Chair44 Visualizer
A single block shape that fills space in only one way, and that way never repeats. This visualizer follows one project from the published tile to a printed set, and then to a method for making the hidden structure of the tiling visible: keep any combination of each tile's nine faces as flat plates, from a single face to the whole tile.
The tile
A tiling covers space with copies of shapes, leaving no gaps and no overlaps. Most tilings people know are periodic: a bathroom floor or a brick wall can be slid by some fixed distance and land exactly on itself. A tile is aperiodic when every tiling it makes refuses to do that, at every distance, in every direction.
In September 2026 Ioannis Tsiokos reported the first known strongly aperiodic monotile in three dimensions, a single shape with that property. Chaim Goodman-Strauss then published a clearer account of it, the source for this visualizer. The tile is called Chair44.
Its shape is plain: a 2 × 2 × 2 cube with one corner cube removed, seven small cubes in all. This shape is the three-dimensional L-tile, or "chair". Eight L-tiles fit together into an L-tile twice as large, so the shape is a rep-tile. On its own the L-tile is not aperiodic; it also stacks into ordinary repeating rows. Chair44 adds colored arrows to its 24 square panels. Two tiles may touch only where their arrows agree, and that one rule is what forces aperiodicity.
The same idea, flattened
The flat version of the chair is made of three squares. Four of them make a chair twice as large, and the four can be split again, without end. The colors show which way each chair faces.
In two dimensions this nesting is easy to see. Chair44 does the same thing in three dimensions with eight pieces per step, which is much harder to see. Making it visible is the subject of the rest of this atlas.
Printing it
Arrows printed on paper cannot be felt. The first goal was a set of blocks that physically enforce the arrow rule, so that a wrong placement will not fit.
Read the markings
The arrows on all 24 panels were read from the views in the paper and written down as data: a color and a direction on one diagonal of each panel.
Check them against the hierarchy
The markings were tested against the 8-into-1 assembly rule. Exactly one way of turning the eight smaller tiles agrees with every arrow, and it stays consistent through 4,096 tiles.
Count legal neighbors
Every way two tiles can touch face to face was tried. Exactly 44 placements obey the arrows, matching the 44 positions listed in Tsiokos's preprint.
Turn arrows into keys
Each color became a raised bump and a matching pocket along the arrow's diagonal. A red arrow fits only into a green one pointing the same way, and blue fits only blue.
Test the solid shapes
The printed geometry was checked directly: all 44 legal placements fit with no interference, and all 1,150 illegal ones collide.
The printed set works: the blocks lock together, and eight of them form a larger copy of the tile. They demonstrate the rule. What they do not show is why the result can never repeat, or how each block is turned inside the larger one. The keyed tile is in the viewer under Build → Keyed tile, with STL export.
Why it never repeats
Eight tiles make a supertile, a tile twice as large. Eight supertiles make a 2-level supertile, four times as large, and the pattern continues upward forever.
Plain L-tiles can do this too, but they do not have to. The arrows remove the choice. Any arrangement the arrows allow, however large, can be grouped into supertiles in exactly one way, those into larger supertiles in exactly one way, and so on at every scale.
Suppose the tiling did repeat when shifted by some distance. The shift would have to carry every supertile onto another supertile, at every level. Supertiles double in size at each level, so eventually they are far larger than the shift. A block that large cannot move a short distance and still land exactly on a block of the same grouping. No shift works, so the tiling never repeats.
The colors on the right show each tile's turn. All 24 possible orientations appear, and their pattern is never periodic.
Reading orientation
In a solid assembly every tile looks the same, so the turn of each one is hidden. The idea pursued here is to keep only some of each tile's faces as thin plates and remove the rest, which opens the inside of the assembly to view. The project began with three faces, one of each kind, but any combination works: all but seven of the 511 possible choices, even a single face, are enough to show exactly how each tile is turned.
The L-tile has nine faces in three kinds: three large squares on the closed back, three L-shaped faces on the front, and three small squares lining the socket. Each kind has one face perpendicular to x, one to y and one to z. The faces are numbered so that the number tells both.
| Kind | ⊥ x | ⊥ y | ⊥ z | Area |
|---|---|---|---|---|
| Large square | 1 | 2 | 3 | 4 |
| L-shape | 4 | 5 | 6 | 3 |
| Small square | 7 | 8 | 9 | 1 |
Area is counted in unit squares; the total is 24, matching the 24 marked panels of Chair44. Faces n, n+3 and n+6 are parallel and share no edges.
