Library · 2. Core Claims · mark M

The construction

Claim

Four equal circles tangent in a square: keep one whole, keep the void they leave, erase the line between them.

Construction

Draw four circles of radius r with centres at the corners of a square of side 2r, so adjacent circles touch. At the centre of the square sits a region belonging to none of them, bounded by four concave arcs.

Take one circle and that void. They meet along a single quarter-arc. Erase it. What is left is one closed curve enclosing both regions.

Fig. 1 — four tangent circles, the void they leave, and the line to erase.

Fig. 2 — the construction in four steps: four tangent circles, one circle and the void kept, the arc they share erased, and the single closed curve that remains

Proof

Before the erasure there are two regions sharing a boundary; after it there is one shape whose boundary is six arcs. Nothing is added and nothing else is removed, which is why every constant in this shelf can be read off the four circles.

The same shape arrives by two other routes — a circle cut from a square with the corners returned, and a figure closed by a run of wave segments. The proof that all three agree is on the explorations shelf; this page records only the first.

Why this page is proposed

This shelf currently opens with the area claim, which presumes the shape already exists. The construction is the only entry here that is definitional rather than derived, and it is what a reader needs first.

It holds one of the four marks still open. If it is admitted, reading order and mark order part company, and the shelf should say so rather than pretend otherwise.

Note

The erasure is a strange move to make on purpose. What made you make it?

Open questions

Whether to admit this mark at all — one of the four open rulings carried over from the handoff.
If admitted, whether the shelf is read in mark order or in reading order.

Checks

What	How	Source
Construction as stated	Four circles radius r, centres on a square of side 2r, adjacent circles tangent	nuna_master_reference.md
The erasure	One circle contributes a 270° major arc; the other three contribute 90° arcs closing the diamond	nuna_master_reference.md
Downstream	Every later claim is measured against this figure; if it were stated wrongly the constants would not come out	reference/geometry-constants.md