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Dervish

Dervish

Togliatti surfaces are quintic surfaces having the maximum possible number of ordinary double points (31).

A related surface sometimes known as the dervish can be defined by

aF+q=0,
(1)

where

F=h_1h_2h_3h_4h_5,
(2)
h_1=x-z
(3)
h_2=cos((2pi)/5)x-sin((2pi)/5)y-z
(4)
h_3=cos((4pi)/5)x-sin((4pi)/5)y-z
(5)
h_4=cos((6pi)/5)x-sin((6pi)/5)y-z
(6)
h_5=cos((8pi)/5)x-sin((8pi)/5)y-z
(7)
q=(1-cz)(x^2+y^2-1+rz^2)^2,
(8)

and

r=1/4(1+3sqrt(5))
(9)
a=-8/5(1+1/(sqrt(5)))sqrt(5-sqrt(5))
(10)
c=1/2sqrt(5-sqrt(5)).
(11)

See also

Algebraic Surface, Quintic Surface, Togliatti Surface

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References

Endraß, S. "Togliatti Surfaces." http://enriques.mathematik.uni-mainz.de/docs/Etogliatti.shtml.Endraß, S. "Flächen mit vielen Doppelpunkten." DMV-Mitteilungen 4, 17-20, 4/1995.Endraß, S. Symmetrische Fläche mit vielen gewöhnlichen Doppelpunkten. Ph.D. thesis. Erlangen, Germany, 1996.Hunt, B. "Algebraic Surfaces." http://www.mathematik.uni-kl.de/~hunt/drawings.html.Nordstrand, T. "Dervish." http://jalape.no/math/dervtxt.

Cite this as:

Weisstein, Eric W. "Dervish." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Dervish.html

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