Letter from the Editor

Welcome to another mammoth double issue of Mathematics Magazine!

The festivities open with a real barn-burner of an article about pseudogravity ballistic trajectories. Long-term space exploration requires the creation of pseudogravity environments to stave off the doleful effects of prolonged weightlessness on human physiology. Such environments can be obtained by rotating the living environment. This poses some interesting problems in physics. Specifically, what is the trajectory of an object that is dropped, thrown, or struck in such an environment? Marc Frantz serves up a scintillating investigation of such questions, showing that the trajectories are frequently both beautiful and mathematically elegant.

Dan Kalman, who has received more MAA writing awards than anyone in history, likewise gives us a contribution to mathematical modeling. He considers the problem of discrete logistic growth. He notes that an approach to this subject based on the Verhulst difference equation has certain advantages over the more traditional approach usually shown to undergraduates. In particular, Kalman’s approach never leads to chaotic behavior. This is an article that deserves a place in undergraduate modeling courses.

Roger Nelsen is the reigning master of Proofs Without Words (and do note his contribution in that regard elsewhere in this issue), so it is nice to see his name in our article section for a change. He takes a visual approach to the problem of understanding certain identities among Pell numbers. His consistently creative approaches to such questions will remind you of why you liked mathematics in the first place.

Nelsen’s article is just the tip of the iceberg for our number theory offerings. Kenneth Stuart Williams sends in part two of the article whose first part appeared in our previous issue. He presents some more examples of the ingenious techniques that can be used to evaluate certain infinite sums. His starting point is the classic Basel problem, which called for summing the reciprocals of the squares. Karen Briggs and Caylee Spivey take their inspiration from modular arithmetic. They consider the question of when the additive and multiplicative inverses of a number are the same with respect to a specific modulus. Samer Seraj likewise contributes an article about modular arithmetic, this time inspired by one of those charming “cryptarithm” brainteasers. Rajib Mukherjee and Manishita Chakraborty present a striking extension of the famous Galileo ratio.

Matthew Roughan’s article considers number theory of a different sort. His subject is the surreal numbers, originally developed by John Conway in the 1970s. One of the things I enjoy about editing Mathematics Magazine is all the fascinating mathematics I learn about from reading the many submissions we receive. I had no idea that numbers could have birthdays, but Roughan’s engaging exposition shows thatthey can.

Geometry fans can start with Hans Humenberger’s exploration of bicentric quadrilaterals, meaning four-sided polygons having both an incenter and a circumcenter. All triangles are bicentric in this sense, and they have a so-called Bevan point, meaning the circumcenter of the triangle formed by the centers of the triangle’s excircles. Can the notion of the Bevan point be extended to bicentric quadrilaterals? Indeed it can, as Humenberger shows. For his part, Quang Hung Tran takes a fresh look at Pascal’s famous theorem about hexagons on a conic. He provides a very clever new proof of this famous result.

There is plenty to learn from the remaining articles as well. Sam Northshield presents numerous variations on the theme of “cubes.” Among other accomplishments, his article presents a short proof of the Siebeck-Marden theorem. If you are unfamiliar with it, consider that a complex, cubic polynomial has three roots, and its derivative has two. The three roots of the polynomial typically define a triangle. This triangle has a unique inscribed ellipse that is tangent to the midpoints of the three sides. According to the theorem, the foci of this ellipse are precisely the roots of the derivative. Very nice!

Zair Ibragimov and Bogdan Suceava consider Ptolemy’s theorem. Their article is a wonderful mix of history, geometry, and astronomy. Matthew Just provides a combinatorial proof for a theorem about the divisibility properties of binomial coefficients. Harvey Diamond considers function iterations and presents a graphical technique for resolving whether a given fixed point is attracting or repelling. Jake Lewis takes note of a problem that appeared on a University of Tokyo entrance examination: Prove that $\pi >3.05$. Establishing this through elementary means is no small task, but Lewis manages to pull it off. Finally, Jan Vrbik and Paul Vrbik round out the proceedings by providing a novel proof of the Desnanot-Jacobi identity from linear algebra.

We also have Problems, Reviews, and Proofs Without Words. That should keep you busy until we do this all again in our October issue.

Three of our last five issues have been full double issues at 168 pages, and the other two issues have also had more than the usual number of pages. This has allowed us to put a considerable dent in our extensive backlog of accepted articles. As a result, authors whose articles are accepted today will have a much more reasonable wait time before seeing their articles in print than has recently been the case. We will continue to increase our page count until the backlog is down to an acceptable level.

Such a substantial increase in our page count requires heroic efforts behind the scenes to make it all happen. Let me once more thank Amanda Gedney, Annie Pettit, and Bonnie Ponce for all of their ongoing unsung efforts to get each issue of the magazine out on time. Receiving too many quality submissions is hardly the worst problem a periodical can have, but it does pose considerable logistical challenges for the entire production staff. If you enjoy reading Mathematics Magazine, then please take a moment to thank them for their efforts.

Jason Rosenhouse, Editor