Library · 1. Introduction · front matter

What nuna is

Claim

Nuna is one closed shape made from four equal circles and the void between them: six arcs of a single radius, no straight lines, and an area equal to the square of its own width.

Construction

Take four circles of radius r in a 2×2 grid, each tangent to its neighbours. At their centre sits a curved four-pointed void — the diamond. Nuna is one whole circle fused with that diamond. The anchor circle contributes a 270° major arc, the head; the other three circles each contribute a 90° arc facing inward, and those three close the diamond into a taper.

Equivalently, and this is how it was first found: inscribe a circle in a square, cut off the four corner pieces, and rearrange them. The shape came out of Aaron Black's book A New Shape, found by hand — a grid of circles with a line erased.

Rotated 45° so the taper points down, nuna stands upright: a round crown, tapering shoulders, and a soft concave scallop at the base where the diamond's lower point sits between two corners. Six arcs, all of radius r, and nothing else. No straight lines, no béziers. Every boundary point lies on one of the four founding circles.

The words

Head or crown, the 270° major arc. Diamond, the curved four-pointed void. Taper or shoulders, where the head narrows into the diamond. Corners, the two lowest points of an upright tile. Arrangement, one nuna placed at a specific position and rotation — a circle fused with one adjacent diamond. Badge, a nuna surrounded by a fixed set of circles with a defined set of arrangements around it; two exist, with 16 and 24 arrangements. Exclusion, the rule that two arrangements cannot both be active if they share an anchor circle or share a diamond.

The lattice

Nuna tiles the plane with no gaps and no overlaps. The grid positions sit on a square lattice turned 45°, with step s = r√2, so P(i, j) = (i·s, j·s). Every lattice point is either a circle position or a diamond position, and a tile always joins one circle to one adjacent diamond — which is why a tile can point up, right, down or left depending on which corner its anchor occupies.

Place a tile at every point of one of the two parity classes and the plane is covered exactly. The arithmetic agrees: each tile holds 4r², and those points occur once per 2s² = 4r² of area. Which parity class carries the circles is not settled here — two project files disagree, and book 2.7 holds both conventions without choosing.

Fig. 1 — four tangent circles, the void between them, and the tile they make.

The numbers, and the one trap

Everything in this table was recomputed from the one exact tile path for this page rather than copied across, and checked against the values the project already holds.

Area 4r² = d². Perimeter 3πr, six quarter-arcs of radius r. Width 2r, which is also d, the diameter of the founding circle. Height r(2+3√2)/2 ≈ 3.1213r, giving an aspect ratio of about 1.5607. The diamond holds (4−π)r² and one corner offcut a quarter of that, about 0.2146r². The isoperimetric ratio is 4/(9π²) ≈ 0.04503, about 56.6 per cent of what a circle of the same perimeter holds.

That the area is exactly the square of the width is the shape's central fact, and it is the one worth carrying away from this page. Book 2.1 proves it.

The trap

The lowest point of an upright nuna is not the bottom of the lower arc. It is the two corners where the lower arcs meet. Measuring from the arc's bottom gives r(2+√2) ≈ 3.414r, which is wrong by about nine per cent. The height is 3.1213r. This one has caught people before, which is why both the seed brief and the handoff brief carry it as a named trap.

Drawing it

The exact path lives in _src/core.py as TILE, with the arrangement table, the hub positions, the exclusion key and the function that draws any combination mark. It is the single source of truth for geometry: import it, scale it with a transform, and never retype it. Approximating these arcs with béziers or polygons is the one thing never to do — the whole shape is its arcs.

Note

Aaron's slot.

Open questions

The parity convention. The Starting Kit says odd and the handoff brief says even, and every coordinate-named dataset inherits the answer. Book 2.7 is Contested until it is ruled on, and this page deliberately does not pick a side.
The master reference says there are 7 diamonds. There are 8. The filing system and the handoff brief both carry this as needing correcting at source, and it is still uncorrected.

Checks

What	How	Source
The construction, in words	Four tangent circles, the void kept, one circle kept whole; equivalently a circle inscribed in a square with its corners cut and rearranged	nuna-seed-brief.md; A New Shape
Six arcs, one radius, no straight lines	Stated in the seed brief and visible in the path: four arc commands, all r = 100	nuna-seed-brief.md; _src/core.py
Area 4r²	Recomputed by shoelace over 12,000 points sampled along the exact path: 40000.0000 against 4r² = 40000	_src/core.py
Perimeter 3πr	Recomputed as path length: 942.4768 against 3πr = 942.4778	_src/core.py
Height r(2+3√2)/2	Recomputed from the sampled bounding box: 312.132 at r = 100, against 312.132	_src/core.py
Width 2r	Recomputed from the same bounding box: 200.000	_src/core.py
Diamond area (4−π)r²	Recomputed from the three kept arcs plus the seam: 8584.075 against (4−π)r² = 8584.073	_src/core.py
The constants as the project already holds them	Every value above matches the table of verified constants	reference/geometry-constants.md; nuna-seed-brief.md
The lattice step s = r√2	141.421 at r = 100, recomputed	_src/core.py; nuna-seed-brief.md
The trap: corners, not the arc bottom	Named as the known trap in both briefs; the sampled bounding box confirms the lowest points are the two corners at y = 212.132	nuna-seed-brief.md; START-HERE.md
Which parity carries the circles	Not settled. Two project files disagree; see book 2.7	—