Ever since I first learned of the Borsuk-Ulam theorem as an undergraduate I wondered if there was a straightforward approach to understanding why it was true. It wasn't until I witnessed the work of youngsters in the Spring of 2008 that I saw it. The following article appeared in the November 2008 issue of FOCUS (volume 28, number 8), the news magazine of the Mathematical Association of America. The proof was also the topic of the October 2008 St. Mark's Institute of Mathematics newsletter.

An Intuitive Approach to the Borsuk-Ulam Theorem

By Alex Bishop, Adam Cimpeanu, Kyle Flood, Bianca Homberg, Steven Homberg, Eric Marriott, Jeffrey Roth, Linus Schultz, William Sherman, Alex Smith, Geoffrey Smith and James Tanton of the St. Mark's Institute of Mathematics.

Each semester I offer extracurricular mathematics courses for math-interested students, ages 11–18, keen to experience mathematics as a creative and organic enterprise. Called research classes, they introduce students to the joys and frustrations of the research experience, of not knowing, of feeling around in the dark, and of finding the fortitude of mind to concentrate on complex issues for sustained periods. (Weeks, not just minutes.) This semester the coauthors of this paper established the classic Borsuk-Ulam Theorem in a way that is slick, intuitive, and accessible. Their success speaks to the joy and value that can be found for all in allowing mathematical creativity to bubble forth, no matter the age and the background experience of the mathematical artists!