Library · 3. Explorations and Theory · mark AC

The cousins

Claim

Seventy-three shapes are made by giving a circle its corners back; nuna is an ordinary member of the family.

Construction

Nuna is made by taking a circle and the void beside it. The cousins are made by a different and more patient move: take the circle and give it back its corners, one at a time.

Cut a circle of radius r from a square of side 2r and four corner pieces fall away, each an r × r cell with a quarter disc removed, each of area (4 − π)r²/4. Return one to the circle and you have a shape; return two, three or four in any legal arrangement and you have a family.

A piece attaches in exactly two ways. An arc join lays its concave arc along the circle's boundary, which requires the arc-corner to sit at the circle's centre and is therefore possible at only four sites. A leg join fixes it to a piece already placed, sharing a complete straight side — nothing less counts.

Fig. 1 — the offcut, four of which close into the diamond.

Fig. 2 — the two joins. Left: the offcut's own concave arc laid along the circle, which is only possible with its corner at the circle's centre. Right: two offcuts sharing one whole straight side

Fig. 3 — three named shapes: 4·01, the circle with all four corners returned, which is the square; 4·52, which is nuna; and d4·03, the diamond, discarded for having no circle to reach

Proof

Three different numbers can reasonably be called the size of this family, and confusing them is easy. A placement is a specific set of cells and turns fixed in the plane: there are 507. Collapse placements that differ only by a quarter turn and 130 classes remain. Collapse further, allowing mirrors as well — the full eight symmetries of the square — and 73 families remain.

The counts by k are 4, 22, 100 and 381 placements; 1, 6, 25 and 98 turn-classes; 1, 4, 14 and 54 families; with 12 shapes discarded for having no path back to a circle.

The two middle columns recover how many families are their own mirror image. If a is the self-mirror count and b the chiral one, a + b = 73 and a + 2b = 130, giving b = 57 and a = 16. The reference's own marks agree, with one wrinkle worth stating: at k = 4 nine families are marked self and one more — 4·01, the square — is marked unique, which is also a one-member orbit. Nine plus the square makes the ten the derivation predicts.

Nuna is one of the sixteen. It is family 4·52, code A0B1E0H0 — one arc join, four leg joins, self-mirror, four placements — which is a strange thing to discover about a shape you thought you had invented.

Where k = 4 is the line

At k = 4 every cousin has area πr² + 4 × (4 − π)r²/4 = 4r² = d². Fifty-four different shapes, one area, the same as nuna's and the same as the square's.

By the counting law of Claim K, only those 54 are candidates for tiling at all. That is the cousins' main contribution to the rest of the library: it converts a search over 73 shapes into a search over 54, by arithmetic rather than by trying.

The discard pile

Twelve shapes are discarded for joining only to each other, with no path back to a circle. One of them is the curved diamond itself: nuna's own void, considered as a free-standing object, fails the reachability test. It is the most important shape in the discard pile.

Note

The cousins were found by looking, not by searching. What were you actually doing when the family appeared?

Which of the seventy-three do you like, and does liking one mean anything?

Open questions

Which cousins tile — alone, with nuna, or as companion pieces only. The starting notes ask for that three-way split and nothing on file has produced it. This is the largest open item on the shelf.
There is no outside check. Nobody had reason to count nuna cousins before this work, so internal consistency is the only test available, and it is a weak one.
The square and nuna each use all four turns exactly once. Coincidence, or a property of the extremes of the k = 4 set?

Checks

What	How	Source
Placements	507, as row sums 4 + 22 + 100 + 381	nuna-cousins-reference.md
Up to turn	130, as 1 + 6 + 25 + 98	nuna-cousins-reference.md
Families	73, as 1 + 4 + 14 + 54	nuna-cousins-reference.md
Discarded	12, as 1 + 2 + 3 + 6	nuna-cousins-reference.md
Self-mirror split	a + b = 73 and a + 2b = 130 give 16 self-mirror and 57 chiral	nuna-cousins-reference.md
Against the marks	At k = 4: nine rows marked self plus 4·01 marked unique — ten one-member orbits, as derived	nuna-cousins-reference.md
Nuna self-mirror	4·52, four placements — agrees with Claim N, derived independently	nuna-cousins-reference.md