TETRAHEDRAL COMPLEXES
Another branch of projective geometry concerns lines. There is a four-fold infinity of lines in space, of which we may form a subset. A subset containing a threefold infinity of lines is called a LINE COMPLEX. An example which is simple to define is the TETRAHEDRAL COMPLEX: given a tetrahedron, a general line in space cuts its four faces in four points:
These four points have a cross ratio which may be any real number. We
may select the set
of lines all of which intersect the tetrahedron in the same cross
ratio. Since there are infinitely
many possible cross ratios we thus select a three-fold infinity of
lines from the four-fold infinity of all possible lines. The
resulting line complex has a definite structure such that through any
point of space it possesses a set
of lines forming a cone, while in any plane of space it possesses a set
of lines enveloping a conic.
Just as we have polarity wrt (with respect to) conics and quadrics, so
we may have
polarity wrt a line complex. This means that if we choose any line u
then the complex determines a line u'
polar to u. This is accomplished by taking the axial pencil of planes
in u, and for each such plane finding the point P polar to u
wrt the conic of the complex in that plane:
The points P in all the
planes of the pencil lie on a straight line u'
which is the polar of u. ( If u happens to be a line of the complex
then it is
self-polar).
We may then find the polar of u', which is a third line u", and so on.
An interesting
question then arises: what figure is formed by such a sequence of polar
lines?
The answer turns out to be quite simple: it is a ruled quadric which is
self-polar wrt the
tetrahedron. This means self-polar in the sense that the faces of the
tetrahedron and their opposite vertices are
harmonic wrt the quadric. Although we started with a discrete set
of lines u,u',u''... it turns out that if we take any line v on a
self-polar quadric Q then its polar line v' wrt the complex also lies
on Q.
Since we could have chosen any cross ratio to define the complex, and
since a quadric Q is
self-polar wrt the tetrahedron irrespective of that cross ratio, we see
that the lines on Q form a
self-polar set for all possible tetrahedral complexes sharing the
same base tetrahedron (such complexes are known as COSINGULAR
COMPLEXES). Of course a given line v of Q will have different lines of
Q as its polar for
different cosingular complexes.
I found this result myself and have not seen it anywhere in the
literature. Has anyone
seen it published elsewhere?
The proof is available from me (via email).