Deepest known
Shown below: the deepest known tile of the selected grid, its coronas growing ring by ring. Click another card, or any shape marked ▸ corona in the lists further down, to show its corona here. Use the ring slider to step through the rings.
Each colour is one corona ring around the centre tile. Black lines are the outer edges of coronas 0 to k, holes filled. Gaps in the outermost ring are holes, which Hh allows there only.
By size
One row per number of cells. The H columns count the free non-tiling shapes of that size with that Heesch number. "Complete" means every shape of that size has been checked; "found so far" rows count only the shapes known at this snapshot, not totals.
▸ Click a number to list those shapes. Then click a shape marked ▸ corona to see its corona above.
| cells | H = 2 | H = 3 | H = 4 | H ≥ 5 | status |
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| corona | shape | cells | H | coordinates | corona runs |
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Complete counts
Free shapes are counted up to rotation and reflection. Tilers have an explicit periodic tiling. The H columns count hole-free non-tilers by Heesch number. "Escalated" counts shapes with a corona 2 that are not yet resolved. A sample row shows the counts in the part done so far, not final totals.
| run | status | free shapes | with holes | hole-free | tilers | H = 0 | H = 1 | H = 2 | H = 3 | H = 4 | escalated |
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Definitions
- Heesch number
- Hh: the largest k for which the tile has a k-corona, holes allowed in the outermost corona only. Copies are placed on the grid, under every rotation and reflection of the grid; all copies are congruent to the one shape.
- A recorded H
- H = k only when corona k was found and corona k+1 was proved impossible by a completed SAT run. A non-tiler is proved by that same refutation.
- Tilers
- A periodic tiling was found on a torus: up to 8 copies per period for squares, 12 for triangles and hexagons, and up to 16 for any shape reaching corona 4.
- Corona checks
- Every corona shown here passed an audit before publication: every copy congruent to the shape, no overlaps, inner coronas without holes, each corona surrounding the one before it, and a completed refutation of the next corona. The named large tiles were also checked for pinches in the inner coronas.
- Coordinates
- Squares: x,y per cell. Triangles: x,y,t, with t = 0 for the up triangle (x,y), (x+1,y), (x,y+1) and t = 1 for the down triangle (x+1,y), (x+1,y+1), (x,y+1), in the basis (1,0), (½,√3⁄2). Hexagons: q,r, the hexagon centred at 3q·(1,0) + 3r·(½,√3⁄2) in the same basis. Kisrhombille: the outline's vertices a,b in the same basis; the cells are the 30-60-90 triangles (drafters) of the kisrhombille grid whose 6-fold points are the a,b with a and b even and (a − b)/2 divisible by 3.