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Compound Of Three Octahedra
In mathematics, the compound of three octahedra or octahedron 3-compound is a polyhedral compound formed from three octahedron, regular octahedra, all sharing a common center but rotated with respect to each other. Although appearing earlier in the mathematical literature, it was rediscovered and popularized by M. C. Escher, who used it in the central image of his 1948 woodcut ''Stars (M. C. Escher), Stars''. Construction A regular octahedron can be circumscribed around a cube in such a way that the eight edges of two opposite squares of the cube lie on the eight faces of the octahedron. The three octahedra formed in this way from the three pairs of opposite cube squares form the compound of three octahedra.. The eight cube vertices are the same as the eight points in the compound where three edges cross each other. Each of the octahedron edges that participates in these triple crossings is divided by the crossing point in the ratio 1:square root of 2, . The remaining octahedron ed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Equilateral Triangle
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the special case of an isosceles triangle by modern definition, creating more special properties. The equilateral triangle can be found in various tilings, and in polyhedrons such as the deltahedron and antiprism. It appears in real life in popular culture, architecture, and the study of stereochemistry resembling the molecular known as the trigonal planar molecular geometry. Properties An equilateral triangle is a triangle that has three equal sides. It is a special case of an isosceles triangle in the modern definition, stating that an isosceles triangle is defined at least as having two equal sides. Based on the modern definition, this leads to an equilateral triangle in which one of the three sides may be considered its base. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Stellation
In geometry, stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original. The word ''stellation'' comes from the Latin ''stellātus'', "starred", which in turn comes from the Latin ''stella'', "star". Stellation is the reciprocal or dual process to '' faceting''. Kepler's definition In 1619 Kepler defined stellation for polygons and polyhedra as the process of extending edges or faces until they meet to form a new polygon or polyhedron. He stellated the regular dodecahedron to obtain two regular star polyhedra, the small stellated dodecahedron and the great stellated dodecahedron. He also stellated the regular oct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Waterfall (M
A waterfall is any point in a river or stream where water flows over a vertical drop or a series of steep drops. Waterfalls also occur where meltwater drops over the edge of a tabular iceberg or ice shelf. Waterfalls can be formed in several ways, but the most common method of formation is that a river courses over a top layer of resistant bedrock before falling onto softer rock, which Erosion, erodes faster, leading to an increasingly high fall. Waterfalls have been studied for their impact on species living in and around them. Humans have had a distinct relationship with waterfalls since prehistory, travelling to see them, exploring and naming them. They can present head of navigation, formidable barriers to navigation along rivers. Waterfalls are religious sites in many cultures. Since the 18th century, they have received increased attention as tourist destinations, sources of hydropower, andparticularly since the mid-20th centuryas subjects of research. Definition and te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Disdyakis Dodecahedron
In geometry, a disdyakis dodecahedron, (also hexoctahedron, hexakis octahedron, octakis cube, octakis hexahedron, kisrhombic dodecahedron) or d48, is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons. It resembles an augmented rhombic dodecahedron. Replacing each face of the rhombic dodecahedron with a flat pyramid results in the Kleetope of the rhombic dodecahedron, which looks almost like the disdyakis dodecahedron, and is topologically equivalent to it. The net of the rhombic dodecahedral pyramid also shares the same topology. Symmetry It has Oh octahedral symmetry. Its collective edges represent the reflection planes of the symmetry. It can also be seen in the corner and mid-edge triangulation of the regular cube and octahedron, and rhombic dodecahedron. The edges of a spherical disdyakis dodecahedron belong to 9 great circles. Three of them form a spherical octahedron ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Stellated Rhombic Dodecahedron
In geometry, the first stellation of the rhombic dodecahedron is a self-intersecting polyhedron with 12 faces, each of which is a non-convex Hexagon. It is a stellation of the rhombic dodecahedron and has the same outer shell and the same visual appearance as two other shapes: a solid, Escher's solid, with 48 triangular faces, and a polyhedral compound of three flattened octahedra with 24 overlapping triangular faces. Escher's solid can tessellate space to form the stellated rhombic dodecahedral honeycomb. Stellation, solid, and compound The first stellation of the rhombic dodecahedron has 12 faces, each of which is a non-convex hexagon. It is a stellation of the rhombic dodecahedron, meaning that each of its faces lies in the same plane as one of the rhombus faces of the rhombic dodecahedron, with each face containing the rhombus in the same plane, and that it has the same symmetries as the rhombic dodecahedron. It is the first stellation, meaning that no other self-intersecti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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George W
