# Trefoil Knot

Curve · (x, y, z) = f(u, t)

`(sin(u + t) + 2 · sin(2u + 2t),  cos(u + t) − 2 · cos(2u + 2t),  −sin(3u + 3t))`

[Open in the app](https://www.wavelace.com/app#p=42) · [This page](https://www.wavelace.com/presets/trefoil-knot)

### What it draws

Sideways and vertically the point sums two circular motions of the same angle `u + t`: a circle of radius `1` turning once, and a circle of radius `2` turning twice the other way. The second expression, `cos(u + t) − 2·cos(2u + 2t)`, is the height of that sum. Their distance from the centre works out as `√(5 − 4·cos(3u + 3t))`, so the strand swings between `1` and `3` from the axis three times a lap.

The third expression, `−sin(3u + 3t)`, lifts the strand out of that plane by up to `1` in depth, three times a lap as well. That is exactly what is needed to turn three would-be crossings into three over-and-unders.

### Why it is a knot

This is the *trefoil*, the simplest closed curve in space that cannot be pulled straight without cutting it. It is the first entry in every knot table, and the only knot with three crossings. It is also the torus knot of type `(2, 3)`, the curve that goes twice round one way and three times the other on the surface of a doughnut. That is where the frequencies `1, 2, 3` come from. Advancing `u` by `2π/3` gives the same curve turned by `2π/3` about the depth axis, so the figure repeats three times as it is drawn. It is chiral as well: its mirror image is a different knot, and no motion in space brings the two together. Systematic tabulation of knots began with Peter Guthrie Tait in the 1870s.

### Standing still

Every angle here is a multiple of `u + t`, so the clock only slides the parametrisation along the knot. The set of drawn points never changes, and the figure sits still while playing. Switch `trace` on and the motion appears, the pen running once round the strand.

### Try

- `Front` looks straight down the depth axis, at the flat three-lobed shadow with its three crossings.
- Turn on `trace` and watch the pen thread over and under itself at each of the three crossings.
- Set the third expression to `0`. The curve drops into that shadow, the crossings become real intersections, and what is left is not a knot.
- Change the third to `−2·sin(3u + 3t)`: the strands part twice as far in depth, and the knot is the same one.

### Read more

- [Trefoil knot](https://en.wikipedia.org/wiki/Trefoil_knot)
- [Torus knot](https://en.wikipedia.org/wiki/Torus_knot)
- [Knot theory](https://en.wikipedia.org/wiki/Knot_theory)
- [Knot (mathematics)](https://en.wikipedia.org/wiki/Knot_(mathematics))