Every polyomino, polyiamond, polyhex, polystick, polytrig and polytwig up to the orders below — enumerated from scratch, verified, and drawn. Each figure page separates the three standard equivalence classes: Free, One-Sided and Fixed.
Edge-joined unit squares of the square lattice — the classic polyform family, home of the twelve pentominoes.
Edge-joined equilateral triangles of the triangular lattice, alternating point-up and point-down.
Edge-joined regular hexagons of the honeycomb tiling; the cell centres form a triangular lattice.
Unit line segments of the square lattice joined end to end — also called polyedges or polyforms of the square grid's edges.
Unit line segments of the triangular lattice joined end to end; six segments meet at every vertex.
Unit line segments of the hexagonal (honeycomb) lattice joined end to end; only three segments meet at each vertex.
Counts of free polyforms — the classification that treats rotations and reflections of a figure as the same thing. Raw data: counts.json.
| Family | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| polyominoes | 1 | 1 | 2 | 5 | 12 | 35 | 108 | 369 | 1,285 | · |
| polyiamonds | 1 | 1 | 1 | 3 | 4 | 12 | 24 | 66 | 160 | 448 |
| polyhexes | 1 | 1 | 3 | 7 | 22 | 82 | 333 | 1,448 | · | · |
| polysticks | 1 | 2 | 5 | 16 | 55 | 222 | 950 | · | · | · |
| polytrigs | 1 | 3 | 12 | 60 | 375 | 2,613 | · | · | · | · |
| polytwigs | 1 | 1 | 3 | 4 | 12 | 27 | 78 | · | · | · |
All six families are enumerated by one algorithm over one data model. A point is a pair of integers; a cell is a sorted tuple of points (the four corners of a square, the three vertices of a triangle, the two endpoints of a segment, the centre of a hexagon); a figure is a set of cells. Triangular and hexagonal geometries use coordinates in thirds of the lattice basis, which places honeycomb vertices — the centroids of lattice triangles — exactly on integer coordinates, so the whole enumeration runs in exact integer arithmetic with no floating point and no rounding.
Figures of order n are grown from those of order n−1 by adding one adjacent cell in every possible way and reducing to a canonical translate. The free and one-sided classes then come from canonicalising over the lattice's symmetry group (D4 for square geometries, D6 for triangular and hexagonal ones), which is a set of integer matrices acting on points. Results are checked with the orbit-counting identities that tie the three counts together through each figure's stabiliser, plus connectivity and canonical-form invariance, plus published reference sequences where they exist.