László Fejes Tóth

László Fejes Tóth (, ; 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. 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). 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, 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".

Fejes Tóth's monograph, Lagerungen in der Ebene, auf der Kugel und im Raum, which was translated into Russian and Japanese, won him the Kossuth Prize in 1957 and the Hungarian Academy of Sciences membership in 1962. another reviewer of Regular Figures, as the foundation of his second chapter in Regular Figures. He emphasized that, at the time of this work, the problem of the upper bound for the density of a packing of equal spheres was still unsolved.

The approach that Fejes Tóth suggested in that work, which translates as "packing [of objects] in a plane, on a sphere and in a space", provided Thomas Hales a basis for a proof of the Kepler conjecture in 1998. The Kepler conjecture, named after the 17th-century German mathematician and astronomer Johannes Kepler, says that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. Hales used a proof by exhaustion involving the checking of many individual cases, using complex computer calculations.

Fejes Tóth received the following prizes: it celebrated the term, "Intuitive Geometry", coined by Fejes Tóth to refer to the kind of geometry, which is accessible to the "man in the street". According to the conference organizers, the term encompasses combinatorial geometry, the theory of packing, covering and tiling, convexity, computational geometry, rigidity theory, the geometry of numbers, crystallography and classical differential geometry.

The University of Pannonia administers the László Fejes Tóth Prize (Hungarian: Fejes Tóth László-díj) to recognize "outstanding contributions and development in the field of mathematical sciences". In 2015, the year of Fejes Tóth's centennial birth anniversary, the prize was awarded to Károly Bezdek of the University of Calgary in a ceremony held on 19 June 2015 in Veszprém, Hungary.

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