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Kobon Triangle Problem
The Kobon triangle problem is an unsolved problem in combinatorial geometry first stated by Kobon Fujimura (1903-1983). The problem asks for the largest number ''N''(''k'') of nonoverlapping triangles whose sides lie on an arrangement of ''k'' lines. Variations of the problem consider the projective plane rather than the Euclidean plane, and require that the triangles not be crossed by any other lines of the arrangement.. Known upper bounds Saburo Tamura proved that the number of nonoverlapping triangles realizable by k lines is at most \lfloor k(k-2)/3\rfloor. G. Clément and J. Bader proved more strongly that this bound cannot be achieved when k is congruent to 0 or 2 (mod 6). The maximum number of triangles is therefore at most one less in these cases. The same bounds can be equivalently stated, without use of the floor function In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kobon Triangles
Kobon (pronounced , or ) is a language of Papua New Guinea. It has somewhere around 90–120 verbs. Kobon has a pandanus language, spoken when harvesting karuka. Geographic distribution Kobon is spoken in Madang Province and Western Highlands Province, north of Mount Hagen. Phonology Vowels Monophthongal vowels are , diphthongs are . and may be and word-initially. () is written and () is written . Only and the diphthongs occur word-initially, apart from the quotative particle, which is variably /a~e~o~ö/. occur syllable-initially within a word. All vowels (including the diphthongs) occur syllable-medially (in CVC syllables), syllable-finally and at the ends of words. Many vowel sequences occur, including some with identical vowels. Consonants Kobon distinguishes an alveolar lateral , a palatal lateral , a subapical retroflex lateral flap ( ), and a fricative trill , though the frication on the latter is variable. Voiced obstruents may be prenasalized aft ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Geometry
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object. Discrete geometry has a large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology. History Although polyhedra and tessellations had been studied for many years by people such as Kepler and Cauchy, modern discrete geometry has its origins in the late 19th century. Early topics studied ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kobon Fujimura
The Kobon triangle problem is an unsolved problem in combinatorial geometry first stated by Kobon Fujimura (1903-1983). The problem asks for the largest number ''N''(''k'') of nonoverlapping triangles whose sides lie on an arrangement of ''k'' lines. Variations of the problem consider the projective plane rather than the Euclidean plane, and require that the triangles not be crossed by any other lines of the arrangement.. Known upper bounds Saburo Tamura proved that the number of nonoverlapping triangles realizable by k lines is at most \lfloor k(k-2)/3\rfloor. G. Clément and J. Bader proved more strongly that this bound cannot be achieved when k is congruent to 0 or 2 (mod 6). The maximum number of triangles is therefore at most one less in these cases. The same bounds can be equivalently stated, without use of the floor function In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arrangement Of Lines
In music, an arrangement is a musical adaptation of an existing composition. Differences from the original composition may include reharmonization, melodic paraphrasing, orchestration, or formal development. Arranging differs from orchestration in that the latter process is limited to the assignment of notes to instruments for performance by an orchestra, concert band, or other musical ensemble. Arranging "involves adding compositional techniques, such as new thematic material for introductions, transitions, or modulations, and endings. Arranging is the art of giving an existing melody musical variety".(Corozine 2002, p. 3) In jazz, a memorized (unwritten) arrangement of a new or pre-existing composition is known as a ''head arrangement''. Classical music Arrangement and transcriptions of classical and serious music go back to the early history of this genre. Eighteenth century J.S. Bach frequently made arrangements of his own and other composers' pieces. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus ''any'' two distinct lines in a projective plane intersect at exactly one point. Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by , RP2, or P2(R), among other notations. There are many other projective planes, both infinite, such as the complex projective plane, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete And Computational Geometry
'' Discrete & Computational Geometry'' is a peer-reviewed mathematics journal published quarterly by Springer. Founded in 1986 by Jacob E. Goodman and Richard M. Pollack, the journal publishes articles on discrete geometry and computational geometry. Abstracting and indexing The journal is indexed in: * ''Mathematical Reviews'' * ''Zentralblatt MATH'' * ''Science Citation Index'' * ''Current Contents''/Engineering, Computing and Technology Notable articles The articles by Gil Kalai with a proof of a subexponential upper bound on the diameter of a polyhedron and by Samuel Ferguson on the Kepler conjecture, both published in Discrete & Computational geometry, earned their author the Fulkerson Prize The Fulkerson Prize for outstanding papers in the area of discrete mathematics is sponsored jointly by the Mathematical Optimization Society (MOS) and the American Mathematical Society (AMS). Up to three awards of $1,500 each are presented at e .... References External link ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floor Function
