Concept

Perimeter

The length of the boundary of a plane region. Compared with the area it encloses it measures roundness, since the isoperimetric inequality makes the disc the shape of least perimeter for its area.

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

The two numbers that make up a width. A Reuleaux polygon with 3 sides, its centre marked as the origin, and the two supporting lines with normals 33° and 213°. The perpendicular distances from the origin to the two lines are marked; they add to the width.

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.

geometry · Constant width

The longest perimeter a polygon of diameter one can have. Reinhardt bound 2n sin(pi/2n) for n=3..16 (reached unless n is a power of two), regular n-gon perimeters at diameter 1, and the best quadrilateral found 3.035273 against 2+4 sin 15 = 3.035276.

The longest fence of diameter one

Among polygons with n sides and diameter one, the one with the most area is regular only when n is odd. Ask for the most perimeter instead and the answer is kinder: Karl Reinhardt proved in 1922 that the longest perimeter is 2n sin(π/2n) whenever n has any odd factor at all, reached by polygons cut from curves of constant width, many of them not regular. Only when n is a power of two does every polygon fall short — and the best quadrilateral, found by searching every one, is an equilateral triangle with one corner bent out.

geometry · Isoperimetric

Named alongside it

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

Constant width Convex hull Convexity Curvature Diameter Duality Fourier series Isoperimetric problem Reuleaux triangle Support function

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