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Concept

Frequency

How many times a repeating pattern completes a cycle per unit of time or distance. A signal is described by the frequencies it contains, and sampling it too slowly folds a high frequency onto a low one.

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

A knot drawn with three sines: 5₂ as a Lissajous knot. Frequencies 2, 3, 7 with phases 0.1, 0.7, 0.3; 7 crossings from above; Alexander polynomial 2t − 3 + 2t⁻¹; determinant 7.

Three sines tie a knot

Let a point move with a different sine wave in each of the three directions of space, the three frequencies whole numbers with no common factor, and it traces a closed curve that is usually knotted. With frequencies 2, 3 and 7 the curve is the knot 5₂. The trefoil, the simplest knot of all, can never be made this way: the half-turn that shifting time by half a period performs forces a condition on the knot's polynomial that the trefoil fails.

analysis · Circular functions Two waves through the same samples. A sine of 9 cycles per unit sampled 10 times per unit coincides at every sample with minus a sine of 1 cycle per unit. Samples: 0.000, -0.588, -0.951, -0.951, -0.588, 0.000, 0.588, 0.951, 0.951, 0.588, -0.000.

A wave known only at its ticks

Read a sine wave nine cycles a second at ten instants a second and the readings describe a wave of one cycle a second instead. Read a wave with nothing faster than 3.7 cycles a second at more than 7.4 instants a second and the readings determine it exactly. Both facts are the same fold of the frequency axis, and it is why wagon wheels turn backwards on film.

analysis · Fourier series

Named alongside it

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

AliasingFourier seriesFourier transformInterpolationInvariantKnotKnot determinantLatticeModular arithmeticPeriodicitySamplingSine

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