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Compound Of Three Cubes
In geometry, the compound of three cubes is a uniform polyhedron compound formed from three cubes arranged with octahedral symmetry. It has been depicted in works by Max Brückner and M.C. Escher. History This compound appears in Max Brückner's book ''Vielecke und Vielflache'' (1900), and in the lithograph print ''Waterfall'' (1961) by M.C. Escher, who learned of it from Brückner's book. Its dual, the compound of three octahedra, forms the central image in an earlier Escher woodcut, ''Stars''. In the 15th-century manuscript ''De quinque corporibus regularibus'', Piero della Francesca includes a drawing of an octahedron circumscribed around a cube, with eight of the cube edges lying in the octahedron's eight faces. Three cubes inscribed in this way within a single octahedron would form the compound of three cubes, but della Francesca does not depict the compound. Construction and coordinates This compound can be constructed by superimposing three identical cubes, and then ro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
UC08-3 Cubes
UC may refer to: Arts and entertainment * ''University Challenge'', a popular British quiz programme airing on BBC Two ** '' University Challenge (New Zealand)'', the New Zealand version of the British programme * Universal Century, one of the timelines of the ''Gundam'' anime metaseries Education In the United States * University of California system ** University of California, Berkeley, its flagship university * University of Charleston, West Virginia * University of Chicago, Illinois * University of Cincinnati, Ohio * Upsala College, East Orange, New Jersey (''defunct since 1995'') * Utica College, Utica, New York * Harvard Undergraduate Council, Harvard College's student government body * University college In other countries * Pontifical Catholic University of Chile * University of Canberra, Australia * University of Cantabria, Spain * University of Canterbury, New Zealand * University of Cebu, Cebu City, Philippines * University of Coimbra, Portugal * University of the Co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedral Symmetry
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedron that is dual polyhedron, dual to an octahedron. The group of orientation-preserving symmetries is ''S''4, the symmetric group or the group of permutations of four objects, since there is exactly one such symmetry for each permutation of the four diagonals of the cube. Details Chiral and full (or achiral) octahedral symmetry are the Point groups in three dimensions, discrete point symmetries (or equivalently, List of spherical symmetry groups, symmetries on the sphere) with the largest symmetry groups compatible with translational symmetry. They are among the Crystal system#Overview of point groups by crystal system, crystallographic point groups of the cubic crystal system. As the hyperoctahedral group of dimension 3 the full oct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference coordinate line is called a ''coordinate axis'' or just ''axis'' (plural ''axes'') of the system, and the point where they meet is its ''origin'', at ordered pair . The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, ''n'' Cartesian coordinates (an element of real ''n''-space) specify the point in an ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Piero Della Francesca
Piero della Francesca (, also , ; – 12 October 1492), originally named Piero di Benedetto, was an Italian painter of the Early Renaissance. To contemporaries he was also known as a mathematician and geometer. Nowadays Piero della Francesca is chiefly appreciated for his art. His painting is characterized by its serene humanism, its use of geometric forms and perspective. His most famous work is the cycle of frescoes ''The History of the True Cross'' in the church of San Francesco in the Tuscan town of Arezzo. Biography Early years Piero was born Piero di Benedetto in the town of 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 how Lisa Gherardini became kno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stars (M
A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth make them appear as fixed points of light. The most prominent stars have been categorised into constellations and asterisms, and many of the brightest stars have proper names. Astronomers have assembled star catalogues that identify the known stars and provide standardized stellar designations. The observable universe contains an estimated to stars. Only about 4,000 of these stars are visible to the naked eye, all within the Milky Way galaxy. A star's life begins with the gravitational collapse of a gaseous nebula of material composed primarily of hydrogen, along with helium and trace amounts of heavier elements. Its total mass is the main factor determining its evolution and eventual fate. A star shines for most of its active life due t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Woodcut
Woodcut is a relief printing technique in printmaking. An artist carves an image into the surface of a block of wood—typically with gouges—leaving the printing parts level with the surface while removing the non-printing parts. Areas that the artist cuts away carry no ink, while characters or images at surface level carry the ink to produce the print. The block is cut along the wood grain (unlike wood engraving, where the block is cut in the end-grain). The surface is covered with ink by rolling over the surface with an ink-covered roller (brayer), leaving ink upon the flat surface but not in the non-printing areas. Multiple colors can be printed by keying the paper to a frame around the woodblocks (using a different block for each color). The art of carving the woodcut can be called "xylography", but this is rarely used in English for images alone, although that and "xylographic" are used in connection with block books, which are small books containing text and images in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Of Three Octahedra
In mathematics, the compound of three octahedra or octahedron 3-compound is a polyhedral compound formed from three 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''. 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: . The remaining octahedron edges cross each other in pairs, within the interior ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Polyhedron
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all can also be constructed as geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a corresponding symmetry class. For example, the regular polyhedrathe (convex) Platonic solids and (star) Kepler–Poinsot polyhedraform dual pairs, where the regular tetrahedron is self-dual. The dual of an isogonal polyhedron (one in which any two vertices are equivalent under symmetries of the polyhedron) is an isohedral polyhedron (one in which any two faces are equivalent .., and vice vers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Waterfall (M
A waterfall is a 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 on to softer rock, which 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 for years, travelling to see them, exploring and naming them. They can present 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 terminology A waterfall is generally d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lithograph
Lithography () is a planographic method of printing originally based on the immiscibility of oil and water. The printing is from a stone (lithographic limestone) or a metal plate with a smooth surface. It was invented in 1796 by the German author and actor Alois Senefelder and was initially used mostly for musical scores and maps.Meggs, Philip B. A History of Graphic Design. (1998) John Wiley & Sons, Inc. p 146 Carter, Rob, Ben Day, Philip Meggs. Typographic Design: Form and Communication, Third Edition. (2002) John Wiley & Sons, Inc. p 11 Lithography can be used to print text or images onto paper or other suitable material. A lithograph is something printed by lithography, but this term is only used for fine art prints and some other, mostly older, types of printed matter, not for those made by modern commercial lithography. Originally, the image to be printed was drawn with a greasy substance, such as oil, fat, or wax onto the surface of a smooth and flat limestone plat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 documen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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