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Mathematics Magazine - December 2012

We begin with tiling; if you like rep-tiles, you will like self-tiling sets.  We step up a dimension to a examine a very special lattice polyhedron, and then we stay with lattice polyhedra for the story of how "mirror pairs'' arise in string theory. All with color illustrations, and there is much more!

This is the issue in which we recognize our referees; when you see someone who is on the list, give them a special thanks.—Walter Stromquist, Editor

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On Self-Tiling Tile Sets
Lee Sallows
A novel species of self-similar tilings is introduced. Results divide naturally into two categories, involving either polyforms or non-polyforms. Some eye-catching findings are presented in what promises to be a rich field of research. The discovery of a class of triangle pairs showing arresting properties will undoubtedly surprise and appeal to many, not least those readers already conversant with rep-tiles.

A Polyhedron Full of Surprises
Hans L. Fetter
The problem of geometric realization for convex polyhedra, which satisfy certain desirable properties, has received quite a bit of attention lately. Interest, mainly, has been on polyhedral representations where either all of the vertex coordinates are small integers, or all of the edge lengths are integers, or all of the edges are tangent to a sphere. In general, it is not easy to construct a convex polyhedron satisfying any of those criteria. We introduce a remarkable polyhedron that satisfies all of them.

From Polygons to String Theory
Charles F. Doran and Ursula A. Whitcher
We describe special kinds of polygons, called Fano polygons or reflexive polygons, and their higher-dimensional generalizations, called reflexive polytopes. Pairs of reflexive polytopes are related by an operation called polar duality. This combinatorial relationship has a deep and surprising connection to string theory: One may use reflexive polytopes to construct “mirror” pairs of geometric spaces called Calabi-Yau manifolds that could represent extra dimensions of the universe. Reflexive polytopes remain a rich source of examples and conjectures in mirror symmetry.

Proof Without Words: Sums of Cubes
Parames Laosinchai


A Short Elementary Proof of $$\sum 1/k^{2}=\pi^{2}/6$$
Daniel Daners
We give a short elementary proof of the well known identity $$\zeta(2)=\displaystyle\sum_{k=1}^{\infty}1/k^{2}=\pi^{2}/6$$. The idea is to write the partial sums of the series as a telescoping sum and to estimate the error term. The proof is based on recursion relations between integrals obtained by integration by parts, and simple estimates.

Backwards Induction and a Formula of Ramanujan
Michael Sheard
Proof by induction is a very versatile technique. A short proof of an elegant formula of Ramanujan involving continued roots provides an opportunity to look at two examples in which induction appears to work backwards.

What Is Special about the Divisors of 24?
Sunil K. Chebolu
What is an interesting number theoretic characterization of the divisors of 24 among all positive integers? This paper will provide one answer in terms of modular multiplication tables and will give 5 different proofs based on the Chinese remainder theorem, Dirichlet’s theorem on primes in an arithmetic progression, the structure theory of units, the Bertrand-Chebyshev theorem, and its generalisations by Erdös and Ramanujan.

Proof Without Words: Runs of Triangular Numbers
Roger B. Nelsen and Hasan Unal

The Limit Comparison Test Needs Positivity
J. Marshall Ash
The limit comparison test for positive series does not extend to general series. An example is given In a certain sense, this is the only possible example. Given a conditionally convergent series, there exists a termwise much smaller series so that the sum of the two series diverges. Given a divergent series with terms tending to zero, there exists a convergent but termwise much bigger series.

Möbius Polynomials
Will Murray
We introduce the Möbius polynomial $$M_{n}(x)=\displaystyle\sum_{d|n}\mu\left(\frac{n}{d}\right) x^{d}$$, which gives the number of periodic bracelets of length n with x possible types of gems, and therefore satisfies $$M_{n}(x)\equiv0\pmod{n}$$ for all $$x\in$$. We derive some key properties, analyze graphs in the complex plane, and then apply Möbius polynomials combinatorially to juggling patterns, irreducible polynomials over finite fields, and Euler’s totient theorem.