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Desmic Surface

Let Delta_1, Delta_2, and Delta_3 be tetrahedra in projective three-space P^3. Then the tetrahedra are said to be desmically related if there exist constants alpha, beta, and gamma such that

alphaDelta_1+betaDelta_2+gammaDelta_3=0.

A desmic surface is then defined as a quartic surface which can be written as

aDelta_1+bDelta_2+cDelta_3=0

for desmically related tetrahedra Delta_1, Delta_2, and Delta_3. Desmic surfaces have 12 ordinary double points, which are the vertices of three tetrahedra in three-space (Hunt 1996).

See also

Kummer Surface, Quartic Surface

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References

Hudson, R. W. H. T. Kummer's Quartic Surface. Cambridge, England: University Press, 1905. Reprinted Cambridge, England: Cambridge University Press, 1990.Hunt, B. "Desmic Surfaces." §B.5.2 in The Geometry of Some Special Arithmetic Quotients. New York: Springer-Verlag, pp. 311-315, 1996.Jessop, C. A Treatise on the Line Complex. Cambridge, England: University Press, 1903.Jessop, C. §13 in Quartic Surfaces with Singular Points. Cambridge, England: Cambridge University Press, 1916.

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Desmic Surface

Cite this as:

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

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