László Fejes Tóth (
Hungarian: Fejes Tóth László, pronounced[ˈfɛjɛʃˈtoːtˈlaːsloː] 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).[1] 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.
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.[3]
After university, he served as a soldier for two years, but received a medical exemption. In 1941 he joined the
University of Kolozsvár (
Cluj).[8] It was here that he became interested in packing problems.[9] 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,[3] 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.[8]
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.[9] By 1953, Fejes Tóth had written dozens of papers devoted to these types of fundamental issues.[8] 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.[3]
Fejes Tóth met his wife in university. She was a chemist. They were parents of three children, two sons—one 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.[3] 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.[8]
Fejes Tóth held the following positions over his career:[2]
Assistant instructor, University of Kolozsvár (Cluj) (1941–44)
Teacher, Árpád High School (1944–48)
Private Lecturer, Pázmány Péter University (1946–48)
According to
J. A. Todd,[11] a reviewer of Fejes Tóth's book Regular Figures,[12] 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
In work dedicated to Fejes Tóth, this compact binary
circle packing was shown to be the densest possible planar packing of discs with this size ratio.[13][14]
A semi-regular
tessellation with three prototiles: a triangle, a square and a hexagon.
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".[11]
Honors and recognition
Imre Bárány credited Fejes Tóth with several influential proofs in the field of discrete and convex geometry, pertaining to packings and
coverings by circles, to convex sets in a plane and to packings and coverings in higher dimensions, including the first correct proof of
Thue's theorem. He credits Fejes Tóth, along with
Paul Erdős, as having helped to "create the school of Hungarian discrete geometry."[6]
Fejes Tóth's monograph, Lagerungen in der Ebene, auf der Kugel und im Raum,[17][18] which was translated into Russian and Japanese, won him the Kossuth Prize in 1957 and the Hungarian Academy of Sciences membership in 1962.[2][8]
William Edge,[19] another reviewer of Regular Figures,[12] cites Fejes Tóth's earlier work, Lagerungen in der Ebene, auf der Kugel und im Raum,[17] 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.[20][21][22][23][24]
Gold Medal of the Hungarian Academy of Sciences (2002)
He received honorary degrees from the
University of Salzburg (1991) and the University of Veszprém (1997).
In 2008, a conference was convened in Fejes Tóth's memory in Budapest from June 30 – July 6;[4] 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".[25] 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.[26]
Partial bibliography
Fejes Tóth, László (1935). "Des séries exponentielles de Cauchy". C. R. Acad. Sci. (in French). 200: 1712–1714.
JFM62.1191.03.
Fejes Tóth, László (1938). "Über einige Extremumaufgaben bei Polyedern". Mat. Fiz. Lapok (in Hungarian and German). 45: 191–199.
JFM64.0732.02.
Fejes Tóth, László (1939). "Über das Schmiegungspolyeder". Mat. Fiz. Lápok (in Hungarian and German). 46: 141–145.
JFM65.0827.01.
Fejes Tóth, László (1938). "Sur les séries exponentielles de Cauchy". Mat. Fiz. Lapok (in Hungarian and French). 45: 115–132.
JFM64.0284.04.
Fejes Tóth, László (1939). "Über zwei Maximumaufgaben bei Polyedern". Tôhoku Math. J. (in German). 46: 79–83.
JFM65.0826.03.
Fejes Tóth, László (1939). "Über die Approximation konvexer Kurven durch Polygonfolgen". Compositio Mathematica (in German). 6. Groningen: 456–467.
JFM65.0822.03.
Fejes Tóth, László (1939). "Two inequalities concerning trigonometric polynomials". J. London Math. Soc. 14: 44–46.
doi:
10.1112/jlms/s1-14.1.44.
JFM65.0254.01.
Fejes Tóth, László (1940). "Über ein extremales Polyeder". Math.-naturw. Anz. Ungar. Akad. Wiss. (in Hungarian and German). 59: 476–479.
JFM66.0905.04.
Fejes Tóth, László (1940). "Eine Bemerkung zur Approximation durch n-Eckringe". Compositio Mathematica (in German). 7. Groningen: 474–476.
JFM66.0902.05.
Fejes Tóth, László (1940). "Sur un théorème concernant l'approximation des courbes par des suites de polygones". Ann. Scuola Norm. Sup., Pisa, Sci. Fis. Mat (in French). 2 (9): 143–145.
JFM66.0902.04.
Fejes Tóth, László (1942). "Die regulären Polyeder, als Lösungen von Extremalaufgaben". Math.-naturw. Anz. Ungar. Akad. Wiss. (in Hungarian and German). 61: 471–477.
JFM68.0341.02.
Fejes Tóth, László (1942). "Das gleichseitige Dreiecksgitter als Lösung von Extremalaufgaben". Mat. Fiz. Lapok. 49: 238–248.
JFM68.0340.04.
Fejes Tóth, László (1942). "Über die Fouriersche Reihe der Abkühlung". Math.-naturw. Anz. Ungar. Akad. Wiss (in Hungarian and German). 61: 478–495.
JFM68.0144.03.
Fejes Tóth, László (1950). "Some packing and covering theorems". Acta Sci. Math. 12A: 62–67.
Fejes Tóth, László (1953), Lagerungen in der Ebene, auf der Kugel und im Raum, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete (in German), vol. LXV, Berlin, New York:
Springer-Verlag, p. 238,
MR0057566
Fejes Tóth, László (1964), Regular Figures, Oxford: Pergamon Press, p. 339
Fejes Tóth, László (1965), Reguläre Figuren (in German), Budapest: Akadémiai Kiadó, p. 316
Fejes Tóth, László (1971), "Lencsék legsűrűbb elhelyezése a síkban", Matematikai Lapok, 22: 209–213
Fejes Tóth, László (1986), "Densest packing of translates of the union of two circles", Discrete and Computational Geometry, 1 (4): 307–314,
doi:10.1007/bf02187703,
Zbl0606.52004
References
^Fejes Tóth, László (1950). "Some packing and covering theorems". Acta Sci. Math. 12A: 62–67.
^
abcd
Kántor-Varga, T. (2010), "Fejes Tóth László", in Horváth, János (ed.), A Panorama of Hungarian Mathematics in the Twentieth Century, I, New York: Springer, pp. 573–574,
ISBN9783540307211
^
Katona, G. O. H. (2005), "Laszlo Fejes Toth – Obituary", Studia Scientiarum Mathematicarum Hungarica, 42 (2): 113
^
abBárány, Imre (2010), "Discrete and convex geometry", in Horváth, János (ed.), A Panorama of Hungarian Mathematics in the Twentieth Century, I, New York: Springer, pp. 431–441,
ISBN9783540307211
^O’Toole, P. I.; Hudson, T. S. (2011). "New High-Density Packings of Similarly Sized Binary Spheres". The Journal of Physical Chemistry C. 115 (39): 19037.
doi:
10.1021/jp206115p.
^
abFejes Tóth, László (1953), Lagerungen in der Ebene, auf der Kugel und im Raum, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete (in German), vol. LXV, Berlin, New York:
Springer-Verlag, p. 238,
MR0057566
János Pach:
A geometriai elrendezések diszkrét bája ("The Discrete Charm of Geometric Arrangements"), a memorial article in KöMaL (High School Mathematics and Physics Journal)