Desmic system

In projective geometry, a desmic system (from Greek δεσμός  'band, chain') 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 Stephanos (1879). 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.

The three tetrahedra given by the equations

• ${\displaystyle \displaystyle (w^{2}-x^{2})(y^{2}-z^{2})=0}$
• ${\displaystyle \displaystyle (w^{2}-y^{2})(x^{2}-z^{2})=0}$
• ${\displaystyle \displaystyle (w^{2}-z^{2})(y^{2}-x^{2})=0}$

form a desmic system, contained in the pencil of quartics

• ${\displaystyle \displaystyle a(w^{2}x^{2}+y^{2}z^{2})+b(w^{2}y^{2}+x^{2}z^{2})+c(w^{2}z^{2}+x^{2}y^{2})=0}$

for a + b + c = 0.

• Borwein, Peter B (1983), "The Desmic conjecture", Journal of Combinatorial Theory, Series A, 35 (1): 1–9, doi:10.1016/0097-3165(83)90022-5, MR 0704251.
• Hudson, R. W. H. T. (1990), Kummer's quartic surface, Cambridge Mathematical Library, Cambridge University Press, ISBN 978-0-521-39790-2, MR 1097176.
• Stephanos, Cyparissos (1879), "Sur les systèmes desmiques de trois tétraèdres", Bulletin des sciences mathématiques et astronomiques, Série 2, 3 (1): 424–456, JFM 11.0431.01.

• Desmic tetrahedra