Barth Sextic

The Barth sextic is a sextic surface in complex three-dimensional projective space having the maximum possible number of ordinary double points, namely 65. The surface was discovered by W. Barth in 1994, and is given by the implicit equation


where phi is the golden ratio.

Barth sextic with w = 1

Taking w=1 gives the surface in 3-space illustrated above, which retains 50 ordinary double points.

Barth sextic nodal surfaces

Of these, 20 nodes are at the vertices of a regular dodecahedron of side length 2/phi and circumradius sqrt(3) (left figure above), and 30 are at the vertices of a concentric icosidodecahedron and circumradius 1 (right figure).

The Barth sextic is invariant under the icosahedral group. Under the map


the surface is the eightfold cover of the Cayley cubic (Endraß 2003).

The Barth sextic appeared on the cover of the March 1999 issue of Notices of the American Mathematical Society (Dominici 1999).

See also

Algebraic Surface, Barth Decic, Cayley Cubic, Ordinary Double Point, Sextic Surface

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Barth, W. "Two Projective Surfaces with Many Nodes Admitting the Symmetries of the Icosahedron." J. Alg. Geom. 5, 173-186, 1996.Update a linkDominici, P. "Flight Through Barth's Sextic.", P. "About the Cover." Notices Amer. Math. Soc. 46, cover and p. 318, No. 3, March 1999.Endraß, S. "Flächen mit vielen Doppelpunkten." DMV-Mitteilungen 4, 17-20, 4/1995.Endraß, S. "Barth's Sextic." Feb. 6, 2003., T. "Barth Sextic."

Cite this as:

Weisstein, Eric W. "Barth Sextic." From MathWorld--A Wolfram Web Resource.

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