A quintic surface is an algebraic surface of degree 5. Togliatti (1940, 1949) showed that quintic surfaces having 31 ordinary double points exist, although he did not explicitly derive equations for such surfaces. Beauville (1978) subsequently proved that 31 double points was the maximum possible, and quintic surfaces having 31 ordinary double points are therefore sometimes called Togliatti surfaces. van Straten (1993) subsequently constructed a three-dimensional family of solutions and in 1994, Barth derived the example known as the dervish.
Examples of quartic surfaces include the dervish, kiss surface, peninsula surface, swallowtail catastrophe surface, and Togliatti surface.
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