In March 2023, four people posted a paper to arXiv with a title of three words: “An aperiodic monotile”. David Smith, a retired print technician from Yorkshire, had found the shape in November 2022 by cutting card at home. He emailed Craig Kaplan at the University of Waterloo and asked him to run it through his software. Joseph Samuel Myers and Chaim Goodman-Strauss then joined Smith and Kaplan on the proof. The paper (arXiv 2303.10798) was published in Combinatorial Theory 4(1) in June 2024. It answered a question that had stayed open for sixty years. A single shape can cover an infinite plane with no gaps and no overlaps, and it can do so only in a pattern that never repeats.

That result is the newest link in a long chain. The idea underneath it matters on a façade as much as in a proof: order without repetition, built from very few parts.

What it is:

• A periodic tiling has translational symmetry. You can shift the whole pattern, without rotating it, and it lands exactly on itself. Brick bond, square grids and hexagonal paving are all periodic.
• A non-periodic tiling has no such shift. Even plain squares can do this if you offset each row by a random amount. But squares can also tile periodically, so being able to tile non-periodically proves nothing about the shape.
• An aperiodic set of tiles is the strong case. In the standard definition, the tiles “can tile, but only non-periodically”. No arrangement of them ever repeats by translation.

So aperiodic order is not randomness. Every patch follows strict local rules: which edge may meet which, which angle may close a vertex. Yet no window of the pattern, however large, repeats across the whole plane. It is a plane that is rule-bound everywhere and periodic nowhere.

Why it works:

Start with what periodic patterns cannot do. The crystallographic restriction theorem says that a lattice in 2D or 3D can only have rotational symmetry of order 1, 2, 3, 4 or 6. The algebraic reason fits on one line: rotation by 2π/n maps a lattice to itself only if 2cos(2π/n) is an integer. For n = 5, 2cos 72° = 0.618…, which is the reciprocal of the golden ratio φ and not an integer. Geometrically, a pentagon of lattice points would breed ever smaller pentagons of lattice points, which no discrete lattice can hold. Fivefold symmetry and translation cannot live in the same pattern.

The way out is to drop translation and keep the local rule. Aperiodic tilings do exactly this, and they rely on two mechanisms. The first is substitution, also called inflation or deflation. Every tile splits into smaller copies of the tile set, so the pattern is a recursion. The splitting ratio is irrational (φ for Penrose), so the scales never line up into a repeating period. The second is cut-and-project. N. G. de Bruijn showed in 1981 that the Penrose rhomb tiling is a slice of a five-dimensional cubic lattice, projected onto a plane at an irrational angle. He also gave the dual method, the pentagrid: overlay five families of parallel lines, and each crossing becomes a rhomb. The tiling obeys Penrose’s matching rules exactly when the five shifts sum to an integer. In a large kite-and-dart tiling, the ratio of kites to darts approaches φ ≈ 1.618.

←TODAY: A one-shape aperiodic tile is peer-reviewed, its outlines are free SVGs, and a Grasshopper definition can lay it out on a façade grid in seconds. →3012: Zurich’s envelopes in 3012 are read like text, never the same sentence twice, and assembled from an alphabet of one or two parts. Fulcrum: Fabrication economy and visual variety stopped being a trade-off the moment the rule, not the panel, became the thing you design.

Origins:

In 1961 the logician Hao Wang introduced square tiles with coloured edges and conjectured that any set that tiles the plane can also tile it periodically. If he had been right, it would be decidable whether a given set of tiles can tile the plane at all. In 1966 Robert Berger showed it is undecidable. His construction encoded the halting problem of a Turing machine into tiles, and the first set used 20,426 Wang tiles. He later reduced that to 104, and Hans Läuchli found a set of 40. In 1971 Raphael Robinson got it down to six.

Roger Penrose reached two in 1973–74: first the kite and dart, then a pair of rhombs. The thin rhomb has angles of 36°/144° and the thick one 72°/108°. On their own, the rhombs can tile periodically. Matching rules, drawn as arcs or notches on the faces, force them into aperiodic order. Robert Ammann, an amateur working largely alone, found several further aperiodic sets around 1977 (the Ammann bars carry his name). In January 1977 Martin Gardner’s “Mathematical Games” column in Scientific American brought Penrose tiles, along with the properties Penrose and John Conway had worked out, to a very wide readership. De Bruijn’s algebra followed in 1981.

Then nature turned out to have done it already. On 8 April 1982, Dan Shechtman of the Technion was on sabbatical at the US National Bureau of Standards, now NIST. He saw a tenfold electron-diffraction pattern in a rapidly cooled aluminium–manganese alloy. Next to sample 1725 in his notebook he wrote “10 Fold ???”. His group leader told him to “go back and read the textbook” and later asked him to leave for “bringing disgrace” on the team. Shechtman did not give up. The paper by Shechtman, Blech, Gratias and Cahn appeared in Physical Review Letters in 1984, more than two years after the observation. Linus Pauling opposed the finding until his death. The 2011 Nobel Prize in Chemistry went to Shechtman alone, “for the discovery of quasicrystals”. In Switzerland, Walter Steurer has studied the structure of aperiodic crystals at the ETH Zurich Laboratory of Crystallography since 1993. In that higher-dimensional view, an icosahedral quasicrystal is a 3D cut through a 6D “hypercrystal”, which is de Bruijn’s projection carried out in metal.

