Desmic system

{{short description|Configuration of 3 tetrahedra in projective geometry}}

File:Reye configuration.svg with the same 12 vertices as a desmic system]]

In projective geometry, a desmic system ({{ety|el|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 {{harvs|txt|authorlink=Cyparissos Stephanos|last=Stephanos|year=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.

Example

The three tetrahedra given by the equations

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

form a desmic system, contained in the pencil of quartics

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

for a + b + c = 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 | 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 | 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