling curve

The Gosper curve, also known as Peano-Gosper Curve, named after Bill Gosper, also known as the flowsnake, is a space-filling curve whose limit set is rep-7. It is a fractal curve similar in its construction to the dragon curve and the Hilbert curve.

In geometry, a tile substitution is a method for constructing highly ordered tilings. Most importantly, some tile substitutions generate aperiodic tilings, which are tilings whose prototiles do not admit any tiling with translational symmetry. The most famous of these are the Penrose tilings. Substitution tilings are special cases of finite subdivision rules, which do not require the tiles to be geometrically rigid.

In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(1/5), or approximately 78.46°.

Quaquaversal 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 that relies essentially on their previous construct, the pinwheel tiling. The rotations relating these tiles belong to the group G(6,4) generated by two rotations of order 6 and 4 whose axes are perpendicular to each other. These rotations are dense in SO(3).

Ammann–Beenker tiling

In geometry, an Ammann–Beenker tiling is a nonperiodic tiling which can be generated either by an aperiodic set of prototiles as done by Robert Ammann in the 1970s, or by the cut-and-project method as done independently by F. P. M. Beenker. Because all tilings obtained with the tiles are non-periodic, Ammann–Beenker tilings are considered aperiodic tilings. They are one of the five sets of tilings discovered by Ammann and described in Tilings and Patterns.

Hadwiger conjecture (combinatorial geometry)

In combinatorial geometry, the Hadwiger conjecture states that any convex body in n-dimensional Euclidean space can be covered by 2n or fewer smaller bodies homothetic with the original body, and that furthermore, the upper bound of 2n is necessary if and only if the body is a parallelepiped. There also exists an equivalent formulation in terms of the number of floodlights needed to illuminate the body.

Penrose tiling Non-periodic tiling of the plane

A Penrose tiling is an example of an aperiodic tiling. Here, a tiling is a covering of the plane by non-overlapping polygons or other shapes, and aperiodic means that shifting any tiling with these shapes by any finite distance, without rotation, cannot produce the same tiling. However, despite their lack of translational symmetry, Penrose tilings may have both reflection symmetry and fivefold rotational symmetry. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated them in the 1970s.

Rauzy fractal

In mathematics, the Rauzy fractal is a fractal set associated with the Tribonacci substitution

Pythagorean tiling Tiling by squares of two sizes

A Pythagorean tiling or two squares tessellation is a tiling of a Euclidean plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides. Many proofs of the Pythagorean theorem are based on it, explaining its name. It is commonly used as a pattern for floor tiles. When used for this, it is also known as a hopscotch pattern or pinwheel pattern, but it should not be confused with the mathematical pinwheel tiling, an unrelated pattern.

Rep-tile Shape subdivided into copies of itself

In the geometry of tessellations, a rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by recreational mathematician Solomon W. Golomb and popularized by Martin Gardner in his "Mathematical Games" column in the May 1963 issue of Scientific American. In 2012 a generalization of rep-tiles called self-tiling tile sets was introduced by Lee Sallows in Mathematics Magazine.

Charles Lewis Radin is an American mathematician, known for his work on aperiodic tilings and in particular for defining the pinwheel tiling and the quaquaversal tiling.

In plane geometry, the einstein problem asks about the existence of a single prototile that by itself forms an aperiodic set of prototiles, that is, a shape that can tessellate space, but only in a nonperiodic way. Such a shape is called an "einstein", a play on the German words ein Stein, meaning one tile. Depending on the particular definitions of nonperiodicity and the specifications of what sets may qualify as tiles and what types of matching rules are permitted, the problem is either open or solved. The einstein problem can be seen as a natural extension of the second part of Hilbert's eighteenth problem, which asks for a single polyhedron that tiles Euclidean 3-space, but such that no tessellation by this polyhedron is isohedral. Such anisohedral tiles were found by Karl Reinhardt in 1928, but these anisohedral tiles all tile space periodically.

The Fibonacci word fractal is a fractal curve defined on the plane from the Fibonacci word.

References

  1. Radin, C. (May 1994). "The Pinwheel Tilings of the Plane". Annals of Mathematics . 139 (3): 661–702. CiteSeerX   10.1.1.44.9723 . doi:10.2307/2118575. JSTOR   2118575.
  2. Radin, C., Conway, J., Quaquaversal tiling and rotations, preprint, Princeton University Press, 1995
  3. Sadun, L. (January 1998). "Some Generalizations of the Pinwheel Tiling". Discrete and Computational Geometry . 20 (1): 79–110. arXiv: math/9712263 . CiteSeerX   10.1.1.241.1917 . doi:10.1007/pl00009379. S2CID   6890001.