Fermat Spiral
r = sqrt(θ) · (1 + 0.2 · sin(t))
Open in the app The dials and keys named below are the app's.
What it draws
√θ is the whole shape. Squared it reads r² = θ, which is Fermat's spiral. The pen leaves the origin and is out at √(2π) ≈ 2.51 after one turn. It reaches √(12π) ≈ 6.14 at the end of the sixth turn, where Turns of θ opens. The second factor, 1 + 0.2·sin(t), multiplies every radius alike, so the spiral keeps its shape and only swells and shrinks by a fifth either way, once every 2π ≈ 6.28 seconds at Speed 1.
Why the turns crowd
A spiral that grows as a square root is the one that grows in area rather than in radius. The disc out to the curve at angle θ has area πr² = πθ, so every radian adds the same π of area, however far out the pen already is. Radius has to pay for that. The gap between one crossing of a bearing and the next falls off as 2.51, 1.04, 0.80, 0.67, 0.59, 0.54, and the arms grow tighter without ever touching.
That is the law a sunflower head follows. Helmut Vogel's 1979 model puts the nth seed at radius c·√n and turns it by the golden angle, 137.5°, from the one before. The square root gives every seed the same area, and the angle keeps any two of them from lining up. The screen shows only the θ ≥ 0 half of the curve; the full Fermat spiral has a mirror branch at r = −√θ.
History
Pierre de Fermat wrote the spiral down in 1636, working out the area it sweeps as an exercise in his method for quadratures.
Try
- Raise Turns of θ to 20 and the arms crowd into a disc, the sunflower packing.
- Add the other branch by hand: −√(θ) · (1 + 0.2·sin(t)) draws the missing half, turned through half a turn.
- Change the root to a plain θ, θ · (1 + 0.2·sin(t)): an Archimedean spiral, whose arms keep a constant gap instead of tightening.
- Deepen the breath to √(θ) · (1 + 0.9·sin(t)): the spiral now shrinks to a tenth of its reach and swells back out to nearly twice it.