Melissa Lenzi

Mysteries of Mathematics

Professor Ulrich Daepp

December 14, 1999

Topology is the study of those properties of geometric figures that are unchanged when the shape of the figure is twisted, stretched, shrunk, or otherwise distorted without breaking. It is sometimes referred to as "rubber sheet geometry" (West 577). Topology is a basic and essential part of any post school mathematics curriculum. Johann Benedict Listing introduced this subject, while Euler is regarded as the founder of topology. Mathematicians such as August Ferdinand Möbius, Felix Christian Klein, Camille Marie Ennemond Jordan and others have contributed to this field of mathematics. The Möbius band, Klein bottle, and Jordan curve are all examples of objects commonly studied. These and other topics prove to be intricate and fascinating mathematical themes.

Topologists are mathematicians who study qualitative questions about geometrical structures. They ask questions like does the structure have any holes in it? Is it all connected, or can it be separated into parts? Topologists are not concerned with size, straightness, distance, angle, or other such properties. An often-cited example is the London Underground map. This will not reliably tell you how far it is from Kings Cross to Picadilly, or even the compass direction from one to the other. However, it will tell you how the lines connect between them, using topological rather than geometric information (What 1).

Furthermore, if one figure can be distorted into another figure without breaking, then the two figures are described as being topologically equivalent to each other. Two examples of topologically equivalent figures are a coffee cup and doughnut, and groups of the letters of the alphabet. First, an object shaped like a doughnut is a torus. A torus can be molded into a coffee cup without tearing it or breaking it. Also, the letters C, I, L, M, N, S, U, V, W, and Z are all topologically equivalent. C can be straightened out to make an I, the I can be bent to make an L, and so on. O and D are topologically equivalent only to each other (West 578).

Topology has an interesting history. As a branch of mathematics, it did not spring full-blown into the minds of some mathematicians. It gradually developed as a number of mathematicians experimented with the distortion of geometric figures. In the 18th and 19th centuries, Euler distorted a map into a network and concluded that the formula v - e + f = 2, where v equals the number of vertices, e equals the number of edges, and f equals the number of faces, holds true for all solids. Then, Antoine-Jean Lhuilier attempted to classify cases in which he discovered that Euler's formula was wrong. Also, Möbius discovered a one-sided surface. Carl Friedrich Gauss, a German mathematician, explored the distortion of knots. Listing published his Census; Bernhard Riemann studied the multiplicities of the roots of equations, while Klein and Fricke developed his ideas. Gustav Kirchhoff wrote on the flow of current in the electrical networks. Many others also made contributions (Flood 105-118).

The result of all these individual efforts was the eventual development of a branch of mathematics devoted exclusively to studying the constant properties of figures under distortion. The first attempt to systematize topology was made by Henri Poincaré, a French mathematician. In 1895, he published five papers developing topology as a purely qualitative, not quantitative, subject. Since their publication, the field of topology has grown at such an explosive rate that today it is one of the major branches of mathematics (West 584).

One of the most well known mathematicians in this field is August Ferdinand Möbius. Möbius was born on November 17, 1790 in Schulpforta, Saxony to Johann Heinrich Möbius, who died when he was only 3 years old. He originally planned to study law, as his family had wanted him too. However, he soon decided to pursue his own interest in mathematics (August 1). The timeline of Möbius' life exhibits a great variety of accomplishments. In 1809 he was a student at the Leipzig University. From 1813-1814, he traveled to Gottigen with Gauss. In 1815, Möbius wrote a Doctoral thesis, The occulations of fixed stars and a Habilitation thesis on Trigonometrical Equations. In 1816 he was appointed Extraordinary Professor of Astronomy in Leipzig. The Leipzig Observatory was developed under his supervision from 1818-1821. In 1820 he married and later had one daughter and two sons. He wrote the The Barycentric Calculus in 1827. Then, in 1829, he was made Corresponding Member, Berlin Academy of Sciences. From 1834-1836, Möbius wrote popular treatises on Halley's comet and the Principles of astronomy. In 1837 he wrote a two-volume textbook of statistics, and Celestial Mechanics. In 1844 he was appointed Full Professor in Astronomy, Leipzig and Director of the Observatory. He wrote The Theory of Circular Transformations in 1855. His most famous discovery, the Möbius band, was discovered in 1858. Möbius died on September 26, 1868 in Leipzig (Flood 7).

Möbius is most remembered for his discovery of the Möbius band. Distortions that break a figure so that the points next to each other do not remain so are not topological. An example of this is a one-sided surface, a Möbius strip, as it has come to be called. Möbius took a closed belt-shaped loop. He cut the belt, gave one end a half twist (180 degrees), and fastened the ends back together. More generally, one can twist a belt by an angle of k x 180 degrees, where k is an odd integer. This gives a generalized Möbius band. Because one end is only half twisted, the points that were originally close to each other on the closed loop are now separated on the band. For this reason, the Möbius band is not topologically equivalent to the original closed belt. The surprising property of the Möbius band is that it is a non-orientable surface. As you go once around the loop inside the Möbius band, you come back with the left arm on the right and vice versa. This means one can no longer talk about left and right. Even though an untwisted loop clearly has two sides and two edges, the Möbius band has only one side and one edge (West 578).