Every choice of faces
Each of the nine faces can be kept or left out, which gives 511 possible choices, from a single face up to the whole closed tile. The project began with three faces, one of each kind (one red, one green, one blue): 27 choices. Allowing any three faces gives 84. Allowing any number gives all 511.
The choices are sorted here by how many faces they keep, and then by how the kept faces touch each other on a single tile. Two faces touch when they share an edge; on the L-tile no two faces meet at only a corner point.
| Faces kept | Choices | Up to a turn | All joined on the tile | No two touching |
|---|
Keeping k faces is the same as leaving out 9 − k, so the counts mirror around the middle. Three faces is the most that can be kept with no two touching; from four faces up, some always touch. A third of a turn about the tile's long diagonal carries x → y → z, so most choices come in groups of three that are the same arrangement turned; each row of cards below is one such group. Seven choices look the same after that turn: whole sets of large squares, L-shapes or small squares, any two of those sets, and the full tile. On those tiles three different orientations look identical, so they are marked orientation hidden.
Three faces: four touch patterns
Each card shows one tile (left) and the 8-tile supertile it builds (right). Select a card to open it in the viewer.
Supertiles
On one tile, the three members of a group are the same arrangement turned. Inside a supertile they come apart. The assembly rule places its eight tiles in a fixed, lopsided way that has no three-fold symmetry, so turning the faces on each tile produces a different supertile. All 511 choices build 511 different supertiles.
The rule does have one mirror symmetry: swapping x and y. Choices related by that swap, such as faces 1, 4, 7 and faces 2, 5, 8, build exact mirror-image supertiles; this holds for all 511. The z direction has no partner, and z-based configurations behave differently from the rest.
The chart measures how the plates hang together when 512 tiles are assembled: the share of all plate area that belongs to the single largest connected structure. Touching within one tile does not decide this. With few faces kept, many choices fall apart. Among three-face choices, chains and triangles usually form one continuous network through the whole block, but several fall apart into small pieces, and some pairs break up almost completely: faces 3, 7, 9 leave no structure larger than 1% of the area. With more faces kept, nearly everything joins into one network. Use the tabs to compare.
Bar: share of plate area in the largest edge-connected structure when 512 tiles are assembled. Pieces: number of separate structures. Rows marked ◆ are three-face choices with one face of each kind. Select a row to open it in the viewer.
Plates and miters
To build these assemblies as objects, each chosen face becomes a flat plate 2.5 mm thick at a scale of 20 mm per small cube. The outer surface of each plate lies exactly on the face, and its thickness goes into its own tile, so plates from neighboring tiles never pass through each other. Where plates from different tiles meet, the joint is cut according to how they meet.
Outside corner
Two plates meet at a right angle, both thickened away from the corner. Each is extended and cut at 45°, so the corner closes with a diagonal seam.
End against side
One plate's end would butt against the other's side. The square corner is shared out along the 45° diagonal instead.
Back-to-back pair
Two tiles' faces share one square from opposite sides, and a third plate meets them. The outer plate is mitered to the third; the inner one butts against it.
Same plane
Plates lying in one plane meet edge to edge. When their thicknesses go to opposite sides, they meet along a line.
Two exceptions remain plain butt joints: edges where three or more plates meet in other arrangements, which become common when many faces are kept, and places where a miter would run into a third plate, which are trimmed back. All 1,022 mitered models in the viewer were checked: no two plates overlap.
Viewer
Choose any of the nine faces, from one to all nine, then the level of assembly and how it is built. Exports contain exactly what is on screen, packed in a .zip file.
Who did what
This project was a conversation between an artist and an AI model. The tile itself, its markings and the proof that it is aperiodic belong to its discoverers.
Ioannis Tsiokos · Chaim Goodman-Strauss
- Tsiokos discovered the Chair44 monotile and its 44 neighbor positions.
- Goodman-Strauss presented it as a marked L-tile and gave the hierarchical proof of aperiodicity.
Andy Huss
- Brought the paper and asked for a printable, playable set.
- Printed and assembled the tiles and confirmed that eight lock into a supertile.
- Proposed marking orientation with three faces, one of each kind, one per axis.
- Required that the chosen faces not touch within a tile, and asked for the faces to be numbered and every configuration enumerated.
- Directed assembly of all 27 supertiles, the grouping into categories, and the choice of faces 3, 6, 9.
- Extended the study to every way of choosing three faces, sorted by how they touch, and then to every choice of one to nine faces.
