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 
is the golden ratio.
Taking 
gives the surface in 3-space illustrated above, which retains 50 ordinary
double points.
Of these, 20 nodes are at the vertices of a regular dodecahedron of side length 
and circumradius 
(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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References
Barth, W. "Two Projective Surfaces with Many Nodes Admitting the Symmetries of the Icosahedron." J. Alg. Geom. 5,
173-186, 1996.Dominici, P. "About the Cover." Notices Amer.
Math. Soc. 46, cover and p. 318, No. 3, March 1999.
Dominici, P. "Flight
Through Barth's Sextic." http://www.mi.uni-erlangen.de/~bauerth/flight/Endraß,
S. "Flächen mit vielen Doppelpunkten." DMV-Mitteilungen 4,
17-20, 4/1995.Endraß, S. "Barth's Sextic." Feb. 6,
2003. https://web.archive.org/web/20031027010006/http://enriques.mathematik.uni-mainz.de/kon/docs/Ebarthsextic.shtml.Nordstrand,
T. "Barth Sextic." http://jalape.no/math/sexttxt.htm.Trott,
M. The
Mathematica GuideBook for Numerics. New York: Springer-Verlag, pp. 330-332,
2006. https://www.mathematicaguidebooks.org/.Referenced
on Wolfram|Alpha
Barth Sextic
Cite this as:
Weisstein, Eric W. "Barth Sextic." From
MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BarthSextic.html
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