Developable roller

A developable roller is a 3d solid with one face such that it can roll along a flat surface with every point on its face touching the surface as it rolls.

Classes
There are several known classes of developable rollers.

Prime polysphericons
The ($p$,$k$)-polysphericon is constructed by taking the solid of rotation of a regular $p$-gon, bisecting it along an $p$-gon, rotating one of the halves by 2$k&pi;/n$, and recombining the two halves along their $p$-gonal faces. A (2$p$,$k$)-polysphericon is a developable roller iff $$\gcd(p,k)=1$$. A ($p$,$l$)-polysphericon where $p$ is odd, is never a developable roller. The result can still have a single developable face, however it will not roll because the solid of rotation has a circular face where the axis of rotation intersects an edge which results in culs-de-sac which prevent the polysphericon from rolling.

The polysphericons which are developable rollers are called prime polysphericons. The prime polysphericons correspond 1-to-1 with the regular polygons (including the degenerate digon) with the $n/k$-gon corresponding to the (2$n$,$k$)-polysphericon.

Platonicons
The platonicons are a class of developable rollers based on the platonic solids.