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Sylvester–Gallai Theorem
The Sylvester–Gallai theorem in geometry states that every finite set of points in the Euclidean plane has a line that passes through exactly two of the points or a line that passes through all of them. It is named after James Joseph Sylvester, who posed it as a problem in 1893, and Tibor Gallai, who published one of the first proofs of this theorem in 1944. A line that contains exactly two of a set of points is known as an ''ordinary line''. Another way of stating the theorem is that every finite set of points that is not collinear has an ordinary line. According to a strengthening of the theorem, every finite point set (not all on one line) has at least a linear number of ordinary lines. An algorithm can find an ordinary line in a set of n points in time O(n\log n). History The Sylvester–Gallai theorem was posed as a problem by . suggests that Sylvester may have been motivated by a related phenomenon in algebraic geometry, in which the inflection points of a cubic curve i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Lines
Ordinary or The Ordinary often refer to: Music * Ordinary (EP), ''Ordinary'' (EP) (2015), by South Korean group Beast * Ordinary (album), ''Ordinary'' (album) (2011), by Every Little Thing * Ordinary (Alex Warren song), "Ordinary" (Alex Warren song) (2025) * Ordinary (Two Door Cinema Club song), "Ordinary" (Two Door Cinema Club song) (2016) * Ordinary (Wayne Brady song), "Ordinary" (Wayne Brady song) (2008) * "Ordinary", song by Train from ''Alive at Last'' (2004) Religion * Ordinary (Catholic Church), a supervisor, typically a bishop, in charge of a territory comparable to a diocese, or a major superior of a religious institute * Ordinary (church officer), an officer of a church or civic authority who by reason of office has ordinary power to execute laws * Ordinary (liturgy), a set of texts in Roman Catholic and other Western Christian liturgies that are generally invariable * Ordinary Time, the parts of the Roman Catholic liturgical year that are outside Advent, Christmastide, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arrangement Of Lines
In geometry, an arrangement of lines is the subdivision of the Euclidean plane formed by a finite set of lines. An arrangement consists of bounded and unbounded convex polygons, the ''cells'' of the arrangement, line segments and rays, the ''edges'' of the arrangement, and points where two or more lines cross, the ''vertices'' of the arrangement. When considered in the projective plane rather than in the Euclidean plane, every two lines cross, and an arrangement is the projective dual to a finite set of points. Arrangements of lines have also been considered in the hyperbolic plane, and generalized to ''pseudolines'', curves that have similar topological properties to lines. The initial study of arrangements has been attributed to an 1826 paper by Jakob Steiner. An arrangement is said to be ''simple'' when at most two lines cross at each vertex, and ''simplicial'' when all cells are triangles (including the unbounded cells, as subsets of the projective plane). There are three kno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Similar Triangle
In Euclidean geometry, two objects are similar if they have the same shape, or if one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (geometry), scaling (enlarging or reducing), possibly with additional translation (geometry), translation, rotation (mathematics), rotation and reflection (mathematics), reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruence (geometry), congruent to the result of a particular uniform scaling of the other. For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, and isosceles triangles are not all similar to each other. This is because two ellipse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof By Contradiction
In logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally valid. More broadly, proof by contradiction is any form of argument that establishes a statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved. In this general sense, proof by contradiction is also known as indirect proof, proof by assuming the opposite, and ''reductio ad impossibile''. A mathematical proof employing proof by contradiction usually proceeds as follows: #The proposition to be proved is ''P''. #We assume ''P'' to be false, i.e., we assume ''¬P''. #It is then shown that ''¬P'' implies falsehood. This is typically accomplished by deriving two mutually ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sylvester Gallai Kelly Proof
