Seeing a Fundamental Theorem

Bruce Torrence
Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture (2014)
Pages 457–460


The Geometric Modulus Principle, developed by Bahman Kalantari, provides a powerful geometric means for understanding the Fundamental Theorem of Algebra. Given a complex polynomial and any point in the complex plane, the principle gives a complete characterization of directions of ascent and descent for the modulus of the polynomial at that point. This characterization has a simple and compelling visual representation. By making a contour plot of the modulus of the polynomial, and including only those level curves that pass through the polynomial's critical points, the Geometric Modulus Principle comes to life. The Fundamental Theorem of Algebra is a straightforward consequence, and the visualizations stand on their own as portraits of the polynomials in question.