Polar Roses

Rhodonea curves: r = cos(k theta)

r = cos(3θ)
Rhodonea curves r = cos(k*theta) produce rose-like shapes. When k = n/d (rational): if n*d is odd, the rose has n*d petals; if either is even, it has 2*n*d petals. Integer k=2 gives 4 petals, k=3 gives 3, k=5 gives 5 — but rational k opens up infinitely intricate families. These curves were studied by Luigi Guido Grandi in 1723, who named them "rhodoneae".