### ARTICLES

**Tropical Mathematics**

by David Speyer and Bernd Sturmfels

163-173

In tropical mathematics, the sum of two numbers is their minimum and the product of two numbers is their usual sum. Many results familiar from algebra and geometry, including the Quadratic Formula, the Fundamental Theorem of Algebra, and Bezout's Theorem, continue to hold in the tropical world. This article explains how to draw tropical curves, how tropical linear spaces differ from their classical counterparts, and why evolutionary biologists might care.

**Envelopes and String Art**

by Gregory Quenell

174-185

The base of a ladder is being pulled away from a wall at a constant rate. In a stroboscopic photograph of this act, the ladder would show up as a family of line segments whose upper edge approximates a curve, which is called their envelope. We discuss an elementary way to identify the envelope of a family of line segments, and consider several examples involving ladders, game theory, and nails driven into a board and connected by strings.

**Leveling with Lagrange: An Alternate View of Constrained Optimization**

by Dan Kalman

186-196

To maximize a function *f(x,y)* subject to the constraint *g(x,y)* = 0, Lagrange multipliers is a standard method. In one version, we define the function *F(x,y,λ)* = *f(x,y)* + *λ g(x,y)*, called a Lagrangian function, and set all the partial derivatives equal to zero. The Lagrange multipliers theorem shows that for a local constrained maximum of *f* to occur at *(x,y*, there must exist a *λ* for which *(x,y,λ )* is a critical point of *F*. This is commonly explained as a transformation of the constrained optimization problem for f into an unconstrained optimization problem for *F*, which is very close to what Lagrange himself said about the multiplier method, but it is a fallacy. In actuality, the Lagrangian function has a saddle point at *(x,y,λ)*, and indeed, *F* has no local extrema at all. Lagrangian leveling is an alternative rationale for the Lagrangian function approach, based on the idea of leveling the graph of *f *at the constrained maximum, without perturbing its graph over the constraint set. This provides additional insights about the Lagrangian function approach.

**NOTES**

**Quartic Polynomials and the Golden Ratio**

by Harald Totland

197-201

We investigate the graph of a quartic polynomial with inflection points and find many nice regularities. Among other things, the golden ratio occurs in several characteristic length ratios. These regularities turn out to be the same for all quartic polynomials with inflection points, a fact that is explained with the help of an affine transformation that maps one graph to another.

**When Cauchy and Hölder Meet Minkowski: A Tour through Well-Known Inequalities**

by Gerhard J. Woeginger

202-207

The article presents concise and simple proofs for several classical inequalities, like the Cauchy inequality and the Minkowski inequality. All proofs proceed in exactly the same way, and they all exploit the concavity of an underlying function in exactly the same way.

### PROOF WITHOUT WORDS

Beyond Extriangles

by M. N. Despande

208

For any triangle

*ABC*, construct squares on each of the three sides. Connecting adjacent square corners creates three extriangles. Iterating this process once more produces four quadrilaterals whose area is five times the original triangle.

**Varignon's Theorem for Octahedra and Cross-Polytopes**

by John D. Pesek, Jr.

209-215

Varignon's theorem states that the midpoints of a quadrilateral form the vertices of a parallelogram. It is not at first clear how this theorem might be generalized to higher dimensions. But if we notice that an octahedron can be regarded as a 3-dimensional version of a quadrilateral and that the midpoints can be thought of as centroids, then we can conjecture that the centroids of the faces of the octahedron are the vertices of the parallelpiped, the 3-dimensional version of a parallelogram. Using vector methods we show that this is true and generalize it to cross-polytopes which are the higher dimensional analogues of octahedra. Noting that the usual centroid is the center of mass of equal weights, we also state a generalization for weighted centroids. This is applied to show that the cross-section of a tetrahedron by a plane parallel to two opposite sides is a parallelogram.

**A Curious Way to Test for Primes Explained**

by David M. Bradley

215-218

In a previous note, Dennis P. Walsh develops a curious primality test based on repeatedly differentiating a certain sum of exponential functions. This note places Walsh's test in context by providing increasingly simpler functions for testing primality. The associated combinatorial interpretations are given, and it is ultimately shown how the tests relate to Lambert's generating series for the divisor function.

**More on the Lost Cousin of the Fundamental Theorem of Algebra**

by Roman Sznajder

218-219

We are discussing two generalizations of the fundamental theorem of algebra. The first, recently discovered by T. Tossavainen, relates to functions being linear combinations of exponential functions. The second one is a result on linear combinations of power functions and appeared in a small monograph by V.N. Malozemov, published in Russian, in the early 1970s. We show that these two ostensibly different generalizations are equivalent.

**Closed Knight’s Tours with Minimal Square Removal for All Rectangular Boards**

by Joe DeMaio and Thomas Hippchen

219-225

A closed knight's tour of a chessboard uses legal moves of the knight to visit every square exactly once and return to its starting position. In 1991 Schwenk completely classified the rectangular chessboards that admit a closed knight's tour. For a rectangular chessboard that does not contain a closed knight's tour, this paper determines the minimum number of squares that must be removed in order to admit a closed knight's tour. Furthermore, constructions that generate a closed tour once appropriate squares are removed are provided.

**PROOF WITHOUT WORDS**

Every Octagonal Number Is the Difference of Two Squares

by Elizabeth Jakubowski and Hasan Unal

225

This Proof Without Words offers a visual illustration of the fact that every octagonal number is the difference of two squares.

### POEM

Universal Language

by Robert Gethner

226

This poem is a meditation on, among other things, the feasibility of using mathematics as a language with which to communicate with extraterrestrial intelligence.