The isoperimetric ratio
Claim
A/P2 = 4/(9π2) ≈ 0.04503 — about 57 per cent of what a circle of the same perimeter encloses.
Construction
The isoperimetric ratio A/P2 is dimensionless: it does not change with r, and it is the standard way to ask how efficiently a closed curve encloses area. The circle maximises it at 1/(4π) ≈ 0.07958, and every other shape scores lower.
Proof
Substitute the two constants already established, A = 4r2 from Claim A and P = 3πr from Claim B.
A/P2 = 4r2 / (9π2r2) = 4/(9π2) ≈ 0.045032
Against the circle: 0.045032 / 0.079577 ≈ 0.5659, so nuna encloses about 56.6 per cent of the area a circle of equal perimeter would. A square scores 1/16 = 0.0625, and nuna against the square is 0.045032 / 0.0625 ≈ 0.72. Nuna is less efficient than both, and markedly so.
What it costs
This is the price of the shape, stated plainly. A nuna tile and a square tile of the same width enclose the same area, by Claim A, but the nuna carries more edge: 3πr against the square's 8r, which is about 18 per cent more. In manufacture that is more cutting per unit area, more glaze on a ceramic edge, more grout in a floor, more perimeter to chip.
The often-quoted “half again” is a different comparison — nuna's 3πr against the founding circle's 2πr, which is exactly 1.5. Against a square of the same width the factor is 1.178.
It is worth having the number in the library rather than leaving a fabricator to discover it. The shape is not defended on efficiency.
Note
A claim that says what the shape is bad at. Whether it belongs on this shelf is your call.
Open questions
- One of the four open marks.
- Whether the applied shelf should carry this figure as a cost line rather than leaving it here among the geometry.
Checks
| What | How | Source |
|---|---|---|
| Arithmetic, recomputed | 4/(9π²) = 0.045032; circle 0.079577; square 0.062500; ratios 0.5659 and 0.7205 | reference/geometry-constants.md |
| Edge, against a square | 3πr = 942.478 against 8r = 800 at equal width: 1.178 | _src/core.py |
| Inputs | A = 4r2 (Claim A) and P = 3πr (Claim B), both checked separately | _src/core.py |
| Sanity | Nuna must fall below both the circle and the square, and it does | reference/geometry-constants.md |