From Pascal's Theorem to d-Constructible Curves

Abstract

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.

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Notes on contributors

Will Traves

WILL TRAVES grew up in Toronto, Canada and moved to the United States during graduate school. His paper [1] in this Monthly, co-authored with Andy Bashelor and Amy Ksir, won both the Lester R. Ford and Merten M. Hasse prizes. He joined the faculty of the United States Naval Academy in 1999 and is a “brown-dot” Project NExT fellow.