Concept

Fourier series

A periodic function written as a sum of sines and cosines at whole-number multiples of one frequency. It turns a condition on a shape into a condition on which frequencies may appear, and differentiating becomes multiplication by the frequency.

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

6 cosines, and a curve with no tangent anywhere. Partial sums of a sum of cosines whose amplitudes shrink geometrically and whose frequencies grow faster. Each term adds finer detail; the curve converges and its slopes do not.

A curve with a corner at every point

Continuity means a curve can be drawn without lifting the pen. Differentiability means it has a tangent. The first was assumed to nearly imply the second until 1872, when Weierstrass exhibited a curve that is continuous everywhere and has a tangent nowhere — and it is a sum of cosines.

analysis · The derivative

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

Named alongside it

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

Constant width Continuity Convergence Convex hull Convexity Counterexample Curvature Difference quotient Differentiability Duality Fractal dimension Perimeter

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