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mathematics, in pictures
Series

Circular functions — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · analysis
  2. A circular sector of area 1.10 and a hyperbolic sector of area 0.80. On the left, the unit circle with the sector from (1, 0) to (cos 2.2, sin 2.2), of area 1.100. On the right, the hyperbola x² − y² = 1 with the sector from (1, 0) to (cosh 1.6, sinh 1.6), of area 0.800. In both, the parameter is twice the shaded area.

    The angle that is really an area

    On a unit circle the angle t is the length of arc the point has walked, and it is also twice the area of the slice it has swept. The two readings agree on the circle and part company on the hyperbola — where arc length leads nowhere and area leads straight to cosh, sinh, and the exponential.

    part 2 · analysis
  3. Lissajous figures for every coprime pair of frequencies up to 4. A 4 by 4 grid of Lissajous figures x = sin(pt + 0.3), y = sin(qt), with the crossing count 2pq − p − q under each and the non-coprime pairs left blank.

    When two circular motions come home

    Drive a point across with one sine wave and up and down with another. If the two frequencies are in a whole-number ratio the point retraces a closed figure whose crossings can be counted in advance — 2pq − p − q of them — and if they are not, it never comes back and fills the square, spending twenty times longer in the corners than in the middle.

    part 3 · analysis

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