Triangles, Ellipses and Cubic Polynomials
By: D. Minda and S. Phelps
There are a number of very interesting geometric connections between the roots of a cubic polynomial and the roots of the derivative. Precisely, suppose p(z) is a monic cubic polynomial whose roots form the vertices of a triangle, Δ, in the complex plane C. It is elementary that the centroid of Δ is the midpoint, or centroid, of the roots of p'(z). The Gauss-Lucas Theorem implies that the roots of p'(z) lie inside Δ. Siebeck's Theorem (1864) asserts that the roots of p'(z) are the foci of the Steiner inellipse for Δ. The Steiner inellipse for a triangle is the unique inscribed ellipse that is tangent to the sides at their midpoints. A result of Coolidge (1913) asserts that the line of best fit for the vertices of the triangle is the line through the roots of p'(z). These known geometric connections are established using elementary properties of complex numbers and affine transformations.
Holomorphic Maps and Pencils of Circles
By: Rémi Langevin and Pawel G. Walczak
Conformal geometry of the plane deals with circles instead of lines. Holomorphic maps are conformal. Is it possible to characterize holomorphic maps using circles? The article provides an answer to this question and illustrates it with several pictures.
Enumerative Algebraic Geometry of Conics
By: Andrew Bashelor, Amy Ksir, and Will Traves
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A recent undergraduate project dealt with Steiner's problem: How many conics are simultaneously tangent to five fixed conics? This challenging problem can be solved by first tackling a collection of easier enumerative problems involving conics, lines, and points. Many beautiful ideas in algebraic geometry make an appearance along the way. Complicated tools like moduli spaces, blowing-up, duality, and cohomology are both natural and accessible when studied in this context. A list of fun problems develops connections to other topics, such as string theory and kissing spheres.
The Sixty-Eighth William Lowell Putnam Mathematical Competition
By: Leonard F. Klosinski, Gerald L. Alexanderson, and Loren C. Larson
A Probalistic Proof of Wallis's Formula for π
By: Steven J. Miller
How to Compute Σ 1/n2 by Solving Triangles
By: Mikael Passare
An Elementary Proof of the Hitting Time Theorem
By: Remco van der Hofstad and Michael Keane
Google's PageRank and Beyond: The Science of Search Engine Rankings
By: Amy N. Langville and Carl D. Meyer
Reviewed by: Daniel T. Kaplan