Pinwheel tiling

Key takeaways

Pinwheel tiling belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Pinwheel tiling to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Pinwheel tiling from memory before moving on to harder problems.

Reference excerpt

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.

Definition

Let T {\displaystyle T} be the right triangle with side length 1 {\displaystyle 1} , 2 {\displaystyle 2} and 5 {\displaystyle {\sqrt {5}}} . Conway noticed that T {\displaystyle T} can be divided in five isometric copies of its image by the dilation of factor 1 / 5 {\displaystyle 1/{\sqrt {5}}} . There are multiple ways of performing this subdivision; in the subdivision used for the pinwheel tiling, four copies of T {\displaystyle T} meet at the midpoint of the middle side of T {\displaystyle T} , and the central copy has no sides parallel to the sides of T {\displaystyle T} .

The pinwheel tiling is obtained by repeatedly inflating T {\displaystyle T} by a factor of 5 {\displaystyle {\sqrt {5}}} and then subdividing each tile in this manner. Conversely, the tiles of the pinwheel tiling can be grouped into groups of five that form a larger pinwheel tiling. In this tiling, isometric copies of T {\displaystyle T} appear in infinitely many orientations because the small angle of T {\displaystyle T} by which its central copy is rotated, arctan ⁡ 1 2 {\displaystyle \arctan {\frac {1}{2}}} , is not a rational multiple of π {\displaystyle \pi } . Radin found a collection of five prototiles, each of which is a marking of T {\displaystyle T} , so that the matching rules on these tiles and their reflections enforce the pinwheel tiling. All of the vertices have rational coordinates, and tile orientations are uniformly distributed around the circle.

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.

One gets a fractal by iteratively dividing T {\displaystyle T} in five isometric copies, following the Conway construction, and discarding the middle triangle (ad infinitum). This "pinwheel fractal" has Hausdorff dimension d = ln ⁡ 4 ln ⁡ 5 = log 5 ⁡ ( 16 ) ≈ 1.7227 {\displaystyle d={\frac {\ln 4}{\ln {\sqrt {5}}}}=\log _{5}(16)\approx 1.7227} .

Use in architecture

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.

References

External links Pinwheel at the Tilings Encyclopedia Dynamic Pinwheel made in GeoGebra

Illustrations

Worked examples

Example 1 — a first encounter with Pinwheel tiling

Start with the simplest possible case. Write down what Pinwheel tiling claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Pinwheel tiling before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Pinwheel tiling ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Pinwheel tiling

In research: Pinwheel tiling appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Pinwheel tiling in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Pinwheel tiling is common in secondary-school and first-year university syllabi. It links to neighbouring topics Aperiodic tilings, Discrete geometry, Triangular tilings, so understanding it makes those chapters shorter.
In everyday life: Look for Pinwheel tiling outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Pinwheel tiling in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Pinwheel tiling means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Pinwheel tiling out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Pinwheel tiling in simple terms?

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.

Why does Pinwheel tiling matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Pinwheel tiling?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Pinwheel tiling.