# Fractal Cosine

Wave · y = f(x, t)

`y = cos(x+t) + cos(3x+t)/2 + cos(9x+t)/4 + cos(27x+t)/8 + cos(81x+t)/16`

[Open in the app](https://www.wavelace.com/app#p=17) · [This page](https://www.wavelace.com/presets/fractal-cosine)

### What it draws

Five cosines are added, each three times as fine and half as tall as the one before it. The wavenumbers are `1, 3, 9, 27, 81`, so crests are `6.28, 2.09, 0.698, 0.233` and `0.0776` apart. The heights are `1, ½, ¼, ⅛` and `1/16`.

The sum can never exceed their total, `1.9375`, and it reaches it. At `x = 0` every term is `cos(t)`, so the origin swings through the full range with period `2π ≈ 6.28` seconds at `Speed` 1. Every wavenumber is odd, so at `x = ±π` every term is `−cos(t)`: those points swing as far, in exact antiphase.

### Why it is fractal

This is a partial sum of the *Weierstrass function*, `Σ aⁿ·cos(bⁿx)` with `a = ½` and `b = 3`. Each term costs half the height and buys three times the detail, so the roughness never smooths out. Carried to infinity the function is continuous everywhere and differentiable nowhere, since `a·b = 1.5` clears the threshold `1` that G. H. Hardy proved sufficient in 1916. Five terms is still a smooth curve, but the structure shows. Magnify it three times across and two up and the same picture returns, one term shorter: `f(x) = cos(x + t) + ½·g(3x)`, where `g` is the same sum without its last term.

The clock is what makes it churn. Every term carries the same `+t` rather than its own, so a term of wavenumber `k` travels at `1/k`. That is one unit per second for the first, then `0.333, 0.111, 0.037` and `0.0123`. The fine ripples all but stand still while the coarse shape moves out from under them.

### History

Weierstrass presented the function in Berlin on 18 July 1872, against the belief that a continuous function must have a derivative except at isolated points. His own proof needed `b` an odd integer and `a·b` above `1 + 3π/2 ≈ 5.71`, which these values do not meet. Bolzano had built such a function around 1831, but it stayed unpublished until 1922.

### Try

- Add a sixth term, `+ cos(243x + t)/32`: crests `0.0259` apart, and the outline barely changes.
- Keep only `cos(x + t) + cos(3x + t)/2 + cos(9x + t)/4`: an ordinary wobbly wave, the roughness gone.
- Give each term its own clock: `cos(x + t) + cos(3x + 3t)/2 + cos(9x + 9t)/4 + cos(27x + 27t)/8 + cos(81x + 81t)/16`. Every term now travels at one unit per second, and the shape slides rigidly instead of boiling.
- Turn `Span of x` down to 1: the longest term flattens into a tilt and the fine detail becomes the shape.

### Read more

- [Weierstrass function](https://en.wikipedia.org/wiki/Weierstrass_function)
- [Karl Weierstrass](https://en.wikipedia.org/wiki/Karl_Weierstrass)
- [Self-similarity](https://en.wikipedia.org/wiki/Self-similarity)
- [Fourier series](https://en.wikipedia.org/wiki/Fourier_series)