In geometry, the Steiner surfaces are certain self-intersecting embeddings (that is to say, immersions) of the real projective plane into three-dimensional space. More particularly, they are linear projections of a six-dimensional embedding called the Veronese surface, which is the image of an ordinary 2-sphere centered at the origin under the map

f(x, y, z) = (x², y², z², yz, xz, xy).

There are ten different types, including the Roman surface and cross-cap.

They are named after Jakob Steiner, who discovered them.