Mathematical construction is the theme of the first two articles in the October issue, for violins and *n*-gons by Stroeker and Milnikel, respectively. These articles are followed by two articles on more discrete topics. Möbius maps and continued fractions are related by Beardon in an article that proves a theorem by Galois. Then, Konvalina uses a combinatorial approach to determine the entries of powers of 2x2 matrices, leading to two combinatorial identities. Mixed in are a proof without words by Nelsen, an anniversary crossword, the problem section, and the reviews. The issue concludes with reports on the most recent olympiads and the Allendoerfer awards —*Michael A. Jones, Editor*

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Vol. 88, No. 4, pp 245 – 319

## Articles

### On the Shape of a Violin

Roel J. Stroeker

For centuries luthiers—that is, instrument makers of violins and other stringed instruments—had no more sophisticated tools at their disposal to define the shape of their instruments than marked ruler and compass. Today, modern aids are available in terms of computational power and expertise in graphic design to assist them in this respect. This raises the following question: How can these powerful computational techniques be applied in the process of searching for a form of the violin both pleasing to the eye and optimal in some mathematical sense? In this paper, I use parametric cubic splines in an attempt to come close to and possibly improve upon—strictly in a mathematical and visual sense—the shape of a violin as laid down by the great masters of the past. The main reasons for choosing the cubic spline are: good approximation properties, simplicity of construction, and most importantly, its unique curvature properties.

To purchase from JSTOR: 10.4169/math.mag.88.4.247

### A New Angle on an Old Construction: Approximating Inscribed *n-*gons

Robert S. Milnikel

To purchase from JSTOR: 10.4169/math.mag.88.4.260

### Editors Past and President, Part II

Tracy Bennett and Michael A. Jones

To purchase from JSTOR: 10.4169/math.mag.88.4.270

### Möbius Maps and Periodic Continued Fractions

A. F. Beardon

We describe some of the relationships that exist between quadratic irrationals, continued fractions, Möbius maps, and hyperbolic geometry, and we illustrate these by giving a simple geometric proof of Galois' result on dual continued fractions.

To purchase from JSTOR: 10.4169/math.mag.88.4.272

### Proof Without Words: Sums of Odd and Even Squares

Roger B. Nelsen

To purchase from JSTOR: 10.4169/math.mag.88.4.278

### A Combinatorial Formula for Powers of 2 × 2 Matrices

John Konvalina

Formulas for the nth power of a 2 × 2 matrix are known and typically depend on either computing the determinant of the matrix or its eigenvalues. In this note we derive a direct combinatorial formula for the nth power of the matrix that does not depend on any auxiliary computation. Applications include the derivation of a combinatorial identity involving the Fibonacci numbers.

To purchase from JSTOR: 10.4169/math.mag.88.4.280

## Problems and Solutions

Proposals, 1976-1980

Quickies, 1053-1054

Solutions, 1946-1950

Answers, 1053-1054

To purchase from JSTOR: 10.4169/math.mag.88.4.285

## Reviews

*The Three Laws of Mathematicians; new tiling pentagon; Benford’s law*

To purchase from JSTOR: 10.4169/math.mag.88.4.294