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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. Early life and career As described in a 1999 interview witIstván Hargittai Fejes Tóth's father was a railway worker, who advanced ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eötvös Loránd University
Eötvös Loránd University (, ELTE, also known as ''University of Budapest'') is a Hungarian public research university based in Budapest. Founded in 1635, ELTE is one of the largest and most prestigious public higher education institutions in Hungary. The 28,000 students at ELTE are organized into nine faculties, and into research institutes located throughout Budapest and on the scenic banks of the Danube. ELTE is affiliated with 5 Nobel laureates, as well as winners of the Wolf Prize, Fulkerson Prize and Abel Prize, the latest of which was Abel Prize winner László Lovász in 2021. The predecessor of Eötvös Loránd University was founded in 1635 by Cardinal Péter Pázmány in Nagyszombat, Kingdom of Hungary (today Trnava, Slovakia) as a Catholic university for teaching theology and philosophy. In 1770, the university was transferred to Buda. It was named Royal University of Pest until 1873, then University of Budapest until 1921, when it was renamed Royal Hungarian Pá ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cluj
Cluj-Napoca ( ; ), or simply Cluj ( , ), is a city in northwestern Romania. It is the second-most populous city in the country and the seat of Cluj County. Geographically, it is roughly equidistant from Bucharest (), Budapest () and Belgrade (). Located in the Someșul Mic river valley, the city is considered the unofficial capital of the Historical regions of Romania, historical province of Transylvania. For some decades prior to the Austro-Hungarian Compromise of 1867, it was the official capital of the Grand Principality of Transylvania. , 286,598 inhabitants live in the city. The Cluj-Napoca metropolitan area had a population of 411,379 people, while the population of the peri-urbanisation, peri-urban area is approximately 420,000. According to a 2007 estimate, the city hosted an average population of over 20,000 students and other non-residents each year from 2004 to 2007. The city spreads out from St. Michael's Church, Cluj-Napoca, St. Michael's Church in Unirii Square, C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Madison, Wisconsin
Madison is the List of capitals in the United States, capital city of the U.S. state of Wisconsin. It is the List of municipalities in Wisconsin by population, second-most populous city in the state, with a population of 269,840 at the 2020 United States census, 2020 census. The Madison metropolitan area had 680,796 residents. Centrally located on an isthmus between Lakes Lake Mendota, Mendota and Lake Monona, Monona, the vicinity also encompass Lakes Lake Wingra, Wingra, Lake Kegonsa, Kegonsa and Lake Waubesa, Waubesa. Madison was founded in 1836 and is named after American Founding Fathers of the United States, Founding Father and President James Madison. It is the county seat of Dane County. As the state capital, Madison is home to government chambers including the Wisconsin State Capitol building. It is also home to the University of Wisconsin–Madison, the flagship campus of the University of Wisconsin System. Major companies in the area include American Family Insurance, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Freiburg
Freiburg im Breisgau or simply Freiburg is the List of cities in Baden-Württemberg by population, fourth-largest city of the German state of Baden-Württemberg after Stuttgart, Mannheim and Karlsruhe. Its built-up area has a population of about 355,000 (2021), while the greater Freiburg metropolitan area ("Einzugsgebiet") has about 660,000 (2018). Freiburg is located at the southwestern foothills of the Black Forest, on the Dreisam River, a tributary of the Elz (Rhine), Elz. It is Germany's southwestern- and southernmost city with a population exceeding 100,000. It lies in the Breisgau, one of Germany's warmest regions, in the south of the Upper Rhine Plain. Its city limits reach from the Schauinsland summit () in the Black Forest to east of the French border, while Switzerland is to the south. The city is situated in the major Baden (wine region), wine-growing region of Baden and, together with Offenburg, serves as a tourist entry-point to the scenic Black Forest. According ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iron Curtain
The Iron Curtain was the political and physical boundary dividing Europe into two separate areas from the end of World War II in 1945 until the end of the Cold War in 1991. On the east side of the Iron Curtain were countries connected to the Soviet Union, and on the west side those that were NATO members. Economic and military alliances developed on each side of the Iron Curtain, and it became a term for the physical barriers of razor wire, Fence, fences, Fortified wall, walls, minefields, and Watchtower, watchtowers built along it. The nations to the east of the Iron Curtain were People's Republic of Poland, Poland, East Germany, Socialist Republic of Czechoslovakia, Czechoslovakia, Hungarian People's Republic, Hungary, Socialist Republic of Romania, Romania, People's Republic of Bulgaria, Bulgaria, People's Republic of Albania, Albania, and the USSR; however, Reunification of Germany, East Germany, Breakup of Czechoslovakia, Czechoslovakia, and the Dissolution of the USSR, USS ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dodecahedron
