August Ferdinand Möbius

German mathematician

• Born: November 17, 1790; Schulpforta, Germany
• Died: September 26, 1868; Leipzig, Germany

Nineteenth-century mathematician August Ferdinand Möbius is best known for his discovery of the Möbius strip, a unique, twisting surface with only one side. Although much of his work took place within the field of astronomy, he made invaluable contributions to a wide range of scientific disciplines.

Primary field: Mathematics

Specialties: Geometry; theoretical astronomy; number theory

Early Life

August Ferdinand Möbius was the only child of his father, a dancing teacher, and his mother, a descendant of the religious reformer Martin Luther. When Möbius was three years old, his father died and his mother relocated to Naumburg, Germany. Until the age of thirteen, Möbius studied at home, where he immersed himself in mathematics. At thirteen, he went to study formally at the University of Göttingen, where he began studying astronomy with mathematician Carl Friedrich Gauss.

After completing a thesis on the occultation of stars, a process in which stars’ visibility is diminished due to their being hidden by another object, Möbius was appointed to a professorship at the University of Leipzig in 1816. He taught both astronomy and mathematics, although his lack of skills as a lecturer prevented him from advancing in a university career as quickly as he would have liked. Nonetheless, he remained at the University of Leipzig for many years, serving additionally as the head observer of the Leipzig Observatory. He married Dorothea Rothe in 1820. The couple had three children. While publishing occasionally in astronomical journals, it was not until several years after settling into the university that Möbius began his best-known work in geometry.

Life’s Work

Möbius began actively publishing theoretical work in mathematics and astronomy early in his career, but it was his publications in Crelle’s Journal, one of the only academic journals at the time devoted to mathematics, that solidified his reputation. In 1827, Möbius published The Calculus of Centers of Gravity, a complex overview of analytical geometry and one of the earliest papers to introduce projective geometry as a serious field of research. The work included a number of equations for analysis of curves and segments in space, including the comparison of shapes, many of which were a part of what Möbius referred to as barycentric calculus. Möbius’s work also explored projective spaces; loosely defined, these are spaces like those in perspective paintings, in which two parallel lines do not run parallel to each other forever but rather meet at a point of infinity. While the field of projective geometry was still relatively new as a focus of geometric interest, Möbius’s work in homogenous coordinates and similar concepts helped paved the way for further research in the field.

This calculus work was just the first of many discoveries that would come to bear the mathematician’s name. In 1828, he published his work on the intersection of two tetrahedra, demonstrating that when seven of the planes overlap with the vertices of the corresponding planes of the other tetrahedra, the eighth plane must do so as well. These tetrahedra have since been known as Möbius tetrahedra, and the particular way they overlap in projective space is called a Möbius configuration.

In 1832, Möbius developed an important contribution to number theory. In line with his thinking on projective spaces, he conceived of what is now known as the Möbius function, a multiplicative set of numbers that follow very particular theoretical rules. Also similar to his geometrical work, the function exists in several examples outside of Möbius’s theories, particularly in the physics of supersymmetry.

The bulk of Möbius’s work was published in Crelle’s Journal, and his collected essays on mathematics would eventually be published as their own volume, with significant impact on geometry and topology. While his interests remained fixed on the mathematical analysis of points in space, he found ways to explore these ideas in related fields. In 1837, he showed how statistics could be used to understand geometry and the particularities of lines and planes in space through his Handbook on Statistics. He published The Elements of Celestial Mechanics in 1843. These works significantly spread Möbius’s reputation as a mathematician, and he received offers to work as a research professor at European universities. However, his commitment to the University of Leipzig and desire to raise his family in that area kept him from accepting any of these offers.

In 1858, Möbius introduced the Möbius strip. His discovery of the shape was simultaneous but independent of a similar discovery by German mathematician Johann Benedict Listing. The Möbius strip is often represented by taking a strip of paper, twisting it one-half turn, and the taping the two ends together. What results is a shape with only one side and one boundary component, so that you can run your finger along the entire edge of the strip without ever crossing an edge. While simple in concept, the ramifications of the Möbius strip are evidenced by its application in other fields. While Möbius did not fully explore all of these applications, his announcement of the shape and documentation of some of its properties served as a major breakthrough in the field of topology.

Apart from some very brief travel, Möbius remained with his home and his family in Leipzig until his death in 1868.

Impact

Shortly after Möbius’s death, his collected mathematical works began to be published in 1885. These works were his fundamental contribution to the development of modern geometry. Möbius was particularly adept at describing theoretical and actual shapes, and it is his exploration of those shapes that most affected the development of mathematics. The particularities of the Möbius strip and the Möbius configuration, for instance, have influenced thinking in topology, computer science, and astronomical mechanics. Graphs charting the behavior of a Möbius configuration in projective space have led to the discovery of multiple, closely related configurations. The Möbius strip is recognized in the behavior of charged particles, in algebraic graphs, and in music theory, allowing these concepts to be carefully unpacked based on Möbius’s topological and geometrical analysis. Everything from conveyor belts to superconductors have been built using the Möbius strip, which allows for unique engineering and mechanical benefits. A belt in the configuration of a Möbius strip, for instance, cycles through its entire surface area (rather than only one side, as with regular belts), effectively doubling its longevity. The strip has also become a focus of popular imagination, appearing regularly in visual art and literature, and drawing the curiosity of countless people to the world of topological mathematics. While these applications are far from Möbius’s original goals, they stand as a testament to the far-reaching significance of his work.

Bibliography

Adams, Colin, and Robert Franzosa. Introduction to Topology: Pure and Applied. New York: Brooks Cole, 2011. Print. Presents an introduction to the modern concepts of organic chemistry, including an explanation of the Brønsted-Lowry definition of acid-base reactions and some contemporary applications of his ideas. Illustrations, bibliography, index.

Fauvel, John, et al. Möbius and His Band: Mathematics and Astronomy in Nineteenth-Century Germany. New York: Oxford University Press, 1993. Print. Introduces modern concepts of topology, with many foundations based in Möbius’s ideas. Illustrations, bibliography, index.

Pickover, Clifford A. The Möbius Strip: Dr. August Möbius’s Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology. New York: Basic, 2007. Print. Overview of the way the Möbius strip has been used by artists and philosophers, with some history of Möbius himself. Illustrations, bibliography, index.