Concept

Reuleaux triangle

The constant-width shape bounded by three circular arcs, each centred at the opposite corner. It is the constant-width shape of least area, and it rolls under a plank as smoothly as a circle while having no centre to turn about.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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.

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 · Constant width

One perimeter of 300, spent five ways. Regular polygons all of the same perimeter, drawn to scale beside the circle of that perimeter, with the area each encloses and the ratio 4πA/L².

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 · Isoperimetric

Named alongside it

The objects these essays reach for when they reach for this one.

Area Constant width Convexity Barbier's theorem Circle Existence proof Integral geometry Optimality Pi Regular polygon Scaling Support function

All concepts