In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Poinsot polyhedron, regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120. Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular: The #Pyritohedron, pyritohedron, a common crystal form in pyrite, has pyritohedral symmetry, while the #Tetartoid, tetartoid has tetrahedral symmetry. The rhombic dodecahedron can be seen as a limiting case of the pyritohedron, and it has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, are space-filling polyhedra, space-filling. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It is a type of parallelepiped, with pairs of parallel opposite faces, and more specifically a rhombohedron, with congruent edges, and a rectangular cuboid, with right angles between pairs of intersecting faces and pairs of intersecting edges. It is an example of many classes of polyhedra: Platonic solid, regular polyhedron, parallelohedron, zonohedron, and plesiohedron. The dual polyhedron of a cube is the regular octahedron. The cube can be represented in many ways, one of which is the graph known as the cubical graph. It can be constructed by using the Cartesian product of graphs. The cube is the three-dimensional hypercube, a family of polytopes also including the two-dimensional square and four-dimensional tesseract. A cube with 1, unit s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedron
In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of irregular octahedra also exist, including both convex set, convex and non-convex shapes. Combinatorially equivalent to the regular octahedron The following polyhedra are combinatorially equivalent to the regular octahedron. They all have six vertices, eight triangular faces, and twelve edges that correspond one-for-one with the features of it: * Triangular antiprisms: Two faces are equilateral, lie on parallel planes, and have a common axis of symmetry. The other six triangles are isosceles. The regular octahedron is a special case in which the six lateral triangles are also equilateral. * Tetragonal bipyramids, in which at least one of the equatorial quadrilaterals lies on a plane. The regular octahedron is a special case in which all thr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedron
In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tetrahedron is the simplest of all the ordinary convex polytope, convex polyhedra. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean geometry, Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid (geometry), pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron, the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such net (polyhedron), nets. For any tetrahedron there exists a sphere (called th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Platonic Solid
In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (identical in shape and size) regular polygons (all angles congruent and all edge (geometry), edges congruent), and the same number of faces meet at each Vertex (geometry), vertex. There are only five such polyhedra: Geometers have studied the Platonic solids for thousands of years. They are named for the ancient Greek philosopher Plato, who hypothesized in one of his dialogues, the ''Timaeus (dialogue), Timaeus'', that the classical elements were made of these regular solids. History The Platonic solids have been known since antiquity. It has been suggested that certain carved stone balls created by the late Neolithic people of Scotland represent these shapes; however, these balls have rounded knobs rather than being polyhedral, the num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polytope
In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -dimensional polytope or -polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a -polytope consist of -polytopes that may have -polytopes in common. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes. Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such a figure a polyschem. The German term ''Polytop'' was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
János Pach
János Pach (born May 3, 1954) is a mathematician and computer scientist working in the fields of combinatorics and discrete and computational geometry. Biography Pach was born and grew up in Hungary. He comes from a noted academic family: his father, (1919–2001) was a well-known historian, and his mother Klára (née Sós, 1925–2020) was a university mathematics teacher; his maternal aunt Vera T. Sós and her husband Pál Turán are two of the best-known Hungarian mathematicians. Pach received his Candidate degree from the Hungarian Academy of Sciences, in 1983, where his advisor was Miklós Simonovits. Since 1977, he has been affiliated with the Alfréd Rényi Institute of Mathematics of the Hungarian Academy of Sciences.Research Fellows Rényi Institute He was Research Professor at the |
