Seven cubes in the shape of a chair, a pyramid on every face, and a 2026 claim of the first single tile that forces space-filling without repetition
The hat settled the einstein problem in the plane in 2023. In September 2026 a preprint claimed the same for space: a seven-cube chair, with a small pyramid on each face, that tiles space but never periodically.
Subdivide the chair again and again, and pull the pieces apart to see that every small chair sits inside exactly one bigger one.
At stage 0 the white pyramids mark the 24 exposed square panels. Chair44 puts a small square-pyramid feature on each one; the pattern of those features is what forces the hierarchy, and it is not reproduced here — these markers are schematic.
Take a 2 × 2 × 2 block of cubes and remove one corner. The seven cubes left form the 3D chair, the three-dimensional version of the L-shaped tromino. Like the tromino, it is a rep-tile: a chair twice the size splits exactly into eight copies of the original. One copy sits in the middle facing the same way, one sits in the far corner facing the same way, and the other six face inward.
Repeat the split and you get chairs inside chairs inside chairs. Run the process backwards forever and you get tilings of all of space in which every chair belongs to a unique supertile, every supertile to a unique larger one, and so on. Such a hierarchical tiling cannot be periodic. A translation that mapped it to itself would have to map supertiles to supertiles at every level, and once the supertiles are bigger than the translation, that is impossible.
The catch is that nothing forces a chair tiling to look like this. The plain chair has plenty of other tilings, as the next demo shows. The goal of the einstein problem is a single tile whose shape alone allows only non-periodic tilings.
Here the same chair fills a box periodically. Switch on the colouring to see the one-line proof.
About 49 chairs touch this box, every one a translate of the same chair. Shift by (7, 0, 0), (−2, 1, 0) or (−4, 0, 1) and the tiling lands on itself.
The two-dimensional einstein problem was settled in 2023, when David Smith, Joseph Myers, Craig Kaplan and Chaim Goodman-Strauss found the hat and then the spectre. Three dimensions had partial answers. The Schmitt–Conway–Danzer biprism tiles space only non-periodically, but its tilings always have a screw symmetry, so it is only weakly aperiodic. Three-dimensional versions of the Socolar–Taylor tile leave a direction in which tilings can repeat.
On 16 September 2026 Ioannis Tsiokos, a software developer in Athens, posted “A Strongly Aperiodic Monotile in Three Dimensions” (arXiv:2609.19214). His tile, Chair44, is the seven-cube chair with a small square-pyramid feature on each of its 24 exposed panels. He says an AI model found the shape. The paper claims that every tiling of space by copies of Chair44, with reflections allowed, has no translational period and a symmetry group of order at most 24, uses only one handedness of the tile, and carries a unique infinite hierarchy of nested supertiles: exactly the chair hierarchy in the first demo.
The proof has two parts. A written geometric argument shows that the pyramid features force every tiling onto a regular lattice of cubes. After that, everything reduces to finite checks. The crucial one is that the tile’s contact rule survives coarsening: group the chairs into supertiles, and the supertiles obey the same rule. These checks were run by two independent programs and kernel-checked in the Lean 4 proof assistant, relying on compiled code for the largest enumerations.
This is a preprint, not yet peer reviewed, and it is the newest result in this module. Formal verification makes a mistake in the finite checks unlikely, but the statement being checked, and the translation from geometry to code, still need human scrutiny. Treat it as a very promising claim rather than a theorem.
Spot a mistake or have an idea?