Concept

Sine — where it appears

The vertical coordinate of a point travelling round the unit circle, as a function of the angle turned through. It is the derivative of nothing simpler than cosine, and it is what a circle looks like from the side as it turns.

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

A circle unrolled into a sine wave. On the left a radius turns through an angle; on the right the height of its tip is plotted against the angle, tracing a sine curve.

A sine wave is a circle seen from the side

Sine is introduced as a ratio in a right triangle, which is true and explains nothing about why its graph is a wave. There is a better picture.

analysis · Circular functions

Partial sums of the square wave. Approximations using 1, 3, 7, 21 terms; the corners sharpen but a fixed overshoot remains.

A square wave built entirely out of round ones

Add enough sine waves together and flat tops and vertical cliffs appear from nothing. Almost — there is a 9% overshoot that never goes away, and it is not a bug.

analysis · Fourier series

Named alongside it

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

Periodicity Pi Continuity Convergence Cosine Derivative Gibbs' phenomenon Harmonics Orthogonality Radian Simple harmonic motion Spectrum

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