Library · 2. Core Claims · mark L

The wave is the silhouette

Claim

The board's wave line and the outer edge of a row of tiles are the same curve.

Construction

The board's frame carries a wave along its inner edge, a line that alternates up and down at the lattice pitch. Separately, a row of tiles laid alternately upright and inverted presents an outer edge that also rises and falls.

These were arrived at independently, one from the frame's construction and one from the tiles. The claim is that they are not merely similar.

Fig. 1 — the outer edge of a row of tiles, and the frame's wave, drawn over it.

Fig. 2 — the silhouette wave: quarter arcs of radius r, their centres alternating above and below the baseline, endpoints one lattice step r√2 apart

Proof

Both curves are chains of 90° arcs of radius r whose endpoints lie on a straight baseline spaced r√2 apart, with the sense of curvature alternating from arc to arc.

That the chord of a 90° arc of radius r is r√2 is immediate: the chord subtends a right angle at the centre, so chord² = r² + r². The endpoint spacing therefore equals the lattice pitch of Claim F, which is why the wave registers with the lattice at every crossing.

Two curves built from arcs of the same radius, through the same angle, with the same endpoints and the same alternation, are the same curve. There is no freedom left in which they could differ.

Consequence

This is why the frame fits the tiles without adjustment, and why a base cut to the wave profile seats falling tiles exactly. The board tool builds both from the same arc commands: its fieldBoundary() notes that each edge cell contributes one 90° arc, convex over a circle and concave over a diamond.

Note

This is the moment two separate threads turned out to be one. When did you notice?

Open questions

There are two different waves in this project and they are easy to confuse. The one here has chord r√2; the generating wave of the three-forms construction is a chain of semicircles with chord 2r. They need distinct names.

Checks

What	How	Source
Junction spacing	All six chords on the tile measure r√2 = 141.4214 at r = 100	_src/core.py
Frame and field built alike	`wavePath()` and `fieldBoundary()` emit arc commands only	nuna-board (2).html
Overlay test	Difference the two constructions; deviation beyond drawing tolerance means one was built wrong	reference/geometry-constants.md