Desmic Surface

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

$$\alpha\Delta_1+\beta\Delta_2+\gamma\Delta_3=0.$$

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

$$a\Delta_1+b\Delta_2+c\Delta_3=0$$

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 3-space (Hunt).

See also Quartic Surface

References

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. §13 in Quartic Surfaces with Singular Points. Cambridge, England: Cambridge University Press, 1916.