|
László Fejes Tóth
László Fejes Tóth ( hu, Fejes Tóth László, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eötvös Loránd University
Eötvös Loránd University ( hu, Eötvös Loránd Tudományegyetem, ELTE) 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 Pázmá ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cluj
; hu, kincses város) , official_name=Cluj-Napoca , native_name= , image_skyline= , subdivision_type1 = Counties of Romania, County , subdivision_name1 = Cluj County , subdivision_type2 = Subdivisions of Romania, Status , subdivision_name2 = County seat , settlement_type = Municipiu, City , leader_title = Mayor , leader_name = Emil Boc , leader_party = National Liberal Party (Romania), PNL , leader_title1 = Deputy Mayor , leader_name1 = Dan Tarcea (PNL) , leader_title2 = Deputy Mayor , leader_name2 = Emese Oláh (Democratic Alliance of Hungarians in Romania, UDMR) , leader_title3 = City Manager , leader_name3 = Gheorghe Șurubaru (PNL) , established_title= Founded , established_date = 1213 (first official record as ''Clus'') , area_total_km2 = 179.5 , area_total_sq_mi = 69.3 , area_metro_km2 = 1537.5 , elevation_m = 340 , population_as_of = 2011 Romanian census, 2011 , population_total = 324,576 , population_foot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Madison, Wisconsin
Madison is the county seat of Dane County and the capital city of the U.S. state of Wisconsin. As of the 2020 census the population was 269,840, making it the second-largest city in Wisconsin by population, after Milwaukee, and the 80th-largest in the U.S. The city forms the core of the Madison Metropolitan Area which includes Dane County and neighboring Iowa, Green, and Columbia counties for a population of 680,796. Madison is named for American Founding Father and President James Madison. The city is located on the traditional land of the Ho-Chunk, and the Madison area is known as ''Dejope'', meaning "four lakes", or ''Taychopera'', meaning "land of the four lakes", in the Ho-Chunk language. Located on an isthmus and lands surrounding four lakes—Lake Mendota, Lake Monona, Lake Kegonsa and Lake Waubesa—the city is home to the University of Wisconsin–Madison, the Wisconsin State Capitol, the Overture Center for the Arts, and the Henry Vilas Zoo. Madison is ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Freiburg
Freiburg im Breisgau (; abbreviated as Freiburg i. Br. or Freiburg i. B.; Low Alemannic: ''Friburg im Brisgau''), commonly referred to as Freiburg, is an independent city in Baden-Württemberg, Germany. With a population of about 230,000 (as of 31 December 2018), Freiburg is the fourth-largest city in Baden-Württemberg after Stuttgart, Mannheim, and Karlsruhe. The population of the Freiburg metropolitan area was 656,753 in 2018. In the south-west of the country, it straddles the Dreisam river, at the foot of the Schlossberg. Historically, the city has acted as the hub of the Breisgau region on the western edge of the Black Forest in the Upper Rhine Plain. A famous old German university town, and archiepiscopal seat, Freiburg was incorporated in the early twelfth century and developed into a major commercial, intellectual, and ecclesiastical center of the upper Rhine region. The city is known for its medieval minster and Renaissance university, as well as for its high stand ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iron Curtain
The Iron Curtain was the political 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. The term symbolizes the efforts by the Soviet Union (USSR) to block itself and its satellite states from open contact with the West, its allies and neutral states. On the east side of the Iron Curtain were the countries that were connected to or influenced by the Soviet Union, while on the west side were the countries that were NATO members, or connected to or influenced by the United States; or nominally neutral. Separate international economic and military alliances were developed on each side of the Iron Curtain. It later became a term for the physical barrier of fences, walls, minefields, and watchtowers that divided the "east" and "west". The Berlin Wall was also part of this physical barrier. The nations to the east of the Iron Curtain were Poland, East Germany, Czechoslovakia, Hungary, Romania, Bulgaria, Albania, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") 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 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, a common crystal form in pyrite, has pyritohedral symmetry, while the 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. There ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and a right rhombohedron a 3-zonohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron. It has cubical or octahedral symmetry. The cube is the only convex polyhedron whose faces are all squares. Orthogonal projections The ''cube'' has four special orthogonal projections, centered, on a vertex, edges, face and normal to its vertex figure. The first and third correspond to the A2 and B2 Coxeter planes. Spherical tiling The cube can also be represented as a spherical tiling, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. A regular octahedron is the dual polyhedron of a cube. It is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations. It is also a triangular antiprism in any of four orientations. An octahedron is the three-dimensional case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan () metric. Regular octahedron Dimensions If the edge length of a regular octahedron is ''a'', the radius of a circumscribed sphere (one that touches the octahedron at all vertices) is :r_u = \frac a \approx 0.707 \cdot a and the radius of an inscribed sphere (tangent to each of the octahedron's faces) is :r_i = \frac a \approx 0.408\cdot a while the midradius, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of 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 nets. For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sphere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Platonic Solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each 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'', 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 numbers of knobs frequently differed from the numbers of vertices of the Platonic solids, there is no ball whose knobs match the 20 vertic ... [...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 mathematicians as ' ... [...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 |
_4029.jpg "Click for more on -> Freiburg")
.jpg "Click for more on -> Tetrahedron")
