Rene Descartes' method for finding tangents (equivalently, subnormals) depends on geometric and algebraic properties of a family of circles intersecting a given curve. It can be generalized to establish a calculus of subnormals, an alternative to the calculus of Newton and Leibniz. Here we prove subnormal counterparts of the well-known…

Given a function defined on a subset of the plane whose partial derivatives never change sign, the signs of the partial derivatives form a two-dimensional pattern. We explore what patterns are possible for various planar domains.

Many classical problems in elementary calculus use Euclidean geometry. This article takes such a problem and solves it in hyperbolic and in spherical geometry instead. The solution requires only the ability to compute distances and intersections of points in these geometries. The dramatically different results we obtain illustrate the effect…

We produce a continuum of curves all of the same length, beginning with an ellipse and ending with a cosine graph. The curves in the continuum are made by cutting and unrolling circular cones whose section is the ellipse; the initial cone is degenerate (it is the plane of the ellipse); the final cone is a circular cylinder. The curves of the…

The use of traditional string figures by the Dr. Schaffer and Mr. Stern Dance Ensemble led to experimentation with polyhedral string constructions. This article presents a series of polyhedra made with six loops of three colors which sequence through all the Platonic Solids.

While the path to mathematical discovery is often intuitive and not simply reducible to the application of a method, neither is it in most cases entirely haphazard. Certain kinds of questions and certain ways of looking at things tend to produce new and interesting results. To illustrate, this paper tells a story of how someone might discover…

How many rods does it take to brace a square in the plane? Once Martin Gardner's network of readers got their hands on it, it turned out to be fewer than Raphael Robinson, who originally posed the problem, thought. And who could have predicted the stunning solutions found subsequently for various generalizations of the problem?

The focus of a parabola rolling without sliding along a straight line traces a catenary. We give some historical notes on this classical kinematical construction, observe that it was rediscovered in a recent article, and give a simpler and more geometric alternative derivation.

When a plane figure is photographed from different viewpoints, lengths and angles appear distorted. Hence it is often assumed that lengths, angles, protractors, and compasses have no place in projective geometry. Here we describe a sense in which certain angles are preserved by projective transformations. These angles can be constructed with…

The movement of groundwater in underground aquifers is an ideal physical example of many important themes in mathematical modeling, ranging from general principles (like Occam's Razor) to specific techniques (such as geometry, linear equations, and the calculus). This article gives a self-contained introduction to groundwater modeling with…

We relate the factorization of an integer N in two ways as N = xy = wz with x + y = w - z to the inscribed and escribed circles of a Pythagorean triangle.

One of Gardner's passions was to introduce puzzles into the classroom. From this point of view, polyomino dissections are an excellent topic. They require little background, provide training in geometric visualization, and mostly they are fun. In this article, we put together a large collection of such puzzles, introduce a new approach in solving…

Given a set of oriented hyperplanes P = {p1, . . . , pk} in R[superscript n], define v : R[superscript n] [right arrow] R by v(X) = the sum of the signed distances from X to p[subscript 1], . . . , p[subscript k], for any point X [is a member of] R[superscript n]. We give a simple geometric characterization of P for which v is constant, leading to…

Two coordinate systems are related here, one defined by the earth's equator and north pole, the other by the orientation of a telescope at some location on the surface of the earth. Applying an interesting though somewhat obscure property of orthogonal matrices and using the cross-product simplifies this relationship, revealing that a surprisingly…

The Isoperimetric Quotient, or IQ, introduced by G. Polya, characterizes the degree of sphericity of a convex solid. This paper obtains closed form expressions for the surface area and volume of any Archimedean polyhedron in terms of the integers specifying the type and number of regular polygons occurring around each vertex. Similar results are…