Desmic system



In projective geometry, a desmic system is a set of three tetrahedra in 3-dimensional projective space, such that any two are desmic (related such that each edge of one cuts a pair of opposite edges of the other). It was introduced by. The three tetrahedra of a desmic system are contained in a pencil of quartic surfaces.

Every line that passes through two vertices of two tetrahedra in the system also passes through a vertex of the third tetrahedron. The 12 vertices of the desmic system and the 16 lines formed in this way are the points and lines of a Reye configuration.

Example
The three tetrahedra given by the equations form a desmic system, contained in the pencil of quartics for a + b + c = 0.
 * $$\displaystyle (w^2-x^2)(y^2-z^2) = 0$$
 * $$\displaystyle (w^2-y^2)(x^2-z^2) = 0$$
 * $$\displaystyle (w^2-z^2)(y^2-x^2) = 0$$
 * $$\displaystyle a(w^2x^2+y^2z^2) + b(w^2y^2+x^2z^2) + c (w^2z^2+x^2y^2) = 0$$