# Mathematics Magazine - December 2004

### ARTICLES

It’s Hip to be Square if You’re a Flexagon
Ethan Berkove and Jeffrey Dumont
335-348
Flexagons are surprising geometric objects folded from strips of paper. In this article, we investigate hexaflexagons, which are folded from triangles, and tetraflexagons, which are folded from squares. For hexaflexagons, which have been well-studied, we consider their collections of motions, showing that in general this collection does not have a group structure. Tetraflexagons are considerably less well-known. Here, we establish some preliminary results and determine some restrictions on how tetraflexagons flex, and what types of tetraflexagons can occur.

Loxodromes: A Rhumb Way to Go
James Alexander
349-356
In the 1500s, the Portuguese mathematician Pedro Nunes developed the mathematics of rhumbs, or loxodromes, which are courses of constant bearing on the earth. Mercator maps have the great advantage for navigation that rhumbs are straight lines. Using this property, formulas for distances and directions along rhumbs are easily developed.

Which Way Did You Say That Bicycle Went?
David L. Finn
357-367
Given a set or tire tracks made by a bicycle, it is usually possible to determine which direction the bicycle went by using some basic facts about the physical construction of a bicycle. However, there are some exceptions, notably straight lines and concentric circles. We show that these are not the only exceptions, by providing simple method for constructing an infinite family of other exceptions.

### NOTES

The Egg-Drop Numbers
Michael Boardman
368-372
A problem in Which Way Did the Bicycle Go? leads to an interesting basic combinatorial question. The resulting analysis can be used as a fruitful setting for undergraduate study of combinatorial methods.

### PROOF WITHOUT WORDS

Sums of Consecutive Powers of n via Self-similarity
Mingjang Chen
373
Â“Queen Anne’s laceÂ” is an example of self-similarity. The number of circles in the lace is nk at the kth iteration. Taking the advantage of the visual effects of self-similarity, we design this work.

When Are Two Subgroups of the Rationals Isomorphic?
Friedrich L. Kluempen and Denise M. Reboli
374-379
A typical undergraduate algebra course uses the group of rationals under addition as an important example. Some of its subgroups are investigated, but the question of when two such subgroups are isomorphic is rarely asked. We present a result due to Baer that answers this question. We define the concepts of height and type in a way that can be understood by undergraduates. This approach leads to an invariant that allows us to decide when two of the rationals are isomorphic.

Surfaces of Revolution in Four Dimensions
Bennett Eisenberg
379-386
Most calculus students are familiar with rotating curves around axes in three dimensions to form surfaces of revolution. They have seen the formulas for the areas of these surfaces and the volumes of the corresponding solids. In this paper, we consider what it would be like to rotate the curves around axes in four dimensions. Using a little imagination we find formulas for the surface areas and volumes for regions generated in this way.

A Vector Approach to Ptolemy’s Theorem
Erwin Just and Norman Schaumberger
386-388
Ptolemy’s Theorem states that the product of the diagonals of a cyclic quadrilateral is equal to the sum of the products of its opposite sides. This note describes a new proof of Ptolemy’s Theorem using a vector approach.

Projected Rotating Polygons
Dimitrios Kodokostas
388-390
Projecting a point M of a circle onto a number of fixed lines through the center of the circle, we can produce many closed polygons (one convex and the rest self-intersected) by joining the projection points in an arbitrary order. We show the surprising fact that as the point M moves around the circle, each of these polygons remains unaltered in shape and size, just rotating around in the interior of the circle with each vertex remaining always on one of the fixed lines.

### PROOF WITHOUT WORDS

Convergence of the Hyperpower Sequence
F. Azarpanah
393
The limit of the sequence, root 2 to the root 2 to the root 2 to the root 2..., is 2. This is shown with a diagram familiar from dynamical systems.

The Probability That a Random Quadratic Is Factorable
Michael J. Bossé and N. R. Nandakumar
390-394
Given a quadratic Ax2 + Bx + C with integer coefficients A, B, and C such that A\ne 0 and -N \= A, B, C \= N, what is the probability that the quadratic is factorable if A, B, and C are chosen randomly? Elementary algebraic understanding quickly leads students to realize that some quadratics—even those with integer coefficients—are not factorable. Slightly more advanced algebraic understanding leads to the notion that few quadratics are factorable. Furthermore, more advanced mathematical intuition leads to belief that as N approaches ∞, the probability of factorability approaches zero. This note provides a proof that will formally confirm this supposition.

### POEM

Constructive Geometry
Ben Vineyard
394
First stanza: When I think of the circle, I am filled with awe. Not the circle of the sewer lid, or the Hula Hoop, But the one that spins in Euclid’s mind: The collection of all points in a plane Equidistant from the center, also a point, That thing Â“which hath no part.Â” So beautiful! But think of the work involved. An infinite number of laborers Each pushing a wheelbarrow filled with tiny black points Placing them precisely, just so many centimeters from the center.

### PROOF WITHOUT WORDS

A Triangular Identity
Roger B. Nelsen
395
A proof without words of the identity t(n+1)2t n2 = t n+ 12t n-12, where tn denotes the nth triangular number.