Steiner surface
Encyclopedia
In geometry
, a branch of mathematics
, the Steiner surfaces, discovered by Jakob Steiner
, are mappings of the real projective plane
into three-dimensional real projective space
. They are the linear projection of the Veronese surface
, an embedding of the real projective plane into five-dimensional projective space, onto three-dimensional projective space.
The Steiner surfaces can be classified into ten different types, including the Roman surface
and the cross-cap
.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Steiner surfaces, discovered by Jakob Steiner
Jakob Steiner
Jakob Steiner was a Swiss mathematician who worked primarily in geometry.-Personal and professional life:...
, are mappings of the real projective plane
Real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...
into three-dimensional real projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
. They are the linear projection of the Veronese surface
Veronese surface
In mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space, and is realized by the Veronese embedding, the embedding of the projective plane given by the complete linear system of conics. It is named after Giuseppe Veronese...
, an embedding of the real projective plane into five-dimensional projective space, onto three-dimensional projective space.
The Steiner surfaces can be classified into ten different types, including the Roman surface
Roman surface
The Roman surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry...
and the cross-cap
Cross-cap
In mathematics, a cross-cap is a two-dimensional surface that is a model of a Möbius strip with a single self intersection. This self intersection precludes the cross-cap from being topologically equivalent to a Möbius strip...
.