Femisphere

The femisphere is a solid that has one single surface, two edges, and four vertices.

Description
The form of the femisphere is reminiscent of that of a sphericon but without straight lines. Instead of that, it has circular arcs of arbitrary radius. For this reason, when rolled over a sphere, it contacts the whole surface area of it in a single revolution.

The area of a femisphere of unit radius is $$ S = 4 \pi$$.

Wooden femispheres can be made by turning them on a wood lathe.