# Twisted Torus

Shape · (x, y, z) = f(u, v, t)

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

[Open in the app](https://www.wavelace.com/app#p=60) · [This page](https://www.wavelace.com/presets/twisted-torus)

### What it draws

This is a torus of centre radius `3` and tube radius `1`, written with the same angle `v + 2u + t` in all three expressions. The bracket `3 + cos(v + 2u + t)` is the distance from the vertical axis, and the width and depth carry it round as `cos(u)` and `sin(u)`. The second expression is the height, `sin(v + 2u + t)`.

### The same donut

Adding `2u + t` to `v` only shifts where the tube section starts. As `v` sweeps a full `2π` it covers the same circle whatever the shift is, so the surface is exactly the plain torus. Every point on screen is still `1` away from the centre circle of radius `3`, at every instant. The outer edge is at `4`, the hole at `2`, and the tube reaches `±1` in height.

What the twist changes is the *grid*, not the shape. The mesh lines are lines of constant `v`, and the `2u` makes such a line wind twice round the tube while it goes once round the ring. It closes back on itself after a single lap, a rope laid on the donut rather than a hoop.

### The twist in time

The `t` is added to every section at once, so all of them turn together at one radian per second at `Speed` 1. The ribbing appears to crawl along the tube and comes back to where it began after `2π ≈ 6.28` seconds, while the surface underneath never moves at all. It is the clearest case in the app of a moving parametrisation over a still object.

### Try

- Push `Mesh` to its top stop. The lit solid is an ordinary donut, and the animation vanishes with the grid that carried it.
- Change every `2u` to `3u`, as in `(3 + cos(v + 3u + t))·cos(u)`: three windings of the tube per lap instead of two.
- Drop the `t` from all three expressions and the twist freezes, leaving a static braid.
- Drop the `2u` instead, keeping `v + t`: the sections all point the same way again and the whole grid rolls round the tube in step.
- `Top` in the deck: from above the outline is the same pair of circles as the plain torus.

### Read more

- [Torus](https://en.wikipedia.org/wiki/Torus)
- [Torus knot](https://en.wikipedia.org/wiki/Torus_knot)
- [Parametric surface](https://en.wikipedia.org/wiki/Parametric_surface)