- Specified flat plates with 45° mitered joints, and the step up to the 2-level supertile.
- Conceived this visualizer and set its audience and scope.
Claude (Anthropic AI model)
- Read the arrow markings from the paper's figures and checked them against the assembly rule.
- Verified the 44 legal placements and designed the bump-and-pocket keys, with clearance and collision tests.
- Wrote the plain-language explanation of aperiodicity.
- Proved which face sets share no edges, counted the 27 → 9 → 5 classes, sorted all 84 three-face choices by touch pattern, and built all 511 choices.
- Built and measured all 27 supertiles, including connectivity at 512 tiles and the x ↔ y mirror finding.
- Wrote the plate and miter construction and checked every model for overlaps.
- Built this page and its viewer.
Glossary
- Tiling
- A covering of space by copies of shapes with no gaps and no overlaps.
- Prototile
- One of the shapes a tiling is built from; every tile is a copy of a prototile.
- Monotile
- A single prototile; every tile in the tiling is a copy of the same shape.
- Periodic
- A tiling that can be slid some fixed distance and land exactly on itself.
- Aperiodic
- A set of tiles whose tilings are never periodic.
- Strongly aperiodic
- Aperiodic even when rotations are taken into account: no tiling has any symmetry that shifts it, including a screw motion (a slide combined with a turn).
- L-tile (chair)
- A 2 × 2 × 2 cube with one corner cube removed: seven unit cubes and nine faces.
- Rep-tile
- A shape that can be divided into smaller copies of itself. Eight L-tiles make an L-tile twice as large.
- Chair44
- The L-tile with colored arrows on its 24 panels; the arrows allow exactly 44 ways for two tiles to touch.
- Marking / matching rule
- Information added to a tile's faces, here colored arrows, that restricts which faces may meet.
- Substitution
- The rule that replaces one tile with eight half-size tiles in fixed positions and turns.
- Supertile
- Eight tiles grouped into a tile twice as large. A 2-level supertile is eight supertiles (64 tiles), and so on.
- Hierarchical tiling
- A tiling in which every tile belongs to exactly one supertile, which belongs to exactly one larger supertile, at every scale.
- Orientation
- How a tile is turned in space. A cube shape can be turned 24 ways, and all 24 occur in Chair44 tilings.
- Face kind
- Large square (faces 1–3, the closed back), L-shape (4–6, the front) or small square (7–9, the socket).
- Configuration
- A choice of faces to keep, written by face numbers, such as 3 · 6 · 9. One face of each kind gives 27 choices, any three faces 84, and any number of faces 511.
- Touch pattern
- How the chosen faces meet on one tile. For three faces: none touching, two touching and one loose, all three connected in a chain, or all three touching each other.
- Orientation hidden
- A choice that looks the same after a third of a turn about the tile’s diagonal (made only of whole sets of three, such as all large squares, or the full tile), so it cannot show how a tile is turned.
- Handedness / mirror pair
- Two configurations that are mirror images of each other, labeled A and B.
- Plate
- A chosen face made solid: a flat slab 2.5 mm thick whose outer surface lies on the face.
- Miter
- A joint where two plates meeting at a right angle are each cut at 45°, so they meet along a diagonal.
- Butt joint
- A joint where one plate's square end meets another plate without an angled cut.
- Back-to-back pair
- Two faces from neighboring tiles occupying the same square from opposite sides.
- GLB / STL
- 3D file formats. GLB keeps colors and named parts (open it in Blender); STL is geometry only, the usual format for 3D printing.
Methods and sources
Geometry was computed in Python with exact integer tile coordinates. Solids were built with the Manifold boolean library and checked pairwise for overlap; every mitered model here has zero overlapping volume. Connectivity was measured on the unit squares of the chosen faces, joined where they share an edge. The page renders with three.js. Scale is 20 mm per unit cube and plate thickness 2.5 mm; rescaling is left to the user's own software.
Touch pattern is decided on a single tile by shared edges. Limitations: the plate models use plain butt joints where three or more plates meet at one edge in uncommon arrangements; with many faces kept this becomes common, so choices with five or more faces are mostly butt-jointed, and a few miter ends are trimmed short where they would reach a third plate. The 512-tile figures come from the connectivity analysis; that level is not yet in the viewer.
Sources: C. Goodman-Strauss, Notes on a strongly aperiodic monotile in E³, arXiv:2609.24779 (2026). I. Tsiokos, A strongly aperiodic monotile in three dimensions, arXiv:2609.19214 (2026).