This Month's Feature Column

People Making a Difference

Introduction

Mathematics has a reputation as a subject of austere beauty - a subject that resonates for some people the way that a beautiful sunrise or sunset, a beautiful symphony, or a beautiful painting might for other people. Yet mathematics also has its applied side. Without mathematics that was developed in the 20th century we would not have the cell phones that are radically changing the way we live our lives in the early 21st century. Against a backdrop of the dual aspects of mathematics, its beauty and applicability, is the sense that the frontiers of mathematics are getting further and further from what an "average person," one not pursuing a career in mathematics or one of its closely allied fields, can master. There has been an increase in the length and complexity of mathematical proofs and in some cases, important theorems have required a computer in an integral way. Examples of this are the results of Wolfgang Haken and Kenneth Appel that confirm the conjecture that every plane map can have its faces colored with 4 or fewer colors and Thomas Hale confirming the maximal density that spheres packed in 3-space can achieve.

Yet because of the work of many individual mathematicians and their passion for some parts of mathematics, as well as the clarity of their insights, based on the work of countless generations of impassioned mathematicians of the past, it is possible to see the dual aspects of mathematics' beauty and applicability more clearly. There are many such people, but here I would like to call attention to the work of the American geometer Victor Klee, who recently died.

Victor Klee was one of America's most prominent geometers.

His recent death (August 2007) is a great loss for the mathematics community. His published works included several books and over 240 research papers. Klee was born in San Francisco in 1925 and attended Pomona College with a major in both mathematics and chemistry. Whereas prior to the 20th century nearly all mathematicians (e.g. Newton, Gauss, Euler, Laplace, etc.) made contributions to not only mathematics but physics or some other branch of science, the pressures of specialization have made this more rare. Although most of Klee's work had a geometric focus, his work spanned a wide range of interests motivated by both theoretical and applied considerations. He received his doctorate degree from the University of Virginia, where he studied with the prominent topologist Edward McShane. His doctoral thesis (1949) was entitled:

Convex Sets in Linear Spaces

Klee's early training and research was in the area of topology-- the study of properties of geometric objects that go beyond the traditional concern with angles, distances, and areas of the Euclidean geometry tradition. Thus, from a topological point of view, a straight line segment and a curved segment are alike, and a square and a (Euclidean) ellipse are alike, but a segment and a circle are not alike. This topological difference means that a circle "separates" a plane into an interior region and an exterior region, while a segment does not. Thus, if one picks any two points in a plane not on a segment (or even a finite number of segments), one can find a curve which joins these two points and has no point in common with the segment. However, for a point inside a circle and one outside the circle, any curve joining the points must have a point in common with the circle. This is the kind of geometric information that interests a topologist and does not require information about distances.

Figure 1: Geometric sets can look different but be topologically equivalent.

Joseph Malkevitch
York College (CUNY)
malkevitch at york.cuny.edu