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. Reye configuration

projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...

, a desmic system () is a set of three

tetrahedra In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

in 3-dimensional

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

, such that any two are desmic (related such that each

edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...

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 A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage and keeps it from marking the user's hand. Pencils create marks by physical abrasion, leaving a trail of ...

quartic surface In mathematics, especially in algebraic geometry, a quartic surface is a surface defined by an equation of degree 4. More specifically there are two closely related types of quartic surface: affine and projective. An ''affine'' 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 In geometry, the Reye configuration, introduced by , is a Configuration (geometry), configuration of 12 Point (geometry), points and 16 Line (geometry), lines. Each point of the configuration belongs to four lines, and each line contains three p ...

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=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

Desmic tetrahedra
Projective geometry