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.

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