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Perles Configuration
In geometry, the Perles configuration is a system of nine points and nine lines in the Euclidean plane for which every combinatorially equivalent realization has at least one irrational number as one of its coordinates. It can be constructed from the diagonals and symmetry lines of a regular pentagon, omitting one of the symmetry lines. In turn, it can be used to construct higher-dimensional convex polytopes that cannot be given rational coordinates, having the fewest vertices of any known example. All of the realizations of the Perles configuration in the projective plane are equivalent to each other under projective transformations. Construction One way of constructing the Perles configuration is to start with a regular pentagon and its five diagonals. These diagonals form the sides of a smaller inner pentagon nested inside the outer pentagon. Each vertex of the outer pentagon is situated opposite from a vertex of the inner pentagon. The nine points of the configuration consis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perles Configuration
In geometry, the Perles configuration is a system of nine points and nine lines in the Euclidean plane for which every combinatorially equivalent realization has at least one irrational number as one of its coordinates. It can be constructed from the diagonals and symmetry lines of a regular pentagon, omitting one of the symmetry lines. In turn, it can be used to construct higher-dimensional convex polytopes that cannot be given rational coordinates, having the fewest vertices of any known example. All of the realizations of the Perles configuration in the projective plane are equivalent to each other under projective transformations. Construction One way of constructing the Perles configuration is to start with a regular pentagon and its five diagonals. These diagonals form the sides of a smaller inner pentagon nested inside the outer pentagon. Each vertex of the outer pentagon is situated opposite from a vertex of the inner pentagon. The nine points of the configuration consis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernst Steinitz
Ernst Steinitz (13 June 1871 – 29 September 1928) was a German mathematician. Biography Steinitz was born in Laurahütte (Siemianowice Śląskie), Silesia, Germany (now in Poland), the son of Sigismund Steinitz, a Jewish coal merchant, and his wife Auguste Cohen; he had two brothers. He studied at the University of Breslau and the University of Berlin, receiving his Ph.D. from Breslau in 1894. Subsequently, he took positions at Charlottenburg (now the Technical University of Berlin), Breslau, and the University of Kiel, Germany, where he died in 1928. Steinitz married Martha Steinitz and had one son. Mathematical works Steinitz's 1894 thesis was on the subject of projective configurations; it contained the result that any abstract description of an incidence structure of three lines per point and three points per line could be realized as a configuration of straight lines in the Euclidean plane with the possible exception of one of the lines. His thesis also contains the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Mathematical Intelligencer
''The Mathematical Intelligencer'' is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common among academic journals. Volumes are released quarterly with a subset of open access articles. Springer also cross-publishes some of the articles in ''Scientific American''. Karen Parshall and Sergei Tabachnikov are currently the co-editors-in-chief. History The journal was started informally in 1971 by Walter Kaufman-Buehler, Alice Peters and Klaus Peters. "Intelligencer" was chosen by Kaufman-Buehler as a word that would appear slightly old-fashioned. An exploration of mathematically themed stamps, written by Robin Wilson, became one of its earliest columns. In 1978, the founders appointed Bruce Chandler and Harold "Ed" Edwards Jr. to serve jointly in the role of editor-in-chief. Prior to 1978, articles of the ''Intelligencer'' were not contained in regular volumes and were sent out ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Journal Of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United States, established in 1878 at the Johns Hopkins University by James Joseph Sylvester, an English-born mathematician who also served as the journal's editor-in-chief from its inception through early 1884. Initially W. E. Story was associate editor in charge; he was replaced by Thomas Craig in 1880. For volume 7 Simon Newcomb became chief editor with Craig managing until 1894. Then with volume 16 it was "Edited by Thomas Craig with the Co-operation of Simon Newcomb" until 1898. Other notable mathematicians who have served as editors or editorial associates of the journal include Frank Morley, Oscar Zariski, Lars Ahlfors, Hermann Weyl, Wei-Liang Chow, S. S. Chern, André Weil, Harish-Chandra, Jean Dieudonné, Henri Cartan, Stephen S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pappus Configuration
In geometry, the Pappus configuration is a configuration of nine points and nine lines in the Euclidean plane, with three points per line and three lines through each point. History and construction This configuration is named after Pappus of Alexandria. Pappus's hexagon theorem states that every two triples of collinear points ''ABC'' and ''abc'' (none of which lie on the intersection of the two lines) can be completed to form a Pappus configuration, by adding the six lines ''Ab'', ''aB'', ''Ac'', ''aC'', ''Bc'', and ''bC'', and their three intersection points , , and . These three points are the intersection points of the "opposite" sides of the hexagon ''AbCaBc''. According to Pappus' theorem, the resulting system of nine points and eight lines always has a ninth line containing the three intersection points ''X'', ''Y'', and ''Z'', called the ''Pappus line''. The Pappus configuration can also be derived from two triangles ''XcC'' and ''YbB'' that are in perspective with e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Configuration
