Barth Decic

Barth decic surface

The Barth decic is a decic surface in complex three-dimensional projective space having the maximum possible number of ordinary double points, namely 345. It is given by the implicit equation


where phi is the golden ratio and w is a parameter.


The case w=1, illustrated in the above plot, has 300 ordinary double points. These 300 points lie symmetrically at the vertices of two 60-faced and two 90-faced solids, illustrated above, with the 60-faces solids corresponding to a truncation of the icosahedron (but not the symmetrically truncated truncated icosahedron).

Surfaces of the Barth decic containing ordinary double points

The concentric groups of points lie at distances 1/phi, sqrt(5-2sqrt(5)), 1, and sqrt((5+sqrt(5))/2) from the origin, illustrated above.

The Barth decic is invariant under the icosahedral group.

See also

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

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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.Endraß, S. "Flächen mit vielen Doppelpunkten." DMV-Mitteilungen 4, 17-20, 4/1995.Endraß, S. "Barth's Decic." Feb. 6, 2003., T. "Batch Decic."

Cite this as:

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

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