In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or equal to , denoted or . For example, , , , and . Historically, the floor of has been–and still is–called the integral part or integer part of , often denoted (as well as a variety of other notations). Some authors may define the integral part as if is nonnegative, and otherwise: for example, and . The operation of truncation generalizes this to a specified number of digits: truncation to zero significant digits is the same as the integer part. For an integer, . Notation The ''integral part'' or ''integer part'' of a number ( in the original) was first defined in 1798 by Adrien-Marie Legendre in his proof of the Legendre's formula. Carl Friedrich Gauss introduced the square bracket notation in hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
OEIS
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the OEIS Foundation in 2009. Sloane is chairman of the OEIS Foundation. OEIS records information on integer sequences of interest to both professional and amateur mathematicians, and is widely cited. , it contains over 350,000 sequences, making it the largest database of its kind. Each entry contains the leading terms of the sequence, keywords, mathematical motivations, literature links, and more, including the option to generate a graph or play a musical representation of the sequence. The database is searchable by keyword, by subsequence, or by any of 16 fields. History Neil Sloane started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics. The database was at first stored on punched cards. H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roberts's Triangle Theorem
Roberts's triangle theorem, a result in discrete geometry, states that every simple arrangement of lines, arrangement of n lines has at least n-2 triangular faces. Thus, three lines form a triangle, four lines form at least two triangles, five lines form at least three triangles, etc. It is named after Samuel Roberts (mathematician), Samuel Roberts, a British mathematician who published it in 1889. Statement and example The theorem states that every simple arrangement of lines, arrangement of n lines in the Euclidean plane has at least n-2 triangular faces. Here, an arrangement is simple when it has no two parallel lines and no three lines through the same point. A face is one of the polygons formed by the arrangement, but not crossed by any of its lines. Faces may be bounded or infinite, but only the bounded faces with exactly three sides count as triangles for the purposes of the theorem. One way to form an arrangement of n lines with exactly n-2 triangular faces is to choose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Geometry
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object. Discrete geometry has a large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology. History Although polyhedra and tessellations had been studied for many years by people such as Kepler and Cauchy, modern discrete geometry has its origins in the late 19th century. Early topics studie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unsolved Problems In Geometry '', an American true crime television program that debuted in 1987
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Unsolved may refer to: * ''Unsolved'' (album), a 2000 album by the American band Karate * ''Unsolved'' (UK TV programme), a 2004–2006 British crime documentary television programme that aired on STV in Scotland * ''Unsolved'' (South Korean TV series), a 2010 South Korean television series * ''Unsolved'' (U.S. TV series), a 2018 American television series *'' Unsolved: The Boy Who Disappeared'', a 2016 online series by BBC Three *''The Unsolved'', a 1997 Japanese video game *''BuzzFeed Unsolved'', a show by BuzzFeed discussing unsolved crimes and haunted places See also *Solved (other) *''Unsolved Mysteries ''Unsolved Mysteries'' is an American mystery documentary television show, created by John Cosgrove and Terry Dunn Meurer. Documenting cold cases and paranormal phenomena, it began as a series of seven specials, presented by Raymond Burr, Ka ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recreational Mathematics
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited to being an endeavor for amateurs, many topics in this field require no knowledge of advanced mathematics. Recreational mathematics involves mathematical puzzles and games, often appealing to children and untrained adults, inspiring their further study of the subject. The Mathematical Association of America (MAA) includes recreational mathematics as one of its seventeen Special Interest Groups, commenting: Mathematical competitions (such as those sponsored by mathematical associations) are also categorized under recreational mathematics. Topics Some of the more well-known topics in recreational mathematics are Rubik's Cubes, magic squares, fractals, logic puzzles and mathematical chess problems, but this area of mathematics incl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