In mathematics, specifically projective geometry, a configuration in the plane consists of a finite set of points, and a finite arrangement of lines, such that each point is incident to the same number of lines and each line is incident to the same number of points. Although certain specific configurations had been studied earlier (for instance by Thomas Kirkman in 1849), the formal study of configurations was first introduced by Theodor Reye in 1876, in the second edition of his book ''Geometrie der Lage'', in the context of a discussion of Desargues' theorem. Ernst Steinitz wrote his dissertation on the subject in 1894, and they were popularized by Hilbert and Cohn-Vossen's 1932 book ''Anschauliche Geometrie'', reprinted in English as . Configurations may be studied either as concrete sets of points and lines in a specific geometry, such as the Euclidean or projective planes (these are said to be ''realizable'' in that geometry), or as a type of abstract incidence geometry. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root Of Two
The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property. Geometrically, the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. The fraction (≈ 1.4142857) is sometimes used as a good rational approximation with a reasonably small denominator. Sequence in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 65 decimal places: : History The Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) gives an approximation of in four sexagesimal figures, , which is accurate to about six ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karl Georg Christian Von Staudt
Karl Georg Christian von Staudt (24 January 1798 – 1 June 1867) was a German mathematician who used synthetic geometry to provide a foundation for arithmetic. Life and influence Karl was born in the Free Imperial City of Rothenburg, which is now called Rothenburg ob der Tauber in Germany. From 1814 he studied in Gymnasium in Ausbach. He attended the University of Göttingen from 1818 to 1822 where he studied with Gauss who was director of the observatory. Staudt provided an ephemeris for the orbits of Mars and the asteroid Pallas. When in 1821 Comet Nicollet-Pons was observed, he provided the elements of its orbit. These accomplishments in astronomy earned him his doctorate from University of Erlangen in 1822. Staudt's professional career began as a secondary school instructor in Würzburg until 1827 and then Nuremberg until 1835. He married Jeanette Dreschler in 1832. They had a son Eduard and daughter Mathilda, but Jeanette died in 1848. The book ''Geometrie der Lage'' (1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Micha Perles
Micah (; ) is a given name. Micah is the name of several people in the Hebrew Bible ( Old Testament), and means "Who is like God?" The name is sometimes found with theophoric extensions. Suffix theophory in '' Yah'' and in ''Yahweh'' results in Michaiah or Michaihu (), meaning ''who is like Yahweh?''New Bible Dictionary, second edition. Tyndale House Publishers, Inc., Wheaton, IL, USA. Suffix theophory in '' El'' results in ''Michael'' (), meaning "who is like god". In German and Dutch, Micah is spelled and the ''ch'' in the name is pronounced either or ; the first is more common in female names, the latter in male names. The name is not as common as Michael or Michiel. Bible *Micah son of Mephibosheth son of Jonathan son of Saul, the first king of Israel () *Micah (prophet), eponymous prophet of the Book of Micah in the Old Testament *Micaiah, a prophet and the son of Imlah, who gave a negative prophecy to Ahab on his request Notable people with the given name "Micah" inclu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steinitz's Theorem
In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs. This result provides a classification theorem for the three-dimensional convex polyhedra, something that is not known in higher dimensions. It provides a complete and purely combinatorial description of the graphs of these polyhedra, allowing other results on them, such as Eberhard's theorem on the realization of polyhedra with given types of faces, to be proven more easily, without reference to the geometry of these shapes. Additionally, it has been applied in graph drawing, as a way to construct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perles Polytope
Perles may refer to * Perles, Aisne, a commune in the Aisne department in Picardie in northern France * Perles-et-Castelet, a commune in the Ariège department in southwestern France *Perles, the French name for Pieterlen, Switzerland * Alfred Perlès (1897–1990), Austrian-British writer *George Perles (1934–2020), American football coach *Joseph Perles (1835–1894), Hungarian rabbi *Micha Perles, Israeli mathematician **Perles configuration In geometry, the Perles configuration is a system of nine points and nine lines in the Euclidean plane for which every combinatorially equivalent realization has at least one irrational number as one of its coordinates. It can be constructed from ... * Tessalon Perles See also * Perle (other) * Perls {{Disambiguation, geo, surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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