Projective Geometry over F1 and the Gaussian Binomial Coefficients
by Henry Cohn
There is no field with only one element, yet there is a well-defined notion of what projective geometry over such a field means. This notion is familiar to experts and plays an interesting role behind the scenes in combinatorics and algebra, but it is rarely discussed as such. The purpose of this article is to bring it to the attention of a broader audience, as the solution to a puzzle about Gaussian binomial coefficients.
A Fresh Look at the Method of Archimedes
by Tom M. Apostol and Mamikon A. Mnatsakania
This paper introduces Archimedean globes, a family of solids that circumscribe a sphere. Cross sections of each globe by planes parallel to the equatorial plane are disks bounded by similar polygons that circumscribe the circular cross sections of the sphere. Like the sphere, which is a limiting case, an Archimedean globe has both volume and surface area two-thirds that of its circumscribing prismatic container. The results are obtained geometrically. The volume and surface area of any Archimedean shell, the region between two Archimedean globes, are also determined. Surprising consequences of these new results are: several families of incongruent solids having both equal volume and equal total surface area; the quadrature of the sine curve; and the centroid of any slice of a spherical surface.
Heesch’s Tiling Problem
by Casey Mann
Heesch’s tiling problem concerns the number of layers that can be formed from copies of a tile T around a centrally placed copy of T (this number is the Heesch number of T). Surprisingly, before 1991 no tiles with finite Heesch number greater than 1 where known. Then tiles with Heesch numbers 2 and 3 were discovered. This paper presents new examples of tiles with Heesch numbers 4 and 5. We go on to discuss the connections of Heesch’s tiling problem to other famous unsolved tiling problems, including the Einstein problem that concerns aperiodic monotiles.
A Construction concerning (lP)' lq
by Gilbert Helmberg
Translating Inequalities between Hardy and Bergman Spaces
by Kehe Zhu
A Simple Proof of the Descartes Rule of Signs
by Xiaoshen Wang
Another Proof That R3 Has No Square Root
by Sam B. Nadler, Jr.
An Interesting Recursion
by Steven H. Weintraub
On the Middle Coefficient of a Cyclotomic Polynomial
by Gregory P. Dresden
Problems and Solutions
Mathematical Modeling of Physical Systems: An Introduction
by Diran Basmadjian
Reviewed by Douglas R. Shier