A Continuum of Tiles

It is tempting to read the hat as a lucky find — one precise polygon out of the uncountably many, sitting alone. It is not. The hat is one point on a continuous path of thirteen-sided polygons that all have the same angles and differ only in the lengths of two classes of edge, and the 2023 paper proves that almost every point on that path is an aperiodic monotile.

Write the two lengths as a and b, and the family is Tile(a, b). The hat is Tile(1, √3). The turtle is Tile(√3, 1). Almost everything between them works — the exceptions are the two ends and the midpoint Tile(1, 1), where the two lengths are equal.

The sliders below move a and b. Nothing else changes: thirteen vertices, the same thirteen angles, and a simple polygon at every setting. The two area readouts are computed by different routes — a closed formula, and a shoelace sum over the polygon’s actual coordinates — and can be watched agreeing.

Jump between the three named members with the buttons, then drag somewhere in between and notice that the result looks entirely reasonable — no worse a shape than the hat, and by the paper’s theorem just as aperiodic. The faint outlines behind the tile are the three named members, so you can see how far you have moved.

Three points on the path do not work. At the ends, one of the two lengths reaches zero and the shape stops being a thirteen-gon: Tile(0, 1) is a chevron and Tile(1, 0) is a comet, and both tile the plane periodically. And at the midpoint sits Tile(1, 1), the spectre’s skeleton, where the extra symmetry of equal lengths lets reflected copies combine into periodic tilings. Every other point strictly between the ends is an aperiodic monotile, the hat and the turtle among uncountably many others.

Which reframes the sixty-year search. The answer was not a needle in a haystack. It was a continuum, and the difficulty was never that einsteins are rare — it was that nobody knew where to look, and the shape, once found, took eight kites and a pair of scissors. Then it took four mathematicians and a hundred pages to prove that what David Smith had noticed was true.

There is no reason to expect this to work. A closed polygon needs its edge vectors to sum to zero. Multiply six of its edges by 1.4 and leave the other seven alone, and the shape should fly open.

It does not, because each class of edge closes on its own. Take the six b-edges, lay them nose to tail, and they come back to the origin. Take the a-edges, counting the double-length one twice, and so do they. Two loops, each shut. Scaling either loop leaves it shut, so the tile stays shut too.

Look at each class separately, then both, then drag the scale. Each chain is a closed loop that grows and shrinks without ever coming apart. The closure gap is around 2 × 10⁻¹⁶ — which is zero, as exactly as double-precision arithmetic can say it.

Because every vertex is linear in a and b, the area is a quadratic form: 2√3·a² + 3ab + √3·b². All three coefficients are positive, so the area is positive everywhere in the quadrant and no member ever degenerates. The tile grows and shrinks smoothly across the whole plane.

Hover across the map and watch the area move continuously. The three circles are the shapes with names; the other several hundred cells are perfectly good tiles that nobody has needed to name. The closed form and the measured area disagree nowhere on the grid by more than floating-point noise.

A family needs something to hold it together, and here it is the angles. They never move. Every member has the same thirteen interior angles, every one a multiple of 30°, four of them reflex. Only the distances between the corners change.

Read the drift column. Walking from a tile with edges seven times as long as another’s moves not a single angle by more than a rounding error. Angles are what determine how tiles can meet, which is why the whole family behaves alike and why proving the result for one point nearly proves it for all of them.