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ARTICLES

**Hammer, Juggling, Rotational Instability**

Carl V. Lutzer

243-250

This article addresses the rotational instability inherent in objects with three distinct dimensions. Assuming only basic knowledge of physics and multivariate calculus, it derives Euler's equations of motion. It proceeds to perform a linear stability analysis on the eigenvectors of the equation, and then to show (using the operator norm on matrices) why the eigenvalues of the moment-of-inertia tensor are certainly positive.

**The Volume Principle**

William C. Dickinson and Kristina Lund

251-261

The dual theorems of Menelaus and Ceva are among the most beautiful and useful results in Euclidean Geometry. Grünbaum and Shephard, in their article, Ceva, Menelaus and the Area Principle, introduce a simple tool, called the Area Principle, and use it to present deservingly elegant proofs of both Ceva's and Menelaus' Theorems and their generalizations. In this paper, we prove the Volume Principle, which is valid for the Euclidean Plane, the hyperbolic plane and the two-dimensional sphere. This tool allows us to extend all the theorems from the Grünbaum and Shephard's article, except for those generalizations which require linearity, into these other geometries.

**How Much Does a Matrix or Rank k Weigh?**

Theresa Migler, Kent Morrison, and Mitchell Ogle

262-271

A matrix with just one nonzero entry has rank one and a matrix with two nonzero entries has rank at most two. These observations lead to the natural question about the relationship between the number of nonzero entries (the weight) of a matrix and its rank.

We find a simple formula for the average weight of an m by n matrix of rank k when the entries come from a finite field of order q. The formula shows precisely how the average weight grows with the rank. However, we would like to determine the complete probability distribution for the weight of rank k matrices of size m by n. For rank one matrices we can describe the weight distribution completely. Using this result we show that the weight is normally distributed in the limit as the matrix dimensions go to infinity.

**Folding Optimal Polygons from Squares**

David Dureisseix

272-280

What is the largest regular n-gon that fits in a unit square? Can it be folded from a square piece of paper using standard moves from origami? The answers are provided in this article, as well as the constructions of optimal 5, 6, 7 and 9-gons. Unlike standard compass-and-straightedge constructions, origami allows the construction of regular polygons that require the roots of cubic polynomials, such as the optimal 7-gon.

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PROOF WITHOUT WORDS

**Proof Without Words: Complex Numbers with Modulus One**

Jean Huang

280

A diagram is used to show that any complex number with modulus one can be expressed as a fraction for some real number t.

**Proof Without Words: A Weighted Sum of Triangular Numbers**

Roger Nelsen

317

A proof without words of the following: If denotes the triangular number, then.

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NOTES

**Eigencircles of 2x2 Matrices**

M. J. Englefield and G. E. Farr

281-289

We show how to associate, to any real 2x2 matrix, a circle (which we call the eigencircle) that links properties of the eigenvalues and eigenvectors of the matrix to simple geometric properties of the circle. Insights given by the eigencircle include: pictures of real or complex eigenvalues, their eigenvectors, and the determinant; a geometric derivation of the angle between eigenvectors; a geometric proof that a real symmetric 2x2 matrix has perpendicular eigenvectors; a geometric proof that the product of real or complex eigenvalues is the determinant; and a geometric view of all 2x2 matrices with the same eigenvalues.

**The Volume Swept Out by a Moving Planar Region**

Robert Foote

289-297

An integral formula is presented that gives the volume swept out by a moving planar region. This result, which goes back at least to Courant, generalizes both Cavalieri's Principle and the Theorem of Pappus. The volume depends on the area of the region and the motion of its centroid relative to the plane of the region.

**Equal Sums of Three Fourth Powers or What Ramanujan Could have Said**

Richard Blecksmith and Simcha Brudno

297-302

This note exploits a 100-year-old formula for expressing the sum of three fourth powers as twice the square of a certain quadratic form. Different representations of this of form yield different ways of expressing the original number as a sum of three fourth powers. As a results, we exhibit a sequence of integers with the property that the number of ways to write the nth term in the list as a sum of three fourth powers tends to infinity with n. The third integer in this list is related to 1729, the famous taxicab number of Ramanujan lore.

**Another Look at an Amazing Identity of Ramanujan**

Michael Hirschhorn and Jung Hun Han

302-304

**Ramsey's Theorem is Sharp**

Solomon Golomb

304-306

**Where the Camera Was, Take Two**

Annalisa Crannell

306-308

In the article "Where the camera was," MATHEMATICS MAGAZINE 77 (2004), 251-259, Byers and Henle approximated the position of a photographer from geometric clues in an old photograph of John M. Greene Hall at Smith College. Here, we give an approach to the problem that is slightly more geometric and less algebraic.