Sylvester or Silvester is a name derived from the Latin adjective ''silvestris'' meaning "wooded" or "wild", which derives from the noun ''silva'' meaning "woodland". Classical Latin spells this with ''i''. In Classical Latin, ''y'' represented a separate sound distinct from ''i'', not a native Latin sound but one used in transcriptions of foreign words. After the Classical period ''y'' was pronounced as ''i''. Spellings with ''Sylv-'' in place of ''Silv-'' date from after the Classical period. Given name * Sylvester of Marsico (c. 1100–1162), Count of Marsico in the Kingdom of Sicily * Silvester Ashioya (born 1948), Kenyan hockey player * Silvester Bolam (1905–1953), British newspaper editor * Silvester Brito (1937–2018), American poet and academic * Sylvester Croom (born 1954), American football coach and former player *Silvester Diggles (1817–1880), Australian musician and ornithologist * Sylvester A. Dineen (1898–1950), American schoolteacher and politician *Silves ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Geometry
Ordered geometry is a form of geometry featuring the concept of intermediacy (or "betweenness") but, like projective geometry, omitting the basic notion of measurement. Ordered geometry is a fundamental geometry forming a common framework for affine, Euclidean, absolute, and hyperbolic geometry (but not for projective geometry). History Moritz Pasch first defined a geometry without reference to measurement in 1882. His axioms were improved upon by Peano (1889), Hilbert (1899), and Veblen (1904). Euclid anticipated Pasch's approach in definition 4 of ''The Elements'': "a straight line is a line which lies evenly with the points on itself". Primitive concepts The only primitive notions in ordered geometry are points ''A'', ''B'', ''C'', ... and the ternary relation of intermediacy 'ABC''which can be read as "''B'' is between ''A'' and ''C''". Definitions The ''segment'' ''AB'' is the set of points ''P'' such that 'APB'' The ''interval'' ''AB'' is the segment ''AB'' and it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated at the University of Cambridge, with student visits to Princeton University. He worked for 60 years at the University of Toronto in Canada, from 1936 until his retirement in 1996, becoming a full professor there in 1948. His many honours included membership in the Royal Society of Canada, the Royal Society, and the Order of Canada. He was an author of 12 books, including '' The Fifty-Nine Icosahedra'' (1938) and '' Regular Polytopes'' (1947). Many concepts in geometry and group theory are named after him, including the Coxeter graph, Coxeter groups, Coxeter's loxodromic sequence of tangent circles, Coxeter–Dynkin diagrams, and the Todd–Coxeter algorithm. Biography Coxeter was born in Kensington, England, to Harold Samuel Coxete ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leroy Milton Kelly
Leroy Milton Kelly (May 8, 1914 – February 21, 2002 for Leroy M Kelly: Holt, Michigan.) was an American whose research primarily concerned . In 1986 he settled a conjecture of by proving that n points in complex 3-space, not all lying on a plane, determine an [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Polyhedral Formula
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by \chi ( Greek lower-case letter chi). The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, homological algebra. Polyhedra The Euler characteristic was classically defined for the surfaces of polyhedra, according to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Dodecahedron
In geometry, the elongated dodecahedron, extended rhombic dodecahedron, rhombo-hexagonal dodecahedron or hexarhombic dodecahedron is a convex dodecahedron with 8 Rhombus, rhombic and 4 hexagonal faces. The hexagons can be made equilateral, or regular depending on the shape of the rhombi. It can be seen as constructed from a rhombic dodecahedron elongation (geometry), elongated by a square prism. Parallelohedron Along with the rhombic dodecahedron, it is a space-filling polyhedron, one of the five types of parallelohedron identified by Evgraf Fedorov that Honeycomb (geometry), tile space face-to-face by translations. It has 5 sets of parallel edges, called zones or belts. : Tessellation * It can tesselate all space by translations. * It is the Wigner–Seitz cell for certain Bravais lattice, body-centered tetragonal lattices. This is related to the rhombic dodecahedral honeycomb with an elongation of zero. Projected normal to the elongation direction, the honeycomb looks li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Segment
In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special case of an ''arc (geometry), arc'', with zero curvature. The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using an overline (vinculum (symbol), vinculum) above the symbols for the two endpoints, such as in . Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (geometry), edge (of that polygon or polyhedron) if they are adjacent vertices, or a diagonal. Wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
