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Fagnano's Problem
In geometry, Fagnano's problem is an optimization problem that was first stated by Giovanni Fagnano in 1775: The solution is the orthic triangle, with vertices at the base points of the altitudes of the given triangle. Solution The orthic triangle, with vertices at the base points of the altitudes of the given triangle, has the smallest perimeter of all triangles inscribed into an acute triangle, hence it is the solution of Fagnano's problem. Fagnano's original proof used calculus methods and an intermediate result given by his father Giulio Carlo de' Toschi di Fagnano. Later however several geometric proofs were discovered as well, amongst others by Hermann Schwarz and Lipót Fejér. These proofs use the geometrical properties of reflections to determine some minimal path representing the perimeter. Physical principles A solution from physics is found by imagining putting a rubber band that follows Hooke's Law around the three sides of a triangular frame ABC, such that it cou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fagnano Problem
In geometry, Fagnano's problem is an optimization problem that was first stated by Giovanni Fagnano in 1775: The solution is the orthic triangle, with vertices at the base points of the altitudes of the given triangle. Solution The orthic triangle, with vertices at the base points of the altitudes of the given triangle, has the smallest perimeter of all triangles inscribed into an acute triangle, hence it is the solution of Fagnano's problem. Fagnano's original proof used calculus methods and an intermediate result given by his father Giulio Carlo de' Toschi di Fagnano. Later however several geometric proofs were discovered as well, amongst others by Hermann Schwarz and Lipót Fejér. These proofs use the geometrical properties of reflections to determine some minimal path representing the perimeter. Physical principles A solution from physics is found by imagining putting a rubber band that follows Hooke's Law around the three sides of a triangular frame ABC, such that it cou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lipót Fejér
Lipót Fejér (or Leopold Fejér, ; 9 February 1880 – 15 October 1959) was a Hungarian mathematician of Jewish heritage. Fejér was born Leopold Weisz, and changed to the Hungarian name Fejér around 1900. Biography Fejér studied mathematics and physics at the University of Budapest and at the University of Berlin, where he was taught by Hermann Schwarz. In 1902 he earned his doctorate from University of Budapest (today Eötvös Loránd University). From 1902 to 1905 Fejér taught there and from 1905 until 1911 he taught at Franz Joseph University in Kolozsvár in Austria-Hungary (now Cluj-Napoca in Romania). In 1911 Fejér was appointed to the chair of mathematics at the University of Budapest and he held that post until his death. He was elected corresponding member (1908), member (1930) of the Hungarian Academy of Sciences. During his period in the chair at Budapest Fejér led a highly successful Hungarian school of analysis. He was the thesis advisor of mathematicians ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Internet Archive
The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, movies/videos, moving images, and millions of books. In addition to its archiving function, the Archive is an activist organization, advocating a free and open Internet. , the Internet Archive holds over 35 million books and texts, 8.5 million movies, videos and TV shows, 894 thousand software programs, 14 million audio files, 4.4 million images, 2.4 million TV clips, 241 thousand concerts, and over 734 billion web pages in the Wayback Machine. The Internet Archive allows the public to upload and download digital material to its data cluster, but the bulk of its data is collected automatically by its web crawlers, which work to preserve as much of the public web as possible. Its web archiving, web archive, the Wayback Machine, contains hu ... [...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 and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to Harold Samuel Coxeter and Lucy (). His father had taken over the family business of Coxeter & Son, manufacturers of surgical instruments and compressed gases (including a mechanism for anaesthetising surgical patients with nitrous oxide), but was able to retire early and focus on sculpting and baritone singing; Lucy Coxeter was a portrait and landscape painter who had attended the Royal Academy of Arts. A maternal cousin was the architect Sir Giles Gilbert Scott. In his youth, Coxeter composed music and was an accomplished pianist at the age of 10. Roberts, Siobhan, ''King of Infinite Space: Donald Coxeter, The Man Who Saved Geometry'', Walker & Company, 2006, He felt that mathematics and music were intimately related, outlining his i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul J
Paul may refer to: *Paul (given name), a given name (includes a list of people with that name) *Paul (surname), a list of people People Christianity *Paul the Apostle (AD c.5–c.64/65), also known as Saul of Tarsus or Saint Paul, early Christian missionary and writer *Pope Paul (other), multiple Popes of the Roman Catholic Church *Saint Paul (other), multiple other people and locations named "Saint Paul" Roman and Byzantine empire *Lucius Aemilius Paullus Macedonicus (c. 229 BC – 160 BC), Roman general *Julius Paulus Prudentissimus (), Roman jurist *Paulus Catena (died 362), Roman notary *Paulus Alexandrinus (4th century), Hellenistic astrologer *Paul of Aegina or Paulus Aegineta (625–690), Greek surgeon Royals *Paul I of Russia (1754–1801), Tsar of Russia *Paul of Greece (1901–1964), King of Greece Other people *Paul the Deacon or Paulus Diaconus (c. 720 – c. 799), Italian Benedictine monk *Paul (father of Maurice), the father of Maurice, Byzan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set TSP Problem
