Library · 2. Core Claims · mark B

Perimeter

Claim

The boundary is 3πr — six quarter-turns of a single radius, and nothing else.

Construction

Walk the boundary of an upright tile from the left seam point. The head circle carries you three quarters of the way around, a 270° arc of radius r, and sets you down at the right seam point. From there the diamond takes over: three quarter-arcs of radius r, each concave, each centred on one of the circles that made the void.

The diamond's fourth arc is missing. It is the seam, the line erased when circle and void were fused, and it is interior now rather than boundary.

Fig. 1 — one 270° head arc and three concave quarter-arcs.

Fig. 2 — the boundary walked twice. Left: the 270° head arc in the accent colour, the three concave quarter arcs in ink, and the erased seam dashed. Right: the same outline read as six quarter turns, with the junctions marked

Proof

The head contributes three quarters of a circumference: (3/4) × 2πr = 3πr/2.

The diamond contributes three of its four quarter-arcs, each of length πr/2, so 3πr/2 again.

P = 3πr/2 + 3πr/2 = 3πr = (3/2)πd

Counted the other way, the 270° head is itself three quarter-turns, so the boundary is six quarter-turns of radius r laid end to end — three convex, three concave — and the perimeter is 6 × πr/2.

Where the turning is

Count the arcs as turning and the books do not balance: three convex quarters at +90° and three concave at −90° sum to zero, while the total signed turning of any closed curve must be a full turn.

The turning is in the corners, and measuring them settles it. The path has four arc commands, not six, because the head is one smooth 270° arc whose own quarter divisions are not corners at all. Of the four real junctions, two have no tangent jump and two reverse completely: the jumps measure 0°, −180°, −180°, 0°, summing to −360°.

So the arcs contribute no net turning and the two cusps contribute all of it. The six-arc reading is the right way to count length and the wrong way to count turning.

Note

Did you count the arcs before or after you measured?

Is the six-arc reading how you think of the shape, or the 270-plus-three reading?

Open questions

The two cusps are where the shape stops being smooth, and nothing in the library says what that costs in fabrication — a 180° reversal is a hard point to cut, cast or stitch around.

Checks

What	How	Source
Perimeter of the exact path	Path length over the four arc commands: 942.4768 at r = 100, against 3πr = 942.4778; the residual is sampling	_src/core.py
Multiple of a quarter-arc	Six arcs of radius r, so the perimeter must be an integer multiple of πr/2 — it is six of them	reference/geometry-constants.md
Turning	Corner jumps measured round the path: 0°, −180°, −180°, 0°, summing to a full turn	_src/core.py
Against the founding circle	3πr : 2πr = 3 : 2, measurable on any drawing	reference/geometry-constants.md