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JUNE/JULY 2003

**A Matter of Gravity**

by Steven G. Krantz

sk@math.wustl.edu

We study the notion of center of gravity of a planar region, and in particular consider the question of when the center of gravity of a planar region actually lies inside that region. Along the way, a number of new results about centers of gravity are proved. Centers of gravity are related to barycentric coordinates. Some new stability results are established. The theorems in this paper use techniques both of analysis and geometry in surprising new ways.

**Simple Proofs of a Geometric Property of Four-Bar Linkages**

by Godfried Toussaint

godfried@cs.mcgill.ca

This paper considers the relationship between the lengths of the two diagonals in the four-bar planar linkage. For convex and crossing linkages, one diagonal increases if and only if the other decreases. On the other hand, for nonconvex simple linkages, one diagonal increases if and only if the other also increases. Several novel and simple proofs of this geometric property are presented. The proofs are simpler than existing published proofs, and some of them illustrate the application of Decartes's principle of instantaneous centers of rotation. The four-bar linkage has many applications, ranging from obtaining complex curves (motions) from circular motion in kinematics to reconfiguring complex linkages that serve as models of robot arms, knots and molecules in polymer physics and molecular biology. The introduction briefly reviews the broad spectrum of applications in a historical context.

**Non-Euclidean III.36**

by Robin Hartshorne

robin@math.berkeley.edu

Some theorems of elementary Euclidean geometry are also valid in non-Euclidean geometries, for example, the triangle congruence theorems *SAS*, *SSS*, *ASA*, and the concurrence of angle bisectors or altitudes of a triangle. Other theorems, such as the angle sum of a triangle being two right angles, the Pythagorean theorem, or the triangle inscribed in a semicircle having a right angle, depend on the Euclidean parallel postulate and are not true in any non-Euclidean geometry.

Proposition 36 of Book III of Euclid's *Elements* states that the square of the tangent line from a point to a circle is equal to the product of the two segments from that point along a secant line to the circle. Euclid's proof makes use of the Pythagorean theorem and so depends on the parallel postulate.

In this article the author shows that Euclid's III.36 can be phrased in terms of area in such a way as to make a single statement valid in Euclidean, hyperbolic and spherical geometry. As a corollary he gives a new proof of the Pythagorean theorem in Euclidean geometry.

**Fractals, Graphs, and Fields**

by Franklin Mendivil

franklin.mendivil@acadiau.ca

In mathematics, surprising connections between different areas continuously arise. In this paper we discuss how a problem related to generating an image of an IFS (Iterated Function System) fractal leads to consideration of an Eulerian cycle in a certain graph (a de Bruijn graph) and then to finding generators for the multiplicative group of a finite field.

**Sums of Squares of Distances in M-Space**

Tom M. Apostol and Mamikon A. Mnatsakanian

apostol@caltech.edu, mamikon@caltech.edu

Steiner's parallel-axis theorem states that the moment of inertia of a rigid body about an arbitrary axis is equal to the moment of inertia about a parallel axis through the center of mass of the body, plus the mass times the square of the distance between the two axes. This paper provides a discrete analog of this theorem with extensions to higher-dimensional space. Applications to geometry include some surprising and nontrivial results concerning the sum of squares of distances. For example, the sum of squares of the distances between any n points on a sphere in m-space reaches its maximum when the centroid of the points is at the center of the sphere. Analogous results are given for the sum of squares of edge lengths of a simplex and the sum of the edge lengths themselves.

**Problems and Solutions**

**Notes**

**Counting Even and Odd Partitions**

by Martin Klazar

klazar@kam.mff.cuni.cz

**The Isoperimetric Inequality and a Theorem of Hardy and Littlewood**

by Dragan Vukotic

dragan.vukotic@uam.es

**And Still One More Proof of the Radon-Nikodym Theorem**

by Anton Schep

schep@math.sc.edu

**A Linked Pair of Sequences Implies the Primes are Infinite**

by Michael Somos and Robert Haas

somos@grail.cba.csuohio.edu

**Another Simple Proof of 1 + 1/2**^{2} + 1/3^{2} + ... = π^{2}/6

James D. Harper

harperj@cwu.edu

**Reviews**

**Analysis for Applied Mathematics**

by Ward Cheney

Reviewed by David G. Schaeffer

dgs@math.duke.edu

**The Hilbert Challenge**

by Jeremy J. Gray

**The Honors Class: Hilbert's Problems and Their Solvers**

by Benjamin H. Yandell

Reviewed by Jeffrey L. Nunemacher

jlnunema@cc.owu.edu