Given a positive integer \(m\), the authors exhibit a group with the...

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**Proof of the Double Bubble Conjecture**

by Frank Morgan

Frank.Morgan@williams.edu

The 2000 proof of the Double Bubble Conjecture, which identifies the double soap bubble that has less area than any other configuration enclosing and separating the same two volumes, uses only paper and pencil, unlike the 1995 computer-assisted proof for the special case of equal volumes by Hass, Hutchings, and Schlafly. The full story involves major contributions by undergraduates, including a recent extension to **R**^{4} by Reichardt et al.

**The Crooked Made Straight: Roberval and Newton on Tangents **

by Paul R. Wolfson

pwolfson@wcupa.edu

The tangent vector to a curve is the velocity vector of a moving point. To find the velocity vector, decompose the motion into simpler motions, find the velocity vectors of those motions, and reconstruct the desired vector. Our clear language of vectors makes this seem simple. In the mid-seventeenth century, however, mathematicians developed this method to treat a larger class of curves that were the result of "composition of motions". Because of the vagueness of their term, Descartes fell into error, and Roberval was unable to make clear why his results were correct. Even Newton erred when he reinvented the method in the 1660's. His quick recovery is interesting to trace. In order to include all examples of the composition of motions, and not just vector sums, Newton used the Galilean principle of relativity of motion. The method he evolved was essential in his first systematic exposition of the calculus.

**Solution of a Calculus Problem on Minimal Volume**

by David Spring

dspring@glendon.yorku.ca

An interesting geometrical problem in **R**^{3 }is to find the minimal area of a triangle that passes through a given point in the first octant and whose vertices lie, respectively, on the positive coordinate axes. This problem was the subject of a book prize in 1997. Our solution of the problem for the book prize, which generalizes to solve the analogous problem in the positive orthant in **R**^{n} , n 3, calculates the vertices of the minimal triangle by an elementary formula in terms of the coordinates of the given point. In all cases the formula can be computed approximately by using Newton's Rule. No closed form solution for the minimal area is known.

**Similarities in Irrationality Proofs for pi, ln2, zeta(2) and zeta(3)**

by Dirk Huylebrouck

dirk.huylebrouck@pi.be

When the irrationality of numbers is discussed, the irrationality of the square root of 2 often constitutes the only example that is given at college level. Yet, proofs for the irrationality of other more famous numbers such as pi and its square are fairly simple as well. The irrationality of pi dominated a good 2000 years of mathematical history, starting with the related circle-squaring problem of the ancient Greeks. The interest in piÆs younger colleagues zeta(2) and zeta(3) started only a few centuries ago, and the latter number resisted until some twenty years ago. However, even very recently, Tanguy Rivoal's results about the zeta numbers could still upset the mathematical community.

Simultaneous presentation of four irrationality proofs has the advantage of showing their similarity as well as the occurrence of another popular number, the golden mean, and its generalizations, called "the metallic means". The reader is seriously warned about common misconceptions about these numbers, such as those about "most elegant" rectangles. Loose statements about numbers that are interpreted as "optimal" solutions do not correspond to the standard mathematical terminology, but the various irrationality proofs provided in this paper do lead to expressions where these metallic means pop up as true optimal solutions.

**The Matrices of Pascal and other Greats**

by Lidia Aceto and Donato Trigiante

aceto@calvino.polito.it

trigiant@cesit1.unifi.it

Historical references to the Pascal matrix go back to the fourteenth century. It is mentioned in an ancient book of Chinese mathematics. Its many properties continue to surprise also for the variety of its applications in such diverse fields as probability, numerical analysis, surface reconstruction, and combinatorics. Recent studies of its properties have taken new directions, which highlight its relations with other important matrices. In this paper such connections are studied in a deeper way, establishing the relations existing between the Pascal matrix and some families of classical polynomials. For example, the Bernstein matrix Be(x), whose entries are the Bernstein polynomials, can be written in the very simple form

where *D(x) *is a diagonal matrix and *P* is the Pascal matrix.

**Leftovers from the Ham Sandwich Theorem**

by Graham Byrnes, Grant Cairns, and Barry Jessup

graham.byrnes@mh.org.au, g.cairns@latrobe.edu.au, and bjessup@uottawa.ca

The classical ham sandwich says that it is always possible to slice the sandwich with one cut so that the ham and both slices of bread are each divided into equal halves. Its traditional proof is a clever use of the Borsuk-Ulam Theorem, which is taught in many introductory courses on algebraic topology. A simple and intuitive alternative would be to take the plane through the centres of gravity of each of the three bodies, but this doesnÆt always work! We explain why this doesnÆt work, and investigate interesting situations where it does.

**Notes** **Some New Irrational Decimal Fractions**

by Patrick Martinez

martinez@bretagne.ens-cachan.fr

**Monic Polynomials in Z(x) with Roots in the Unit Disc**

by Pantelis A. Damianou

damianou@ucy.ac.cy

**Pythagorean Triples of Polygonal Numbers**

by Egon Scheffold

scheffold@mathematik.th-darmstadt.de

**A Symmetry of Power Sum Polynomials and Bernoulli Numbers**

by Hans J. H. Tuenter

htuenter@ssb.yorku.ca

**Heronian Triangles are Latttice Triangles **

by Paul Yiu

yiu@fau.edu

**Evolution of. . . **

**On the Emotional Assumptions Without Which One Could Not Effectively Investigate the Laws of Nature **

by Vi. P. Vizgin

translated by Abe Shenitzer

**Problems and Solutions**

**Reviews **

**The Moment of Proof: Mathematical Epiphanies.**

By Donald C. Benson

Reviewed by Susan G. Staples

s.staples@tcu.edu

**Mathematics Education Research: A Guide for the Research Mathematician.**

By Curtis McKnight

Reviewed by George E. Andrews

andrews@math.psu.edu

**Telegraphic Reviews**