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 The dials and keys named below are the app's.
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.