From Pascal's Theorem to d-Constructible Curves

We prove a generalization of both Pascal’s Theorem and its converse, the Braikenridge–Maclaurin Theorem: If two sets of k lines meet in k2 distinct points, and if dk of those points lie on an irreducible curve C of degree d , then the remaining k(k − d) points lie on a unique curve S of degree k − d. If S is a curve of degree k − d produced in this manner using a curve C of degree d , we say that S is d-constructible. For fixed degree d , we show that almost every curve of high degree is not d-constructible. In contrast, almost all curves of degree 3 or less are d-constructible. The proof of this last result uses the group structure on an elliptic curve and is inspired by a construction due to Möbius. The exposition is embellished with several exercises designed to amuse the reader. Dedicated to H.S.M. Coxeter, who demonstrated a heavenly syzygy: the sun and moon aligned with the Earth, through a pinhole. (Toronto, May 10, 1994, 12:24:14)