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Dodecahedral Number
In mathematics, a dodecahedral number is a figurate number that represents a dodecahedron. The ''n''th dodecahedral number is given by the formula = The first such numbers are: 0, 1, 20, 84, 220, 455, 816, 1330, 2024, 2925, 4060, 5456, 7140, 9139, 11480, … . History The first study of dodecahedral numbers appears to have been by René Descartes, around 1630, in his ''De solidorum elementis''. Prior to Descartes, figurate numbers had been studied by the ancient Greeks and by Johann Faulhaber, but only for polygonal numbers, pyramidal numbers, and cubes. Descartes introduced the study of figurate numbers based on the Platonic solids and some semiregular polyhedra; his work included the dodecahedral numbers. However, ''De solidorum elementis'' was lost, and not rediscovered until 1860. In the meantime, dodecahedral numbers had been studied again by other mathematicians, including Friedrich Wilhelm Marpurg in 1774, Georg Simon Klügel in 1808, and Sir Frederick Pollock in 185 ...
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Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ...
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Cube (algebra)
In arithmetic and algebra, the cube of a number is its third exponentiation, power, that is, the result of multiplying three instances of together. The cube of a number is denoted , using a superscript 3, for example . The cube Mathematical operation, operation can also be defined for any other expression (mathematics), mathematical expression, for example . The cube is also the number multiplied by its square (algebra), square: :. The ''cube function'' is the function (mathematics), function (often denoted ) that maps a number to its cube. It is an odd function, as :. The volume of a Cube (geometry), geometric cube is the cube of its side length, giving rise to the name. The Inverse function, inverse operation that consists of finding a number whose cube is is called extracting the cube root of . It determines the side of the cube of a given volume. It is also raised to the one-third power. The graph of a function, graph of the cube function is known as the cubic para ...
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Dodeca2024
Numeral or number prefixes are prefixes derived from numerals or occasionally other numbers. In English and many other languages, they are used to coin numerous series of words. For example: *triangle, quadrilateral, pentagon, hexagon, octagon (shape with 3 sides, 4 sides, 5 sides, 6 sides, 8 sides) * simplex, duplex (communication in only 1 direction at a time, in 2 directions simultaneously) * unicycle, bicycle, tricycle (vehicle with 1 wheel, 2 wheels, 3 wheels) * dyad, triad, tetrad (2 parts, 3 parts, 4 parts) * twins, triplets, quadruplets (multiple birth of 2 children, 3 children, 4 children) * biped, quadruped, hexapod (animal with 2 feet, 4 feet, 6 feet) * September, October, November, December ( 7th month, 8th month, 9th month, 10th month) * binary, ternary, octal, decimal, hexadecimal (numbers expressed in base 2, base 3, base 8, base&nb ...
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Sir Frederick Pollock, 1st Baronet
Sir Jonathan Frederick Pollock, 1st Baronet, PC (23 September 1783 – 28 August 1870) was a British lawyer and Tory politician. Background and education Pollock was the son of saddler to HM King George III David Pollock, of Charing Cross, London, and the elder brother of Field Marshal Sir George Pollock, 1st Baronet. An elder brother, Sir David Pollock, was a judge in India. The Pollock family were a branch of that family of Balgray, Dumfriesshire; David Pollock's father was a burgess of Berwick-upon-Tweed, and his grandfather a yeoman of Durham. His business as a saddler was given the official custom of the royal family. Sir John Pollock, 4th Baronet, great-great-grandson of David Pollock, stated in Time's Chariot (1950) that David was, 'perhaps without knowing it', Pollock of Balgray, the senior line of the family (Pollock of Pollock or Pollock of that ilk) having died out. Pollock was educated at St Paul's School and Trinity College, Cambridge. He was Senior Wrangler ...
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Georg Simon Klügel
Georg Simon Klügel (August 19, 1739 – August 4, 1812) was a German mathematician and physicist. He was born in Hamburg, and in 1760 went to the University of Göttingen where he initially studied theology before switching to mathematics. Georg Christoph Lichtenberg was a fellow student. His doctoral thesis ''Conatuum praecipuorum theoriam parallelarum demonstrandi recensio'', published in 1763 with Abraham Gotthelf Kästner as doctoral advisor, was a study of 30 attempted proofs of the parallel postulate. It was influential at the time and much cited. Klügel edited the ''Hannöversche Magazin'' for 2 years from 1766, before becoming professor of mathematics at the University of Helmstedt. In 1788 he succeeded Wenzeslaus Johann Gustav Karsten to the chair of mathematics and physics at the University of Halle. He died in Halle in 1812. He remained in correspondence with Lichtenberg throughout his career. Klügel made an exceptional contribution to trigonometry, unifying ...
