On the integrability of the shift map on twisted pentagram spirals

In this paper we prove that the shift map defined on the moduli space of twisted pentagram spirals of type $(N, 1)$ possesses a non-standard Lax representation with an associated monodromy whose conjugation class is preserved by the map. We prove this by finding a coordinate system in the moduli space of twisted spirals, writing the map in terms of the coordinates and associating a natural parameter-free non-standard Lax representation. We then show that the map is invariant under the action of a $1$-parameter group on the moduli space of twisted $(N,1)$ spirals, which allows us to construct the Lax pair. We also show that the monodromy defines an associated Riemann surface that is preserved by the map. We use this fact to generate invariants of the shift map

Y-meshes and generalized pentagram maps

We introduce a rich family of generalizations of the pentagram map sharing the property that each generates an infinite configuration of points and lines with four points on each line. These systems all have a description as $Y$-mutations in a cluster algebra and hence establish new connections between cluster theory and projective geometry. Our framework incorporates many preexisting generalized pentagram maps due to M. Gekhtman, M. Shapiro, S. Tabachnikov, and A. Vainshtein and also B. Khesin and F. Soloviev. In several of these cases a reduction to cluster dynamics was not previously known.Comment: 48 pages, 22 figures, to appear in Proceedings of the London Mathematical Societ

An In-depth Investigation of the Divine Ratio

The interesting thing about mathematical concepts is that we can trace their development or discoveries throughout history. Most cultures of the ancient world had some form of mathematics, and these basic skills developed into what we now call modern mathematics. The divine ratio is similar in that it was used in many different sections of history. The divine ratio, sometimes called the golden ratio or golden section, has been found in very diverse areas. The mathematical concepts of the golden ration have been found throughout nature, in architecture, music as well as in art. Phi is an astonishing number because it has inspired thinkers in many disciplines, more-so than any other number has in the history of mathematics. This paper investigates how the golden ratio has influenced civilizations throughout history and has intrigued mathematicians and others by its prevalence

The Vanishing Square, The Fibonacci Sequence, and The Golden Ratio

The Golden Ratio has generally been revered as the most aesthetically pleasing relationship that exists between two numbers. The Fibonacci Sequence is a famous collection of integers occurring unusually often in nature, with many amazing properties, and is intimately related to the Golden Ratio. An intriguing puzzle (“The Vanishing Square”) will set the stage. Then we will examine the origins of both the Golden Ratio and the Fibonacci Sequence and discover how their histories are intertwined. But….does the Golden Ratio really live up to its reputation

Linear difference equations, frieze patterns and combinatorial Gale transform

We study the space of linear difference equations with periodic coefficients and (anti)periodic solutions. We show that this space is isomorphic to the space of tame frieze patterns and closely related to the moduli space of configurations of points in the projective space. We define the notion of combinatorial Gale transform which is a duality between periodic difference equations of different orders. We describe periodic rational maps generalizing the classical Gauss map

Information Literacy, a New Mandate: Undergraduate Assignment for Open-Ended Research and Creative Design

Long gone are the days of a math professor’s contribution being only to mathematics. Today, typical mission statements include goals of writing across the curriculum, higher order thinking, and always creativity and self-expression. This writing assignment focused on developing information literacy, the art of finding meaning and authenticity in the deluge of information that is available today. Like other components peripheral to math education, information literacy weaves techniques of analytical thinking and decision making into a standard curriculum