# Desmic system

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Desmic_system
> Markdown URL: https://mediated.wiki/source/Desmic_system.md
> Source: https://en.wikipedia.org/wiki/Desmic_system
> Source revision: 1342856229
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{short description|Configuration of 3 tetrahedra in projective geometry}}
thumb|right|Two desmic tetrahedra. The third tetrahedron of this system is not shown, but has one vertex at the center and the other three on the plane at infinity.  
[[File:Reye configuration.svg|thumb|left|The [Reye configuration](/source/Reye_configuration) with the same 12 vertices as a desmic system]]

In [projective geometry](/source/projective_geometry), a '''desmic system''' ({{ety|el|δεσμός |band, chain}}) is a set of three [tetrahedra](/source/tetrahedron) in 3-dimensional [projective space](/source/projective_space), such that any two are '''desmic''' (related such that each [edge](/source/Edge_(geometry)) of one cuts a pair of opposite edges of the other). It was introduced by {{harvs|txt|authorlink=Cyparissos Stephanos|last=Stephanos|year=1879}}.  The three tetrahedra of a desmic system are contained in a [pencil](/source/Pencil_(mathematics)) of [quartic surface](/source/quartic_surface)s.

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](/source/Reye_configuration).

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

==References==
*{{Citation | last1=Borwein | first1=Peter B | authorlink = Peter Borwein | title=The Desmic conjecture | doi=10.1016/0097-3165(83)90022-5 | mr=704251 | year=1983 | journal=[Journal of Combinatorial Theory](/source/Journal_of_Combinatorial_Theory) | series = Series A | volume=35 | issue=1 | pages=1–9| doi-access= }}.
*{{Citation | authorlink=R. W. H. T. Hudson | last1=Hudson | first1=R. W. H. T. | title=Kummer's quartic surface | publisher=[Cambridge University Press](/source/Cambridge_University_Press) | series=Cambridge Mathematical Library | isbn=978-0-521-39790-2 | mr=1097176 | year=1990|url=https://archive.org/details/184605691}}.
*{{Citation | last1=Stephanos | first1=Cyparissos | title= Sur les systèmes desmiques de trois tétraèdres | url=http://www.numdam.org/item?id=BSMA_1879_2_3_1_424_1 |jfm=11.0431.01 | year=1879 | journal=Bulletin des sciences mathématiques et astronomiques |series=Série 2 | volume=3 | issue=1 | pages= 424–456 }}.

== External links ==
* [http://members.home.nl/fg.marcelis/desmic.htm Desmic tetrahedra]

Category:Projective geometry

---
Adapted from the Wikipedia article [Desmic system](https://en.wikipedia.org/wiki/Desmic_system) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Desmic_system?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.