The combination-shape families
Claim
The 717 connected combinations reduce to 362 families under the eight symmetries of the square — a figure the project disputes with itself.
Construction
The badge asks a legality question: which tiles can stand together around a centre without wanting the same circle or the same diamond? The answer is 1,157. But legality is a rule about conflict, not about contact, and an arrangement can be perfectly legal while its ink falls into separate pieces.
Asking which arrangements make a single unbroken shape gives 717. Reduce those by turn and mirror and 362 families remain, twenty of which the reference marks unique: their orbit has one member, so no turn or mirror of them produces a different arrangement in the legal set.
The sixteen tiles around the centre are served by eight distinct circles and eight distinct diamonds, which is why conflict is possible: three tiles want the diamond at (0, −1), three want (−1, 0), three want (1, 0).
Proof
Tier by tier, the three counts behave very differently. Legal arrangements run 16, 98, 288, 417, 274, 64; connected shapes 16, 40, 100, 223, 274, 64; families 5, 17, 53, 112, 141, 34.
Tier two is where the two questions disagree most: 98 legal pairs, of which only 40 hold together. A pair is the smallest arrangement that can fail and the sparsest, so that is where the distinction earns its keep.
At tiers five and six every legal arrangement is connected. Above four tiles legality implies wholeness and the connectivity test stops doing any work. Why the threshold falls exactly there is observed, not derived.
362, or 317
The technical reference gives 362 families. A figure of 317 also circulates in this project, attached to the same dataset. They cannot both be right, and the gap of 45 is too large to be a rounding of anything.
A weaker reduction cannot explain it. Collapsing by rotation only, without mirrors, would push the count up from 362 as chiral pairs separated; and no larger symmetry group than the eight of the square is available on a square lattice.
A stricter definition of contact would explain it. If point joins do not count as connection — if only a shared arc makes a shape whole — then some arrangements drop out, the 717 falls, and the family count falls with it. That is the one change that moves the number in the right direction, and it is exactly the discretionary decision in the construction above.
The test is cheap: re-run the enumeration with point joins excluded and see whether the count lands on 317. If it does, the two numbers answer two different questions and both should be published with their contact rule attached. If it does not, one of them is simply wrong. Until then this page uses 362 and marks it provisional.
Seven diamonds, or eight
The master reference records seven used diamonds on the 16-point badge. The arrangement table yields eight, and the exclusion relations derived from eight reproduce the badge's own exclusion key exactly. Eight is almost certainly right and the master reference has a slip.
Note
The 717 came later than the 1,157. What made you go looking for the smaller number?
Twenty shapes have no cousin at all. Does that mean anything to you, or only to the count?
Open questions
- 362 or 317. Unresolved, and nothing downstream should cite either as settled. The index entry for this book leads with 317 while the page leads with 362; whichever way the test comes out, the two should be brought into line.
- Whether point joins count as contact. A decision, not a result.
- Why no two-tile family is a one-member orbit — forty connected pairs, seventeen families, and not one symmetric enough. Forced, or an accident of this badge?
- The master reference's diamond count needs correcting at source.
Checks
| What | How | Source |
|---|---|---|
| Legal arrangements | 1,157, recomputed from the exclusion key | _src/core.py |
| Connected shapes | 717, as 16 + 40 + 100 + 223 + 274 + 64 | nuna-combination-shapes-reference.md |
| Families | 362, as 5 + 17 + 53 + 112 + 141 + 34 — disputed, see above | nuna-combination-shapes-reference.md |
| Unique families | 20, as 1 + 0 + 6 + 1 + 8 + 4 — one-member orbits | nuna-combination-shapes-reference.md |
| Distinct diamonds | 8; the master reference says 7 | nuna-combination-shapes-reference.md |
| Point joins count | A decision recorded in the reference's contact definition, not a result | nuna-combination-shapes-reference.md |