The counting law
Claim
Q − O = 4 − k. A shape carrying fewer than four offcuts holds spare quarter-discs that nothing can absorb.
Construction
This claim belongs to the cousin family — the shapes made by returning one to four corner offcuts to a circle. It is stated here because it is atomic and because it constrains which shapes can tile in company. It is an argument rather than a verified result, and the page keeps it marked as one.
Let k be the number of offcuts attached to a shape, Q the spare quarter-discs it presents, and O the offcut notches it can receive.
Proof
Each offcut returned to the circle consumes one quarter-disc of the shape's own surplus and supplies one notch, so bookkeeping across the figure gives Q − O = 4 − k directly.
At k = 4, Q = O: the shape is balanced, its surplus and its appetite matched. Below four the difference is positive — quarter-discs with nothing to fill them — and the argument is that no company can absorb them.
Nuna itself is k = 4 and balanced. Note what that does and does not give: balance is a necessary condition for tiling, not a sufficient one. It rules shapes out; it cannot rule one in. Nuna tiles, and it is balanced, but the law is not the proof of the first from the second.
The law is this project's own. It is stated in the library's working notes and used in the cousins book, which hedges it in the same way — if the law holds, only the k = 4 families are candidates. No source file derives it.
Consequence
This turns a large search into a smaller one. Of the 73 cousin families only the 54 at k = 4 are candidates for tiling, and the rest are ruled out before any packing is attempted. The 12 discarded shapes — offcuts joined only to each other, with no circle — are excluded on the same grounds.
Note
This one came out of the cousins work rather than the shape itself. Worth noting when a law arrives from a side road.
Open questions
- No file on hand carries the brute-force packing runs that would confirm every k < 4 shape fails. Until those exist the law is an argument, not a result.
- The law itself appears in no reference file — only in this library's own writing. It needs either a derivation somebody else could check, or demotion to a conjecture.
Checks
| What | How | Source |
|---|---|---|
| Area at k = 4 | πr2 + 4 × (4 − π)r2/4 = 4r2 = d2 for every four-piece cousin | nuna-cousins-reference.md |
| Shortfall below k = 4 | (4 − k)(4 − π)r²/4, by the same arithmetic | nuna-cousins-reference.md |
| Family counts | 73 families, of which 54 sit at k = 4 | nuna-cousins-reference.md |
| The law itself | No source file states it. It appears only in this library's own writing, hedged there as conditional | — |