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Friedrich Wilhelm Marpurg
Friedrich Wilhelm Marpurg (21 November 1718 – 22 May 1795) was a German music critic, music theorist and composer. Described as "one of Germany's leading mid 8th-entury music critics," he was friendly and active with many figures of the Enlightenment. Life Little is known of Marpurg's early life. According to various sources, he studied "philosophy" and music. It is clear that he enjoyed a strong education and was friendly with various leading figures of the Enlightenment, including Winckelmann and Lessing. In 1746, he travelled to Paris as the secretary for a General named either Rothenberg or Bodenberg. There, he became acquainted with intellectuals including the writer and philosopher Voltaire, the mathematician d'Alembert and the composer Jean-Philippe Rameau. After 1746, he returned to Berlin where he was more or less independent. Marpurg's offer to write exclusively for Breitkopf & Härtel was declined by the firm in 1757. In 1760, he received an appointment to the R ...
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Semiregular Polyhedron
In geometry, the term semiregular polyhedron (or semiregular polytope) is used variously by different authors. Definitions In its original definition, it is a polyhedron with regular polygonal faces, and a symmetry group which is transitive on its vertices; today, this is more commonly referred to as a uniform polyhedron (this follows from Thorold Gosset's 1900 definition of the more general semiregular polytope). These polyhedra include: *The thirteen Archimedean solids. ** The elongated square gyrobicupola (also called a pseudo-rhombicuboctahedron), a Johnson solid, has identical vertex figures (3.4.4.4) but because of a twist it is not vertex-transitive. Branko Grünbaum argued for including it as a 14th Archimedean solid. *An infinite series of convex prisms. *An infinite series of convex antiprisms (their semiregular nature was first observed by Kepler). These semiregular solids can be fully specified by a vertex configuration: a listing of the faces by number of sides, ...
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Platonic Solid
In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (identical in shape and size) regular polygons (all angles congruent and all edge (geometry), edges congruent), and the same number of faces meet at each Vertex (geometry), 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 (dialogue), 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 num ...
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Pyramidal Number
A pyramidal number is the number of points in a pyramid with a polygonal base and triangular sides. The term often refers to square pyramidal numbers, which have a square base with four sides, but it can also refer to a pyramid with any number of sides. The numbers of points in the base and in layers parallel to the base are given by polygonal numbers of the given number of sides, while the numbers of points in each triangular side is given by a triangular number. It is possible to extend the pyramidal numbers to higher dimensions. Formula The formula for the th -gonal pyramidal number is :P_n^r= \frac, where , . This formula can be factored: :P_n^r=\frac=\left(\frac\right)\left(\frac\right)=T_n \cdot \frac, where is the th triangular number. Sequences The first few triangular pyramidal numbers (equivalently, tetrahedral numbers) are: : 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ... The first few square pyramidal numbers are: : 1, 5, 14, 30, 55, 91, 140, 20 ...
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Figurate Number
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The ancient Greek mathematicians already considered triangular numbers, polygonal numbers, tetrahedral numbers, and pyramidal numbers, ReprintedG. E. Stechert & Co., 1934 and AMS Chelsea Publishing, 1944. and subsequent mathematicians have included other classes of these numbers including numbers defined from other types of polyhedra and from their analogs in other dimensions. Terminology Some kinds of figurate number were discussed in the 16th and 17th centuries under the name "figural number". In historical works about Greek mathematics the preferred term used to be ''figured number''. In a use going back to Jacob Bernoulli's Ars Conjectandi, the term ''figurate number'' is used for triangular numbers made up of successive integers, tetrahedral numbers made ...
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Polygonal Number
In mathematics, a polygonal number is a Integer, number that counts dots arranged in the shape of a regular polygon. These are one type of 2-dimensional figurate numbers. Polygonal numbers were first studied during the 6th century BC by the Ancient Greeks, who investigated and discussed properties of Pronic number, oblong, Triangular Number, triangular, and Square number, square numbers. Definition and examples The number 10 for example, can be arranged as a triangle (see triangular number): : But 10 cannot be arranged as a square (geometry), square. The number 9, on the other hand, can be (see square number): : Some numbers, like 36, can be arranged both as a square and as a triangle (see square triangular number): : By convention, 1 is the first polygonal number for any number of sides. The rule for enlarging the polygon to the next size is to extend two adjacent arms by one point and to then add the required extra sides between those points. In the following diagrams, ...
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Johann Faulhaber
Johann Faulhaber (5 May 1580 – 10 September 1635) was a German mathematician, specifically, a calculator ('':de:Rechenmeister, Rechenmeister''). Biography Born in Ulm, Faulhaber was a trained weaver who later took the role of a surveyor of the city of Ulm. He collaborated with Johannes Kepler and Ludolph van Ceulen. In 1620, while in Ulm, René Descartes, Descartes probably corresponded with Faulhaber to discuss algebraic solutions of polynomial equations. He worked as fortification engineer in various cities (notably Basel, where he was fortification engineer from 1622 to 1624, and Frankfurt), and also worked under Maurice, Prince of Orange in the Netherlands. He also built water wheels in his home town and geometrical instruments for the military. Faulhaber made the first publication of Henry Briggs (mathematician), Henry Briggs's Logarithm in Germany. He is also credited with the first printed solution of equal temperament. He died in Ulm. Faulhaber's major contribution w ...
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