A stereographic projection is technically mapping a sphere onto a plane. Photographers like Alexandre Duret-Lutz, Masato, Seb Perez-Duarte, Boltron, and Masakazu Matsumoto have taken a 360° panorama photograph that covers -90° (the ground) to 90° (the sky) and then mapped that onto a plane.
Regardless of how it is done, the effects are beautiful.
Masakazu Matsumoto provides a FAQ on the technique on his Flickr set.
via Pink Tentacle