# Seashell

Shape · (x, y, z) = f(u, v, t)

`((1 − v/(2 · π)) · cos(2v) · (1 + cos(u)) + 0.2 · cos(2v),  2v/π − 1 +(1 − v/(2 · π)) · sin(u),  (1 − v/(2 · π)) · sin(2v) · (1 + cos(u)) + 0.2 · sin(2v))`

[Open in the app](https://www.wavelace.com/app#p=63) · [This page](https://www.wavelace.com/presets/seashell)

### What it draws

The parameter `v` coils the shell and `u` goes round the tube. The pair `cos(2v)` and `sin(2v)` in the width and depth points the tube outwards from the vertical axis. The doubled angle means two complete whorls as `v` runs `0 → 2π`. The second expression is the height, `2v/π − 1 + (1 − v/(2π))·sin(u)`, whose first part climbs steadily from `−1` to `3`, four units of rise over the two turns.

Two things happen at once, and the factor `1 − v/(2π)` does both. It shrinks from `1` to `0`, so the tube tapers to a point at the tip. It is also the radius of the tube's circular section: writing that section as `(a + a·cos(u), a·sin(u))` with `a = 1 − v/(2π)` gives a circle of radius `a` whose centre sits `a + 0.2` out from the axis.

So the aperture at `v = 0` reaches from radius `0.2` to `2.2`, and one turn later only to `1.2`. The trailing `0.2` keeps every whorl clear of a thin central column, the columella a real shell winds around.

### How shells are built

Henry Moseley set out the geometry in 1838. A shell can be treated as one cross-section curve which stays similar to itself while it grows and revolves about a fixed axis. Growth in a real shell is geometric, each whorl a fixed multiple of the one before, which is why the profile is a logarithmic spiral and the coil could in principle go on forever. This preset shrinks by a constant amount per turn instead, a linear taper, so it runs out of shell and closes at a sharp point after exactly two whorls.

### Try

- Set `Span of v (×π)` to 4. Past `v = 2π` the factor `1 − v/(2π)` turns negative, and the tube flares open again above the tip into a second, inverted cone.
- Change every `2v` to `3v`, as in `(1 − v/(2π))·cos(3v)·(1 + cos(u)) + 0.2·cos(3v)`: three tighter whorls over the same rise.
- Change both `0.2` to `0.6` for a fatter columella, with the whorls held further off the axis.
- Slow the climb: write the height as `v/π − 1 + (1 − v/(2π))·sin(u)` and the whorls press together into a flatter, more discoid shell.
- `Side` in the deck: the profile shows the four units of rise and the taper along it.

### Read more

- [Logarithmic spiral](https://en.wikipedia.org/wiki/Logarithmic_spiral)
- [Seashell](https://en.wikipedia.org/wiki/Seashell)
- [Surface of revolution](https://en.wikipedia.org/wiki/Surface_of_revolution)
- [Nautilus](https://en.wikipedia.org/wiki/Nautilus)