SRC Research Report 134a
SRC Research Report 134aThe complete quadrilateral is a configuration with six points and four lines in the projective plane. It is a classical result that, given the six roots of three quadratic polynomials, we can use the complete quadrilateral (or its dual, the complete quadrangle) to test geometrically whether or not those three quadratics are linearly dependent. But does there exist some configuration that provides an analogous geometric test for the linear dependence of four cubic polynomials?
Yes, there is such a cubic analog of the complete quadrilateral; it has twelve points, two lines, and thirteen planes in 3-space. In fact, the complete quadrilateral and its cubic analog are just two members of a large family of intriguing configurations that are defined in this monograph, using the notion of a "budget". The study of these "budget configurations" combines old-fashioned geometry with modern combinatorics -- from null systems to matroids.
Why didn't the classical geometers discover the budget configurations long ago? Perhaps because they required their configurations to have a certain numeric symmetry that a typical budget configuration doesn't have.
A videotape (134b) accompanies report 134a:
The videotape animates the cubic analog of the complete quadrilateral, as well as some other budget configurations that live in the plane or in 3-space. By doing so, it tries both to give non-mathematicians some hint of these discoveries and to lure mathematicians into reading the monograph.
Back to the SRC Research Reports main page.