In combinatorial optimization, the set TSP, also known as the generalized TSP, group TSP, One-of-a-Set TSP, Multiple Choice TSP or Covering Salesman Problem, is a generalization of the traveling salesman problem (TSP), whereby it is required to find a shortest tour in a graph which visits all specified subsets of the vertices of a graph. The subsets of vertices must be disjoint, since the case of overlapping subsets can be reduced to the case of disjoint ones. The ordinary TSP is a special case of the set TSP when all subsets to be visited are singletons. Therefore, the set TSP is also NP-hard. There is a transformation for an instance of the set TSP to an instance of the standard asymmetric TSP. The idea is to connect each subset into a directed cycle with edges of zero weight, and inherit the outgoing edges from the original graph shifting by one vertex backwards along this cycle. The salesman, when visiting a vertex ''v'' in some subset, walks around the cycle for free and exits ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Altitudes And Orthic Triangle SVG
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context (e.g., aviation, geometry, geographical survey, sport, or atmospheric pressure). Although the term ''altitude'' is commonly used to mean the height above sea level of a location, in geography the term elevation is often preferred for this usage. Vertical distance measurements in the "down" direction are commonly referred to as depth. In aviation In aviation, the term altitude can have several meanings, and is always qualified by explicitly adding a modifier (e.g. "true altitude"), or implicitly through the context of the communication. Parties exchanging altitude information must be clear which definition is being used. Aviation altitude is measured using either mean sea level (MSL) or local ground level (above ground level, or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lami's Theorem
In physics, Lami's theorem is an equation relating the magnitudes of three coplanar, concurrent and non-collinear vectors, which keeps an object in static equilibrium, with the angles directly opposite to the corresponding vectors. According to the theorem, :\frac=\frac=\frac where ''A'', ''B'' and ''C'' are the magnitudes of the three coplanar, concurrent and non-collinear vectors, V_A, V_B, V_C, which keep the object in static equilibrium, and ''α'', ''β'' and ''γ'' are the angles directly opposite to the vectors. : Lami's theorem is applied in static analysis of mechanical and structural systems. The theorem is named after Bernard Lamy. Proof As the vectors must balance V_A+V_B+V_C=0, hence by making all the vectors touch its tip and tail we can get a triangle with sides A,B,C and angles 180^o -\alpha, 180^o -\beta, 180^o -\gamma. By the law of sines then \frac=\frac=\frac. Then by applying that for any angle \theta, \sin (180^o - \theta) = \sin \theta we obtain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hooke's Law
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that distance—that is, where is a constant factor characteristic of the spring (i.e., its stiffness), and is small compared to the total possible deformation of the spring. The law is named after 17th-century British physicist Robert Hooke. He first stated the law in 1676 as a Latin anagram. He published the solution of his anagram in 1678 as: ("as the extension, so the force" or "the extension is proportional to the force"). Hooke states in the 1678 work that he was aware of the law since 1660. Hooke's equation holds (to some extent) in many other situations where an elasticity (physics), elastic body is Deformation (physics), deformed, such as wind blowing on a tall building, and a musician plucking a string (music), string of a guitar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermann Schwarz
Karl Hermann Amandus Schwarz (; 25 January 1843 – 30 November 1921) was a German mathematician, known for his work in complex analysis. Life Schwarz was born in Hermsdorf, Silesia (now Jerzmanowa, Poland). In 1868 he married Marie Kummer, who was the daughter to the mathematician Ernst Eduard Kummer and Ottilie née Mendelssohn (a daughter of Nathan Mendelssohn's and granddaughter of Moses Mendelssohn). Schwarz and Kummer had six children, including his daughter Emily Schwarz. Schwarz originally studied chemistry in Berlin but Ernst Eduard Kummer and Karl Theodor Wilhelm Weierstrass persuaded him to change to mathematics. He received his Ph.D. from the Universität Berlin in 1864 and was advised by Kummer and Weierstrass. Between 1867 and 1869 he worked at the University of Halle, then at the Swiss Federal Polytechnic. From 1875 he worked at Göttingen University, dealing with the subjects of complex analysis, differential geometry and the calculus of variations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Giulio Carlo De' Toschi Di Fagnano
Giulio Carlo, Count Fagnano, Marquis de Toschi (26 September 1682 — 18 May 1766) was an Italian mathematician. He was probably the first to direct attention to the theory of elliptic integrals. Fagnano was born in Senigallia (at the time spelled "Sinigaglia"), and also died there. Life Giulio Fagnano was born to Francesco Fagnano and Camilla Bartolini in Senigallia (at the time spelled "Sinigaglia") in 1682. Fagnano had twelve children. One, Giovanni Fagnano, was also well-known as a mathematician. Another of Fagnano's children became a Benedictine nun. In 1721, Fagnano was made a count by Louis XV; in 1723, he was appointed ''gonfaloniere'' of Senigallia and elected to the Royal Society of London; in 1745 he was made a marquis of Sant' Onofrio. Mathematical work Fagnano made his higher studies at the Collegio Clementino in Rome, and there won great distinction — except in mathematics, to which his aversion was extreme. Only after his college course did he take up the stu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
