A stereographic projection maps these figures onto a plane.
Hosohedra  (HOSE-oh-HEE-dra)  are tessellations of these objects by lunes. Extending Girard's theorem to polygons named after these figures can be used to prove Euler's Formula. In three dimensions, the maximum number of these figures that can touch each other without intersection is the kissing number. The complex plane together with a point at (*)  infinity is often modeled as one of these figures named for Riemann. A stereographic projection maps these figures onto a plane. The Kepler problem concerns the densest packing of these figures. Azimuthal and polar angles appear in a coordinate system named for these objects, and great circles contain these objects' diameters. A ball is the interior of, for 10 points, what round three-dimensional figures with volume "four-thirds pi  r- cubed"?
Hosohedra  (HOSE-oh-HEE-dra)  are tessellations of these objects by lunes.
sphere s [or  2-spheres ; accept  spherical  coordinates; prompt on  ball s until read; do not accept or prompt on "circles"]