The alphabets

For one radius r and one rule, every seed of this kind was grown, each exactly once:

• Symmetric: the tiles are placed 8-way symmetrically about the circle at the centre, (0, 0): every turn and mirror of the square maps the seed onto itself.
• Small: every circle of the seed lies within r places of the centre on the square measure (|x| ≤ r and |y| ≤ r). A tile's diamond may lie one place further out.
• Legal: no two tiles share a place, and the seed has at least one tile.

Each seed was grown on a wrapping board 64 places across until it died, held still, or repeated a beat. No seed reached the board's edge and none ran past 3,000 generations, so neither limit changed any result. Ornaments are counted once each: turns and mirrors are folded together, and a pulsing ornament is one ornament whichever phase it is in.

• Other symmetries. Seeds with 4-way, mirror, diagonal or turning symmetry, or none, are not enumerated; volume one and volume two hold many of those.
• Larger seeds. Nothing with a circle beyond the radius. Radius 8 has 905,607,229 seeds and has not been attempted.
• Other rules. Radius 7 is enumerated under the Default rule only. None of the 16-neighbour or direction-order rules in volume two are enumerated.
• Other routes to the same size. Exhausted is a statement about seeds, not about every ornament of a given size: an ornament no bigger than these can still grow from a seed of another kind, and the tiers of volume one grow from settled ornaments, not from small seeds.
• Seeds that die are counted, not listed.

Each alphabet is a JSON file beside this page, fetched only when it is opened; radius 7 is one file of 9.6 MB and loads only when asked for. An entry is its number, its tile count, its beat (1 for still) and its seed code.

A seed code is a number in base 36. Written in base 5 it has one digit per circle orbit (12 at radius 5, 16 at radius 6, 20 at radius 7), orbits taken in order of x from −r, then y. A digit of 0 leaves the orbit empty; k places its eight tiles facing the k-th of the orbit's symmetric directions, in the order down, right, up, left. At radius 7 the directions whose eight tiles would overlap each other are left out of that list; at radii 5 and 6 they are kept and such seeds were simply skipped. Ornaments are numbered by tile count, smallest first.