Rose 20
r = 2.6 · cos(10θ)
Open in the app The dials and keys named below are the app's.
What it draws
This is the plainest rose there is, and nothing in it depends on the clock, so the curve stands still. cos(10θ) reaches 1 at the ten bearings θ = 0, π/10, 2π/10, …, and 0 halfway between them, π/20 = 9° to either side, where the curve returns to the origin. That gives ten petals 18° wide where the cosine is positive. Where it is negative the radius comes out negative and is plotted in the opposite direction, filling the ten gaps with ten more.
Twenty petals of 18° each is 360° exactly, so they tile the disc edge to edge, each one just touching its neighbours at the centre.
How many petals
For r = a·cos(nθ) with n a whole number, an even n gives 2n petals and an odd n gives n. The reason is the reflection through the origin. For odd n, r(θ + π) = −r(θ), and a negative radius at θ + π is the same point as a positive one at θ, so the negative half lands on top of the positive half. For even n the two miss each other and the count doubles. Ten is even, so this rose has twenty, and one turn of θ draws every one of them.
History
Guido Grandi studied these curves in the 1720s and called them rhodonea, after the rose. His Flores geometrici of 1728 is the study they are named from.
Try
- Make the count odd, 2.6·cos(9θ): nine petals, not eighteen.
- Set it turning with 2.6·cos(10θ + t), one tenth of a radian a second at Speed 1.
- Give the count a half, 2.6·cos(10.5θ), and raise Turns of θ to 2: the curve needs both turns to close, and the petals interleave.
- Turn on trace and follow the pen: out to a tip, back through the centre, out to the tip opposite.