Take a 2×2×2 block of cubes and knock one corner cube out. The seven that remain make a shape called the chair, and copies of it fill space easily — too easily, in fact, because they can do it in a repeating pattern. Now cover every one of its 24 flat square faces with eight tiny bumps and dents, 192 in all, arranged so that two faces fit together only when the bumps of one drop into the dents of the other. This is Chair44.
The claim made for it is strong: copies of this one solid still fill space, but the bumps rule out every repeating arrangement. A single tile that can only tile without repeating is called an aperiodic monotile, and in the plane the first one was found in 2023. Chair44 is a proposed answer in three dimensions.
Every solid in the view is the same chair, in one of the 24 ways a cube can be turned, and never a mirror image. Mirror copies are not banned — the paper allows them as tiles — but they never mix into the same tiling: part of the claim is that every tiling is homochiral, all of one handedness, and the patches grown here use rotations alone. Colour by Rotation to see the poses; colour by Supertile to see the eight groups the patch was grown in. The chairs are drawn slightly shrunk so you can see them as separate solids; set the gap to 1 and they close up into a solid block with no gaps at all.
The matching rule is drawn three ways. True size is the honest one and looks like nothing: the pyramids are a ten-thousandth of a cell tall, far too small to see. Enlarged pyramids is the convention the paper's own figures use — bases five times over, heights exaggerated — so the bumps and dents become visible. Arrows is Chaim Goodman-Strauss's redrawing of the same rule as one flat arrow per face in three colours: blue meets blue, green meets red. A whole arrow is mirror-symmetric, so it looks the same from either side of its face and can match itself; a half arrow is not, so green and red are one marking seen from its two sides, which is why one may only ever meet the other.
Not by stacking on a lattice. The chair is a rep-tile: eight chairs, turned in the right ways, assemble into a chair twice as large. So a patch is grown by substitution — start with one chair, replace it by its eight children, replace each of those by eight, and so on. The buttons give 1, 8, 64, 512, 4,096, 32,768 and 262,144 chairs, filling a chair 1, 2, 4, 8, 16, 32 and 64 cells across. Every chair is the same solid, so the deep patches cost almost nothing to hold — one matrix and a colour each — and what runs out first is the triangles your graphics card can redraw while you turn the patch around. Past that point the page keeps the depth you asked for and draws the rule more plainly, from pyramids to arrows to the bare chair, and says so underneath.
The Built slider takes the patch apart and puts it back in the order the substitution made it, one chair at a time, and it is worth winding slowly. What appears is not a wall growing sideways but a chair filling in, then a bigger chair made of those, then a bigger one again: colour by supertile and each block of eight arrives as a chair the same shape as the one it sits in. That nesting is the whole argument for aperiodicity, and it is easier to watch than to read.
This is where the aperiodicity is supposed to come from, and the same argument works in the plane for the hat. If every tiling has to be built this way, then any patch you point at sits inside a bigger chair, which sits inside a bigger one still, forever. A repeating pattern has a shift that maps it exactly onto itself, and a hierarchy like that leaves no room for one: the shift would have to respect every level of the nesting at once, and no non-zero shift can.
The page also checks the patch against the paper's atlas. Two chairs that share a face can be in one of 2,388 relative positions if you only require their flat bodies to fit; the features cut that to 44. Every face contact in the patch you are looking at is one of those 44, and the line under the view says so — computed here, not asserted.
The heights are not decorative. Each bump is one of twelve depths, positive for a bump and negative for a dent, and two features can only mate as a bump into a dent of exactly the same depth. Work out which marks are forced to meet which inside the eight-child dissection, close that set under further refinement, and require a bump wherever a dent has to be: the resulting 372 equations split the 192 marks into exactly twelve balanced groups. Numbering those groups 1 to 12 is the table. Each depth is carried by eight bumps and eight dents, so they cancel and the solid still has volume exactly 7 — which this page checks on the tile it draws.
Not fully, and it matters. In the plane, the question was open for decades until David Smith, Joseph Myers, Craig Kaplan and Chaim Goodman-Strauss found the “hat” in 2023. In three dimensions the Schmitt–Conway–Danzer biprism has long been known to tile space only non-periodically, but it has tilings with a screw symmetry of infinite order, so it is not aperiodic in the strong sense. Chair44 is presented as a solid that closes that gap.
That claim is a preprint, not a refereed result. The accompanying machine proof is checked only modulo a named hook per machine-decided step, and the written geometric arguments stand as exposition. So treat the aperiodicity as claimed rather than established. Nothing on this page depends on it: the solid, the eight-child dissection, the patches, the volume and the 44-contact check are all finite computations, and they are checked here and against the project's own generator, vertex by vertex.
Computed in your browser by a port of the project's Chair44 engine, checked against it vertex by vertex on every run of the site's tests.