George Walker Bush (born July 6, 1946) is an American politician and businessman who was the 43rd president of the United States from 2001 to 2009. A member of the Bush family and the Republican Party (United States), Republican Party, he is the eldest son of the 41st president, George H. W. Bush, and was the 46th governor of Texas from 1995 to 2000. Bush flew warplanes in the Texas Air National Guard in his twenties. After graduating from Harvard Business School in 1975, he worked in the oil industry. He later co-owned the Major League Baseball team Texas Rangers (baseball), Texas Rangers before being elected governor of Texas 1994 Texas gubernatorial election, in 1994. Governorship of George W. Bush, As governor, Bush successfully sponsored legislation for tort reform, increased education funding, set higher standards for schools, and reformed the criminal justice system. He also helped make Texas the Wind power in Texas, leading producer of wind-generated electricity in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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] [Amazon] |
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Chameleon
Chameleons or chamaeleons (Family (biology), family Chamaeleonidae) are a distinctive and highly specialized clade of Old World lizards with 200 species described as of June 2015. The members of this Family (biology), family are best known for their distinct range of colours, being capable of colour-shifting camouflage. The large number of species in the family exhibit considerable variability in their capacity to change colour. For some, it is more of a shift of brightness (shades of brown); for others, a plethora of colour-combinations (reds, yellows, greens, blues) can be seen. Chameleons are also distinguished by their zygodactylous feet, their prehensility, prehensile tail, their laterally compressed bodies, their head casques, their projectile tongues used for catching prey, their swaying gait, and in some species crests or horns on their brow and snout. Chameleons' eyes are independently mobile, and because of this the chameleon’s brain is constantly analyzing two sepa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Max Brückner
Johannes Max Brückner (5 August 1860 – 1 November 1934) was a German geometer, known for his collection of polyhedral models. Education and career Brückner was born in Hartau, in the Kingdom of Saxony, a town that is now part of Zittau, Germany. He completed a Ph.D. at Leipzig University in 1886, supervised by Felix Klein and Wilhelm Scheibner, with a dissertation concerning conformal maps. After teaching at a grammar school in Zwickau, he moved to the gymnasium in Bautzen. Brückner is known for making many geometric models, particularly of stellated and uniform polyhedra, which he documented in his book ''Vielecke und Vielflache: Theorie und Geschichte'' (''Polygons and polyhedra: Theory and History'', Leipzig: B. G. Teubner, 1900). The shapes first studied in this book include the final stellation of the icosahedron and the compound of three octahedra, made famous by M. C. Escher's print ''Stars''. Joseph Malkevitch lists the publication of this book, which docu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Piero Della Francesca
Piero della Francesca ( , ; ; ; – 12 October 1492) was an Italian Renaissance painter, Italian painter, mathematician and List of geometers, geometer of the Early Renaissance, nowadays chiefly appreciated for his art. His painting is characterized by its serene humanism, its use of geometric forms and Perspective (graphical), perspective. His most famous work is the cycle of frescoes ''The History of the True Cross'' in the Basilica of San Francesco, Arezzo, Basilica of San Francesco in the Tuscany, Tuscan town of Arezzo. Biography Early years Piero was born Piero di Benedetto in the town of Sansepolcro, Borgo Santo Sepolcro, modern-day Tuscany, to Benedetto de' Franceschi, a tradesman, and Romana di Perino da Monterchi, members of the Florentine and Tuscan Franceschi noble family. His father died before his birth, and he was called Piero della Francesca after his mother, who was referred to as "la Francesca" due to her marriage into the Franceschi family (similar to Lisa d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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De Quinque Corporibus Regularibus
''De quinque corporibus regularibus'' (sometimes called ''Libellus de quinque corporibus regularibus'') is a book on the geometry of polyhedra written in the 1480s or early 1490s by Italian painter and mathematician Piero della Francesca. It is a manuscript, in the Latin language; its title means '' he little bookon the five regular solids''. It is one of three books known to have been written by della Francesca. Along with the Platonic solids, ''De quinque corporibus regularibus'' includes descriptions of five of the thirteen Archimedean solids, and of several other irregular polyhedra coming from architectural applications. It was the first of what would become many books connecting mathematics to art through the construction and perspective drawing of polyhedra, including Luca Pacioli's 1509 '' Divina proportione'' (which incorporated without credit an Italian translation of della Francesca's work). Lost for many years, ''De quinque corporibus regularibus'' was rediscovered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Piero Della Francesca - Libellus De Quinque Corporibus Regularibus - P41b (cropped)
Piero is an Italian given name. Notable people with the name include: *Piero Angela (1928–2022), Italian television host * Piero Barucci (born 1933), Italian academic and politician *Piero Cassano (born 1948), Italian keyboardist, singer and composer, a founding member of the Genoan band Matia Bazar *Piero del Pollaiuolo (c. 1443–1496), Italian painter *Piero della Francesca (c1415–1492), Italian artist of the Early Renaissance * Piero De Benedictis (born 1945), Italian-born Argentine and Colombian folk singer * Piero Ciampi (1934–1980), Italian singer *Piero di Cosimo (1462-1522), also known as Piero di Lorenzo, Italian Renaissance painter *Piero di Cosimo de' Medici (1416–1469), ''de facto'' ruler of Florence from 1464 to 1469 *Piero Ferrari (born 1945), Italian businessman * Piero Focaccia (born 1944), Italian pop singer * Piero Fornasetti (1913–1988), Italian painter * Piero Gardoni (1934–1994), Italian professional footballer * Piero Golia (born 1974), Italian conc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |