Math gives you a flower
One equation draws every petal. Change one number and the flower changes.
r = cos(kθ)Every petal in this flower comes from one equation in polar form: go around the centre at angle θ, and draw a point at distance
r = cos(kθ)
Counting petals
The video starts with k = 4 and 8 petals, then slides k to 5, then 6, then back to 4. The number of petals follows a simple rule:
| k | Petals |
|---|---|
| 4 (even) | 8 |
| 5 (odd) | 5 |
| 6 (even) | 12 |
For odd k, the curve retraces itself on the second half-turn, so each petal is drawn twice and you see k petals. For even k, the negative values of the cosine draw a new set of petals between the old ones, so you see 2k.
Between whole numbers
While k slides from one whole number to the next, the curve stops closing neatly and the petals smear and overlap. The flower snaps back into shape whenever k lands on a whole number.
These curves are called rose curves, or rhodonea, named by the Italian mathematician Guido Grandi around 1725. The video draws 18 nested copies so the petals glow.
