- Dear Hyacinthists,
an ancient notion is the desmic configuration of three tetrahedrons :
the tetrahedral pole of any face of one of them wrt any of the two
other tetrahedrons is the opposite vertex.
We can construct any desmic system of tetrahedrons in the following
way starting from a tetrahedron DABC and a point D'' (not on the
faces of ABCD) :
A' is the harmonic conjugate of D'' wrt A and the common point of
the line AD'' and the plane BCD; similarly, we get B', C', D'.
If U et V are the points of the lines DA and BC such as the line UV
goes through D'', A'' is the harmonic conjugate of D'' wrt U,V.
Similarly, we get B'', C''.
DABC, D'A'B'C', D''A''B''C'' is an harmonic system of tetrahedrons
(and that's the only way to get such a system)
Now, the projective image of these 12 points on any plane is clearly
a desmic configuration of quadrangles.
Conversely, starting from a desmic system of quadrangles in a plane
P, for instance
the following system of tetrahedrons is desmic
and the desmic system of quadrangles is the projection of the
tetrahedral system upon P from the point [U+u,V+v,W+w,T]
There should be something wrong because I remember that, in an
ancient Hyacinthist discussion, it appeared that a desmic system of
quadrangles could not proceed from a 3D configuration.
I seriously wonder if it is true?