The Mathematical Association of America Maryland-District of Columbia-Virginia Section

The remarkable fact that \(2^n \sqrt{2-\sqrt{2+\sqrt{2+⋯+\sqrt{2}}}} \rightarrow \pi\) as \(n \rightarrow \infty\) where n is the number of nested radicals, inspires an obvious question. What other similar sorts of results hold? In this talk I will discuss a general framework for results like the mysterious pattern in the context of iterated functions, defining a collection of sequences analogous to the one with nested radicals above. Results include a sufficient condition for convergence, limits that arise in connection with Möbius functions, and a continued fraction analog of the nested radical example. Excerpted from a paper available here: https://arxiv.org/pdf/2509.21409

Certain geometric shapes were revolutionary and impactful throughout the history of mathematics. These shapes motivated new mathematical solutions to problems and inspired and enriched the development of new mathematical areas. For example, right triangles and the Pythagorean theorem solved an uncountable (!) number of problems. Likewise, Thales’ use of similar triangles allowed him to measure the height of the Great Pyramid without direct measurement. Conic Sections helped unlock the secrets of the universe once in the hands of Kepler and Newton. Similarly, Huygens’ catenary curve appeared in unexpected places. Furthermore, Pascal’s research on the cycloid curve, the so-called “Helen of Geometry,” was directly responsible for Leibniz’s independent invention of calculus. Although Euler invented graph theory, he did not see how his polyhedral formula for the Platonic solids connects to graph theory, where it still has implications. Moreover, specific shapes such as the Saccheri quadrilateral, orthogonal circles, and the pseudosphere validated Non-Euclidean geometry. In conclusion, Mandelbrot’s radical departure from Euclidean geometry gave us intricate fractal shapes such as the Sierpinski Triangle, Mandelbrot Set, and Julia Sets. This presentation will highlight a sample of shapes that have been particularly central in the history of mathematics, along with illustrating some of their applications.

Shut the Box is a dice game in which players roll dice and choose which numbered tiles to close, aiming to minimize the sum of the remaining tiles. We model the game as a stochastic process and use dynamic programming to determine optimal strategies for several common rule variations. This work is based on an undergraduate independent research project.

This is an art exhibition of works by students in Rebin Muhammad's Islamic Geometric Patterns course.

In this continuation of Part 1, topics include an analog of eigenvectors and eigenvalues under the operation of function composition, complex dynamical systems and Koenigs functions, and inverse Chebyshev polynomials. Regarding the last of these, we can define the Chebyshev \(n\)th root (denoted \(\sqrt[\langle n \rangle]{x}\)) and obtain beautiful analogs of the original nested radical identity. For example, $$\lim_{n \to \infty} 25^n \left( 5 - \sqrt[5]{5 + \sqrt[5]{5 + \sqrt[5]{5 + \cdots + \sqrt[5]{5}}}} \right) = \frac{5\pi^2}{8}.$$