The final link is the einstein, from the German ein Stein, “one stone”. In 2010 the Socolar–Taylor tile was a single aperiodic tile, but it is not a connected shape. Smith’s hat is connected: eight kites cut from a hexagonal grid, with no irrational angles anywhere. Every tiling of the hat uses some mirrored copies. In May 2023 the same four authors posted “A chiral aperiodic monotile” (arXiv 2305.17743, Combinatorial Theory 4(2), September 2024). It introduces the Spectre, a version with modified edges that tiles without any mirror image. PAZ worked through what that means for production in One shape, no repeat — and the bill for building it. In a hat tiling roughly 12.73% of the tiles are mirrored, so a hat façade needs two moulds. The Spectre needs one.

In Science in 2007, Peter Lu and Paul Steinhardt argued that girih craftsmen used five equilateral tiles and self-similar subdivision. They point to the Darb-i Imam shrine in Isfahan (1453) as showing nearly perfect quasi-crystalline Penrose patterns. Historians of Islamic geometry have contested how far that intent can be read into the work. Either way, the tiles show a substitution rule in the hands of masons five centuries before de Bruijn wrote it down.

In practice:

Verified built examples are few. Federation Square in Melbourne, by LAB Architecture Studio (Donald Bates and Peter Davidson) with Bates Smart, opened on 26 October 2002. Its façades are clad in a pinwheel tiling, the Radin–Conway pattern built from one right triangle with sides 1, 2 and √5. Five triangles in zinc, sandstone or glass make up one panel on an aluminium frame, and five panels make a unit that is lifted into place. One triangle shape, prefabricated off site, gives a surface that LAB said would not be “a regularly repeating flat surface”. Polster and Ross later found that it is not a perfect pinwheel tiling, because some tiles cannot be grouped correctly. That kind of honest gap appears wherever an infinite rule meets a finite building. At the Andrew Wiles Building in Oxford (Rafael Viñoly Architects, opened October 2013), Penrose himself designed the entrance paving. It uses two granite rhombs with 30 mm stainless-steel arc inserts, so the matching rule is visible in the floor. Helsinki’s Keskuskatu was repaved in a Penrose tiling in 2014.

For a Swiss office, the case comes down to this. Few tile types mean few moulds, few CNC programmes and simple stock. Non-repetition gives a façade or a public floor variety without a thousand unique panels. The trade-off is plain: aperiodic patterns never meet a rectangular boundary cleanly, so every edge, corner and opening brings back the bespoke cut pieces that the tile set was meant to remove. Count those boundary pieces before you sell the elegance.

The tooling already exists, and it was built by people who shared it. Roly Hudson published a C# substitution script for Grasshopper in 2015. Menno Deij-van Rijswijk released the PenroseTiling plug-in for Rhino in 2014. Craig Kaplan’s browser tools export hat and Spectre patches as SVG, with source code under BSD-3. Christian Lawson-Perfect collected outlines ready for laser cutting. Kaplan’s position on the shape is simple: “It belongs to everyone.” All of this rests on the habit David Rutten built into Grasshopper in 2007: the rule is the model, and the geometry is just its output.

Atelier: For a Zürich studio working on a ventilated rainscreen or a courtyard paving, aperiodic tiling turns “how many unique panels?” into a question you can answer in the brief. With AI layout tools now proposing patterns, the office needs a rule it can audit, not an image it cannot rebuild. Monday move: give one designer the job of generating a Penrose patch over a real façade elevation in Grasshopper and reporting three numbers: full tiles, boundary cuts and distinct cut shapes.

Hack:

Deflate Robinson triangles into a Penrose tiling. In GhPython, treat points as complex numbers. A thin triangle (0) splits into one thin and one thick, and a thick triangle (1) splits into two thick and one thin, each time at the golden ratio:

phi = (1 + 5 ** 0.5) / 2
def split(tris):
    return [t for c, A, B, C in tris for t in (
        [(0, C, A + (B - A) / phi, B), (1, A + (B - A) / phi, C, A)] if c == 0 else
        [(1, B + (C - B) / phi, C, A), (1, B + (A - B) / phi, B + (C - B) / phi, B), (0, B + (C - B) / phi, B + (A - B) / phi, A)])]

Seed it with a ring of ten thin triangles around the origin, call split five times, and draw each triple as a closed polyline. Then count the tiles by colour: the ratio of thick to thin converges to φ. That number is your mould schedule, worked out before anyone opens a supplier catalogue.

The façades that age well are the ones whose rule can be read back from the pattern. Write the substitution rule into the project file next to the geometry, then build the first patch and count its edges.

Sources & Further Reading

• PAZ context: One shape, no repeat — and the bill for building it