The Barth-sextic is a sextic surface in complex three-dimensional projective space having the maximum possible number of ordinary double points (65). Of these, 20 nodes are at the vertices of a regular dodecahedron of side length , and 30 are at the midpoints of the edges of a concentric dodecahedron of side length , where is the golden ratio. The surface was discovered by W. Barth in 1994, and is given by the implicit equation where is the golden ratio, and w is a parameter (Endraß, Nordstrand), taken as w = 1 in the above plot.

The Barth sextic is invariant under the icosahedral group. Under the map
[cetner][/cetner]
the surface is the eightfold cover of the Cayley cubic (Endraß).