László Fejes Tóth (Hungarian: Fejes Tóth László, pronounced [ˈfɛjɛʃ ˈtoːt ˈlaːsloː]; 12 March 1915 – 17 March 2005) was a Hungarian mathematician who specialized in geometry. He proved that a lattice pattern is the most efficient way to pack centrally symmetric convex sets on the Euclidean plane (a generalization of Thue's theorem, a 2-dimensional analog of the Kepler conjecture). He also investigated the sphere packing problem. He was the first to show, in 1953, that proof of the Kepler conjecture can be reduced to a finite case analysis and, later, that the problem might be solved using a computer. He was a member of the Hungarian Academy of Sciences (from 1962) and a director of the Alfréd Rényi Institute of Mathematics (1970-1983). He received both the Kossuth Prize (1957) and State Award (1973). Together with H.S.M. Coxeter and Paul Erdős, he laid the foundations of discrete geometry.
Early life and career As described in a 1999 interview with István Hargittai, Fejes Tóth's father was a railway worker, who advanced in his career within the railway organization ultimately to earn a doctorate in law. Fejes Tóth's mother taught Hungarian and German literature in a high school. The family moved to Budapest, when Fejes Tóth was five; there he attended elementary school and high school—the Széchenyi István Reálgimnázium—where his interest in mathematics began. Fejes Tóth attended Pázmány Péter University, now the Eötvös Loránd University. As a freshman, he developed a generalized solution regarding Cauchy exponential series, which he published in the proceedings of the French Academy of Sciences—1935. He then received his doctorate at Pázmány Péter University, under the direction of Lipót Fejér. After university, he served as a soldier for two years, but received a medical exemption. In 1941 he joined the University of Kolozsvár (Cluj). It was here that he became interested in packing problems. In 1944, he returned to Budapest to teach mathematics at Árpád High School. Between 1946 and 1949 he lectured at Pázmány Péter University and starting in 1949 became a professor at the University of Veszprém (now University of Pannonia) for 15 years, where he was the primary developer of the "geometric patterns" theory "of the plane, the sphere and the surface space" and where he "had studied non grid-like structures and quasicrystals" which later became an independent discipline, as reported by János Pach. The editors of a book dedicated to Fejes Tóth described some highlights of his early work; e.g. having shown that the maximum density of a packing of repeated symmetric convex bodies occurs with a lattice pattern of packing. He also showed that, of all convex polytopes of given surface area that are equivalent to a given Platonic solid (e.g. a tetrahedron or an octahedron), a regular polytope always has the largest possible volume. He developed a technique that proved Steiner's conjecture for the cube and for the dodecahedron. By 1953, Fejes Tóth had written dozens of papers devoted to these types of fundamental issues. His distinguished academic career allowed him to travel abroad beyond the Iron Curtain to attend international conferences and teach at various universities, including those at Freiburg; Madison, Wisconsin; Ohio; and Salzburg.
Fejes Tóth met his wife in university. She was a chemist. They were parents of three children, two sons—one (Gábor) a professor of mathematics at the Alfréd Rényi Institute of Mathematics, the other a professor of physiology at Dartmouth College—and one daughter, a psychologist. He enjoyed sports, being skilled at table tennis, tennis, and gymnastics. A family photograph shows him swinging by his arms over the top of a high bar when he was around fifty. Fejes Tóth held the following positions over his career:
Assistant instructor, University of Kolozsvár (Cluj) (1941–44) Teacher, Árpád High School (1944–48) Private Lecturer, Pázmány Péter University (1946–48) Professor, University of Veszprém (1949–64) Researcher, then director (in 1970), Mathematical Research Institute (Alfréd Rényi Institute of Mathematics) (1965–83) In addition to his positions in residence, he was a corresponding member of the Saxonian Academy of Sciences and Humanities, Akademie der Wissenschaften der DDR, and of the Braunschweigische Wissenschaftlische Gesellschaft.
Work on regular figures According to J. A. Todd, a reviewer of Fejes Tóth's book Regular Figures, Fejes Tóth divided the topic into two sections. One, entitled "Systematology of the Regular Figures", develops a theory of "regular and Archimedean polyhedra and of regular polytopes". Todd explains that the treatment includes:
Plane Ornaments, including two-dimensional crystallographic groups Spherical arrangements, including an enumeration of the 32 crystal classes Hyperbolic tessellations, those discrete groups generated by two operations whose product is involutary Polyhedra, including regular solids and convex Archimedean solids Regular polytopes
The other section, entitled "Genetics of the Regular Figures", covers a number of special problems, according to Todd. These problems include "packings and coverings of circles in a plane, and ... with tessellations on a sphere" and also problems "in the hyperbolic plane, and in Euclidean space of three or more dimensions." At the time, Todd opined that those problems were "a subject in which there is still much scope for research, and one which calls for considerable ingenuity in approaching its problems".
Honors and recognition
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