Roman Surface / Steiner surface

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The Roman surface or Steiner surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion of the projective plane; however, the figure resulting from removing six singular points is one. The origin is a triple point, and each of the xy-, yz-, and xz-planes are tangential to the surface there. The other places of self-intersection are double points, defining segments along each coordinate axis which terminate in six pinch points. The entire surface has tetrahedral symmetry. It is a particular type (called type 1) of Steiner surface, that is, a 3-dimensional linear projection of the Veronese surface. The Roman surface has four bulbous "lobes", each one on a different corner of a tetrahedron. A Roman surface can be constructed by splicing together three hyperbolic paraboloids and then smoothing out the edges as necessary so that it will fit a desired shape (e.g. parametrization). Jacob Steiner discovered the Steiner surface, also known as The Roman surface, in 1844 in Rome. This surface is named after Jacob Steiner. The Steiner surface is relatively interesting but difficult to visualize without 3D views. Here we use a method that sets markers (spheres) at surface points which can then be done with most software. The Steiner surface is a self-intersecting real projective plane, which has evolved into three-dimensional space and has an unusually high degree of symmetry. The origin is a triple point, and each xy-, yz-, xz- plane is tangential to the surface there. The other places of self-intersection are double points, defining segments along each coordinate axis which terminate in six pinch points. The entire surface has tetrahedral symmetry. It is a particular type (called type 1) of Steiner surface, that is, a 3-dimensional linear projection of the Veronese surface. The Roman surface has four bulbous "lobes", each one on a different corner of a tetrahedron. A Roman surface can be constructed by splicing together three hyperbolic paraboloids and then smoothing out the edges as necessary so that it will fit a desired shape (e.g. parametrization).