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.

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 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.

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…

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…

We provide several constructions, both algebraic and geometric, for determining the ratio of the radii of two circles in an Apollonius-like packing problem. This problem was inspired by the art deco design in the transom window above the Shadek Fackenthal Library door on the Franklin & Marshall College campus.

Viviani's theorem states that the sum of distances from any point inside an equilateral triangle to its sides is constant. Here, in an extension of this result, we show, using linear programming, that any convex polygon can be divided into parallel line segments on which the sum of the distances to the sides of the polygon is constant. Let us say…

After Peano gave an arithmetic construction, Hilbert developed a geometric construction for space-filling curves. This paper describes the key idea of Hilbert's construction, here called "the nested-squares criterion," implicit in Hilbert's writing but, once explicated, generalizes to a whole class of space-filling curves that correspond to a…

The "box problem" from introductory calculus seeks to maximize the volume of a tray formed by folding a strictly rectangular sheet from which identical squares have been cut from each corner. In posing such questions, one would like to choose integral side-lengths for the sheet so that the excised squares have rational or integral side-length.…

For a given initial speed of water from a spigot or jet, what angle of the jet will maximize the visual impact of the water spray in the fountain? This paper focuses on fountains whose spigots are arranged in circular fashion, and couches the measurement of the visual impact in terms of the surface area and the volume under the fountain's natural…

If you take a circle with a horizontal diameter and mark off any two points on the circumference above the diameter, then these two points together with the end points of the diameter form the vertices of a cyclic quadrilateral with the diameter as one of the sides. We refer to the quadrilaterals in question as diametric. In this note we consider…

Given that the sine and cosine functions of a real variable can be interpreted as the coordinates of points on the unit circle, the author of this article asks whether there is something similar for complex variables, and shows that indeed there is.

We answer a geometric question that was raised by the carpenter in charge of erecting helical stairs in a 10-story hospital. The explanation involves the equations of lines, planes, and helices in three-dimensional space. A brief version of the question is this: If A and B are points on a cylinder and the line segment AB is projected radially onto…

Logarithmic spirals are among the most fascinating curves in the plane, being the only curves that are equiangular, and the only ones that are self-similar. In this article, we show that in three dimensions, these two properties are independent. Although there are surfaces that have both properties, there are some that are equiangular, but not…