Record
Every figure on this site was computed and checked, except the ones marked otherwise. Nothing in the three-dimensional list is offered as a theorem.
It is not known whether these cubes have been enumerated before. No literature search has been done. They sit in the general area of polycube box-packing. Clarke’s Poly Pages and Sillke’s tables are the places to look. This site does not claim priority.
From Wichmann and Rigby
| Claim | Status |
|---|---|
| The name, and the flat puzzle: a square cut into four congruent polyominoes by a quarter turn about the centre. | Wichmann–Rigby |
| On a 4 × 4 square there are two: a T pinwheel and an L pinwheel. | Wichmann–Rigby |
Computed
| Claim | Status |
|---|---|
| At order 2 there are 65 dissections and 53 shapes. Twelve shapes are used twice. | Computed |
| 38: a single piece slides straight out, and the cube then comes apart in seven straight slides. 32 of those open one way and 6 open two ways. 18 more: no single piece moves, but a pair can leave together and the rest then separates. 9 cannot be separated by any straight axis motion of any subset: 16, 34, 47, 48, 55, 59, 61, 64, 65. Rotations and dog-leg paths were not tested. | Computed |
| 51 seam patterns. 14 pairs are identical from every side, and in those pairs the pieces are not congruent. No twin pair shares a seam pattern. Each piece shows 12 of the 96 surface squares. | Computed |
| Seven twin pairs are related by a rotation, so one printed set builds both. Five are mirrors only. Of the seven, three also come apart: (8, 24), (25, 44), (53, 57). | Computed |
| Whenever one piece of those 38 can slide free, the cube comes apart in seven straight slides. All eight pieces can leave at once, with no two meeting. Checked along the whole travel. | Computed |
| The pinwheel directions, by piece 0–7: +y, −y, +x, +x, −x, −x, +y, −y. The vertical burst: −z, +z, +z, −z, −z, +z, +z, −z. | Computed |
| Two shapes are a tetromino two layers thick: #1, the T pinwheel, and #13, the L pinwheel. In #1, layers z = 0 and z = 1 match, and so do z = 2 and z = 3. | Computed |
| 34 of the 65 have no 2 × 2 × 1 slab. | Computed |
| Surface areas 28, 30, 32, 34 occur 5, 9, 19, 32 times. Cycle ranks 0, 1, 2, 3 occur 32, 19, 9, 5 times. Bounding boxes 2×2×3, 2×2×4, 2×3×3, 2×3×4, 2×4×4 occur 8, 21, 15, 18, 3 times. | Computed |
| In the plane, through order 7, no Yemeni polyomino has two inequivalent embeddings. | Computed |
Observed
| Claim | Status |
|---|---|
| The 38 where a single piece slides free are exactly the 38 whose piece can be turned so that nothing hangs. The other 27 are exactly the 27 that always leave one or two cubes unsupported. No exceptions among the 65. | Observed |
Open
Order 3, a 6 × 6 × 6 box, has not been enumerated. It is not within reach. The search was profiled: branching around 8 per level, an estimated 1016 nodes or more. There is no count for order 3.
Whether anyone has enumerated the three-dimensional objects before is not known.
The distributions
| Surface area | Count | Cycle rank | Count | Opens along | Count |
|---|---|---|---|---|---|
| 28 | 5 | 0 — a tree | 32 | nothing (locked) | 27 |
| 30 | 9 | 1 | 19 | 1 direction | 32 |
| 32 | 19 | 2 | 9 | 2 directions | 6 |
| 34 | 32 | 3 | 5 |
Bounding boxes, in cells: 2×2×3 → 8, 2×2×4 → 21, 2×3×3 → 15, 2×3×4 → 18, 2×4×4 → 3.