Generator

A Reuleaux triangle

A generator in the geometry library, called 25 times across 5 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

reuleaux is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

A Reuleaux triangle. A curve of constant width on 3 vertices, with 6 pairs of parallel supporting lines drawn across it. Every pair is 180.1 apart.

Rolling without rising

Constant width is not a property of the circle

Area at equal width: the triangle least, the circle most

Area falling to the wall where convexity fails

A wobbling function with a flat sum

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

the 3-sided curve is built at the common width ×4
the 3-sided Reuleaux curve has one width at every angle ×4
the integral of h round a 3-sided curve is π times the width ×3
a Reuleaux polygon needs an odd number of sides ×1
all the shapes have one width ×1
and is convex ×1
and it holds for the solid of revolution, measured ×1
and it is the same in every direction ×1
and its radius of curvature stays positive, so it is convex ×1
and lands between the ball and the conjectured minimum ×1
and the circle's is a quarter of π times it ×1
and the drawn boundary really is that long ×1
and the drawn points confirm it, direction by direction ×1
and the largest is the edge times (√6 − 1)/√2, across two opposite curved edges ×1
and the Reuleaux triangle the least ×1
and the smooth family gets only part of the way to the triangle ×1
and the support function itself is not flat, so the flat sum says something ×1
and the sweep runs past the point where convexity fails ×1
and they do so in every direction, not only the one drawn ×1
at the width the function was given ×1
between one and four smooth amplitudes, each at most twelve ×1
between thirty and a hundred and sixty samples per angle ×1
between twenty and two hundred samples ×1
between two and five odd side counts, each at most eleven ×1
Blaschke's relation holds for the ball, which is the check that it is stated right ×1
convexity fails exactly where h + h″ first vanishes ×1
every arc bulges away from the centre ×1
every harmonic in the support function is odd ×1
every shape in the table has the same width ×1
ground and plank stay one width apart ×1
so the solid is not of constant width ×1
the area falls the whole way, so the minimum is on the constraint boundary ×1
the boundary is sampled between five hundred and twenty thousand times ×1
the built curve has constant width ×1
the circle encloses the most ×1
the circle is convex ×1
the four corners are equally spaced ×1
the generating curve has the stated width ×1
the harmonic is odd, between three and seven ×1
the least is about nine tenths of the most ×1
the plotted sum is flat at the width ×1
the smallest width is the edge itself ×1
the smooth curve has the common width ×1
the spun triangle encloses less than the ball ×1
the triangle's area is (π − √3)/2 times the square of the width ×1
the two support distances in opposite directions add to the width ×1
the view is one the family draws ×1
the width does not depend on how far it has rolled ×1
the width is positive ×1
the width is the arc radius ×1
though it misses by only a few per cent ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Geometry

Round is not the only way to be the same width

A shape that measures the same in every direction sounds like a description of a circle. It is not — there are infinitely many others, one of them is on a coin in most people's pockets, and a drill built from one cuts a nearly square hole.

Geometry

The least area a width can hold

Barbier's theorem says every curve of constant width has the same perimeter, which removes perimeter as a way of telling the family apart. Area is not like that — the circle holds the most and the Reuleaux triangle the least — and the reason the minimiser has corners is a constraint rather than a preference.

Geometry

The most area a fence can hold

One length of boundary, and the question of what shape to bend it into. The answer is a circle, everybody knows it, and the argument that convinced the nineteenth century turned out to prove something slightly different.

Geometry

The same question in space

Intersect four balls at the corners of a tetrahedron and the result is not of constant width — it misses by two and a half per cent, computed exactly. Repairing it gives a body that is, and whether that body is the smallest of its kind has been open for a century.

Geometry

The shape described from outside

A convex shape can be given by its boundary or by the family of lines that touch it, and the second description turns the constant-width condition into one line of arithmetic — after which the perimeter falls out, and curves with no corners at all can simply be written down.

The whole library · What the figures prove