Desmic system explained

In projective geometry, a desmic system 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 . 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

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External links

• Desmic tetrahedra