In the 2005 American Mathematical Monthly article "Geometry and Number Theory on Clovers" (JSTOR link), Cox and Shurman state as their Theorem 2:
Cardioid Theorem. Let $n$ be a positive integer. Then the cardioid can be divided into $n$ arcs of equal length using a straightedge and compass.
I'm calling the property that a curve may be divided into arbitrarily-many arcs of equal length using a straightedge and compass the "Cardioid Property". My question:
What other polar curves have the Cardioid Property? I have found next-to-nothing on this topic in literature.
Cox and Shurman note that the circle and (Bernoulli) lemniscate can be divided into $n$ equal-length arcs via straightedge and compass iff $n=2^a p_1 p_2\cdots p_r$ where the $p_i$ are distinct Fermat primes. (These results are originally due, respectively, to Gauss and Abel.) The authors focus on "origami" constructions on "$m$-clovers" $r^{m/2} = \cos\frac{m}{2}\theta$ (particularly, $m=3$ and $m=4$ (the lemniscate)).