Library · 3. Explorations and Theory · mark AE

The three forms

Claim

A grid with a line erased, a circle given its corners back, and a run of waves all produce the same shape — but the wave construction fails as usually stated.

Construction

A New Shape names three ways to arrive at the figure: a grid of circles with a line erased; a circle cut from a square with its corners redistributed; and wavy vertical and wavy horizontal segments laid across one another.

They are not interchangeable in what they are good for. The circle grid is the origin story and explains nothing else. The square is the proof, because that is where the area falls out. The waves are the generalisation, and the only one of the three that extends beyond a single tile.

The wave of the third form is a chain of semicircles of radius r, alternating side, with period 4r, baselines 2r apart. Each semicircle is exactly half of one circle of the grid.

Fig. 1 — the two waves. Semicircles above, quarter-arcs below; the chords differ.

Fig. 2 — the three constructions, each with the outline it produces drawn over it. The wave panel shows the construction only: in phase, as drawn here, it does not give a tiling

Proof

Each form produces a closed curve, and a curve built from circular arcs is determined by its arcs. In the first form the circle contributes 270° and the void three of its four quarter-arcs. In the second the circle keeps 270° after one arc join and the returned pieces present three concave arcs outward. In the third, with the offset below, each circle is drawn for three quarters and the void's fourth arc coincides with the circle's missing quarter.

All three give a 270° convex arc of radius r followed by three concave quarter-arcs of radius r, with the junctions fixed by the lattice. There is no freedom left in which they could differ, and it follows that all three give the same area and the same perimeter.

Why the obvious wave version fails

Every circle sits on one horizontal wave and one vertical wave. The horizontal gives it a top or a bottom, the vertical a left or a right; half plus half overlapping in a quarter leaves three quarters drawn and exactly one quarter missing. Nuna's construction erases exactly one quarter-arc per circle, so this looks like a free tiling.

It is not. With all the waves in phase, every void at one parity receives two seams while every void at the other parity receives none. A void with two tiles claiming it is not a tiling, and a void with none is a hole.

Shifting the vertical waves by half a period from one column to the next makes the seam assignment a bijection. Offsetting by row does the same, and offsetting both also works. Three of the four phasings give a tiling; the one the construction is usually drawn with is the one that does not.

Two different waves

There are two objects in this project called the wave and they are not the same object. The generating wave is semicircles of radius r, period 4r, chord 2r — the one laid in crossing families to produce the grid. The silhouette wave is quarter-circles of radius r, endpoints r√2 apart — the board's frame line and the outer edge of a row of tiles, which is Claim L.

Any drawing that uses one where the other belongs is off by a factor of √2 in its spacing: close enough to look almost right, wrong enough to fail against the tiles.

Note

Which form did you find first, and which one do you actually think in?

The waves came from the original book. Did you know then that they had to close?

Open questions

Whether the three working offsets give the same tiling up to symmetry or genuinely different ones.
What else the waves give. The book says offsetting produces a further family of tessellations; with the phasing understood that family can be enumerated, and it can be asked whether it is the one the badge already counts.
The construction needs correcting wherever it is stated without the offset.
The two waves need different names in the project's own vocabulary.

Checks

What	How	Source
Same six arcs from all three forms	Three convex, three concave, same places; the arcs determine the curve	_src/core.py
Junction chords	All six measure r√2, recomputed from the tile path	_src/core.py
Junctions lie on the cell	Bounding box s × 2s; a third route to the area, A = 2s² = 4r²	_src/core.py
In-phase waves fail	Seam map over a 14×14 block: 60 interior voids at two seams, none at one	reference/geometry-constants.md
Offset waves succeed	Same test: 121 interior voids, all at exactly one	reference/geometry-constants.md
The three forms are named in the source	The origin section lists all three	A New Shape