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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

form a desmic system, contained in the pencil of quartics

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