Felix Christian Klein has also significantly contributed to the field of topology. He is best known for his work in non-Euclidean geometry, for his connections between geometry and group theory, and for results in function theory. Klein was born on April 25, 1849 in Düsseldorf, Prussia (now Germany). He attended the Gymnasium in Düsseldorf and then the University of Bonn where he studied mathematics and physics from 1865-1866. He started his career with the intention of becoming a physicist. He was appointed to the post of laboratory assistant to Plücker, who held a chair of mathematics, in 1866. Plücker died leaving his Neue Geometrie des Raumes, which assessed the foundations of line geometry, for Klein to complete. This work led him to become acquainted with Clebsch. Then, in 1871, Klein became a lecturer at Göttingen. He was appointed professor at Erlangen, in Bavaria in southern Germany, in 1872. Clebsch "regarded him as likely to become the leading mathematician of his day." Therefore, Klein held a chair from the young age of 23 (Felix 1). In 1875 he was offered a chair at the Technische Hochschule at Munich. In that same year, Klein married Ann Hegel, the granddaughter of the philosopher George Wilhelm Friedrich Hegel. After serving as chair of geometry at Leipzig and at the University of Göttingen in 1886, he taught at Göttingen. In 1913 this mathematician retired due to ill health, but continued to teach mathematics in his home during World War I. He sought to re-establish Göttingen as the foremost mathematics research center in the world. He used his mathematical and management abilities to allow Clebsch's journal, Mathematische Annalen, to surpass Crelle's Journal in importance (Felix 2).

Klein made several mathematical discoveries throughout his career. During his time in Göttingen in 1871, he contributed to geometry by publishing two papers On the So-called Non-Euclidean Geometry. Through these he proved that it was possible to consider Euclidean geometry and Non-Euclidean geometry in special cases as a projective surface with a specific conic section adjoined. In addition, his Erlanger Program (1872) significantly influenced mathematical development. This contained the "synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations." It also gave a unified approach to geometry that is now the standard accepted view (Yaglom 129).

Klein is also remembered for his work with the surface that is now named after him, the Klein Bottle. A Klein Bottle is a one-sided closed surface. It must be constructed in the Euclidean 3-space. It needs a four dimensional space to live in. Like the Möbius band, it too is a non-orientable surface. It can be pictured as a cylinder looped back through itself to join with its other end.

Finally, Klein was elected chairman of the International Commission on Mathematical Instruction at the Rome International Mathematical Congress of 1908. This is of highest significance. Under his guidance the German branch of the Commission published many volumes on the teaching of mathematics at all levels in Germany. He was also elected a member of the Royal Society in 1885 and received the Copely medal in 1912. Felix Klein died on July 22, 1925 (Felix 4).

Another interesting topological concept is the Jordan Curve. Simply stated, the Jordan Curve Theorem states that any continuous simple closed curve in a plane separates the plane into two disjoint regions, the inside and the outside. Here simple means the curve does not self-intersect. Closed means the starting point is identical to the end point. For a long time this was considered so obvious that no one bothered to state it or prove it. Marie Ennemond Camille Jordan, a French mathematician, was the first to state this as a theorem in his famous textbook, Cours d'Analyze de l'Ecole Polytechnique in 1887. However, the proof he gave was incorrect. Veblen gave the first correct proof of the theorem in 1905. Schonflies and Brouwer later generalized his theorem. The Jordan-Schönflies Curve Theorem states that for any simple closed curve in the plane, there is a homoeomorphism of the plane that takes that curve into the standard circle. The Jordan-Brouwer Separation Theorem states that any imbedding of the n-1 dimensional sphere into n-dimensional Euclidean space separates the Euclidean space into two disjoint regions (Jordan 1).

This French mathematician not only concentrated his research in topology, but also analysis, and group theory, publishing Traité des substitutions et des equations algébriques, in 1870. Born in Lyon, he studied engineering at the Ecole Polytechnique in Paris, and joined their staff as a mathematician in 1873. He also gave lectures at the College de France. Influenced by Galois, he was interested in the theory of finite groups. He was motivated by the studies of crystal structures and classified groups of Euclidean motions. In topology, Jordan investigated symmetries in polyhedra and of course the Jordan Curve. Both Felix Klein and Sophus Lie were students of Jordan (Marie 1).

In my opinion, the field of topology definitely adds variety and deviation from the typical, more commonly thought of mathematics, dealing strictly with equations and numbers. The mysteries of surfaces discovered through topology prove to be fascinating and challenging to the minds of mathematicians and non-mathematicians alike. Prior to learning about this field, I never would have believed that a surface could be made from a surface with two sides and two edges, and yet itself have only one side and one edge. It is fun and mind boggling to examine the Jordan Curve, Möbius Band, Klein Bottle, and other surfaces, and to try to solve the problems and puzzles they create.

As much as I marveled at topological concepts, I originally viewed this field as interesting, but really possessing very little relevance and value to the mathematical world. However, while researching I discovered one very useful application of topology. The Möbius Band is indeed more than an intriguing novelty; it has practical value. One functional use of the Möbius Band has been in the design of driving belts such as fan belts and conveyor belts. Ordinarily, friction wears the belts out more quickly on the inside than on the outside. However, belts made with a half twist like a Möbius Strip, have only one side or surface. Thus, they wear uniformly and more slowly (West 579). I am sure that there are also many other practical uses of topology in our present society.

Overall, I have found the study of topology to be very intriguing. Marie Ennemond Camille Jordan, Felix Christian Klein, and August Ferdinand Möbius, among others, have greatly contributed to this field and to mathematics in general. Topology's ability to look at space in a completely spatial way, and nevertheless, to make definite statements about the nature of the object, characterizes this field as different from any other. Finally, I feel that the mathematicians of today are challenged to use their knowledge of topology to find more practical uses of the Möbius Band, Jordan Curve, and Klein Bottle.

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Peer reviewed by Michael Passarelli