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Truncated Icosidodecahedron
In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron,Wenninger Model Number 16 great rhombicosidodecahedron,Williams (Section 3-9, p. 94)Cromwell (p. 82) omnitruncated dodecahedron or omnitruncated icosahedronNorman Woodason Johnson, "The Theory of Uniform Polytopes and Honeycombs", 1966 is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces. It has 62 faces: 30 squares, 20 regular hexagons, and 12 regular decagons. It has the most edges and vertices of all Platonic and Archimedean solids, though the snub dodecahedron has more faces. Of all vertex-transitive polyhedra, it occupies the largest percentage (89.80%) of the volume of a sphere in which it is inscribed, very narrowly beating the snub dodecahedron (89.63%) and small rhombicosidodecahedron (89.23%), and less narrowly beating the truncated icosahedron (86.74%); it also has by far the greatest volume (206.8 cubic un ... [...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] |
Antiprism
In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation . Antiprisms are a subclass of prismatoids, and are a (degenerate) type of snub polyhedron. Antiprisms are similar to prisms, except that the bases are twisted relatively to each other, and that the side faces (connecting the bases) are triangles, rather than quadrilaterals. The dual polyhedron of an -gonal antiprism is an -gonal trapezohedron. History At the intersection of modern-day graph theory and coding theory, the triangulation of a set of points have interested mathematicians since Isaac Newton, who fruitlessly sought a mathematical proof of the kissing number problem in 1694. The existence of antiprisms was discussed, and their name was coined by Johannes Kepler, though it is possible that they were previously known to Archimedes, as they satisfy the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norman Johnson (mathematician)
Norman Woodason Johnson () was a mathematician at Wheaton College, Norton, Massachusetts. Early life and education Norman Johnson was born on in Chicago. His father had a bookstore and published a local newspaper. Johnson earned his undergraduate mathematics degree in 1953 at Carleton College in Northfield, Minnesota followed by a master's degree from the University of Pittsburgh. After graduating in 1953, Johnson did alternative civilian service as a conscientious objector. He earned his PhD from the University of Toronto in 1966 with a dissertation title of ''The Theory of Uniform Polytopes and Honeycombs'' under the supervision of H. S. M. Coxeter. From there, he accepted a position in the Mathematics Department of Wheaton College in Massachusetts and taught until his retirement in 1998. Career In 1966, he enumerated 92 convex non-uniform polyhedra with regular faces. Victor Zalgaller later proved (1969) that Johnson's list was complete, and the set is now known a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Omnitruncation (geometry)
In geometry, an omnitruncation is an operation applied to a regular polytope (or honeycomb) in a Wythoff construction that creates a maximum number of facets. It is represented in a Coxeter–Dynkin diagram with all nodes ringed. It is a ''shortcut'' term which has a different meaning in progressively-higher-dimensional polytopes: * Uniform polytope truncation operators ** For regular polygons: An ordinary truncation, t_\ = t\ = \. *** Coxeter-Dynkin diagram ** For uniform polyhedra (3-polytopes): A cantitruncation, t_\ = tr\. (Application of both cantellation and truncation operations) *** Coxeter-Dynkin diagram: ** For uniform polychora: A runcicantitruncation, t_\. (Application of runcination, cantellation, and truncation operations) *** Coxeter-Dynkin diagram: , , ** For uniform polytera (5-polytopes): A steriruncicantitruncation, t0,1,2,3,4. t_\. (Application of sterication, runcination, cantellation, and truncation operations) *** Coxeter-Dynkin diagram: , , ** ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Williams (geometer)
Robert Edward Williams (born 1942) is an American designer, mathematician, and architect. He is noted for books on the geometry of natural structure, the discovery of a new space-filling polyhedron, the development of theoretical principles of Catenatic Geometry, and the invention of the ''Ars-Vivant Wild-life Protector System'' for repopulating the Western Mojave Desert in California, USA with desert tortoises. Biography—life, theories, and work Robert Williams was born in Cincinnati, Ohio, the son of Robert Finley Williams and Edna Rita Brotherton. His father was the oldest member of the Williams Brothers, a quartet of musical entertainers, who appeared on recordings, radio, and television, from the late 1930s to the present. Williams's work was originally inspired by the design principles in natural structure systems promoted by R. Buckminster Fuller. He was introduced to the work of Fuller by designer Peter Pearce in 1963. He finished graduate studies in structural design ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnus Wenninger
Father Magnus J. Wenninger Order of Saint Benedict, OSB (October 31, 1919Banchoff (2002)– February 17, 2017) was an American mathematician who worked on constructing polyhedron models, and wrote the first book on their construction. Early life and education Born to German people, German immigrants in Park Falls, Wisconsin, Joseph Wenninger always knew he was going to be a priest. From an early age, it was understood that his brother Heinie would take after their father and become a baker, and that Joe, as he was then known, would go into the priesthood. When Wenninger was thirteen, after graduating from the parochial school in Park Falls, Wisconsin, his parents saw an advertisement in the German newspaper ''Der Wanderer'' that would help to shape the rest of his life. The ad was for a preparatory school in Collegeville, Minnesota, associated with the Benedictine College of Saint Benedict and Saint John's University, St. John's University. While admitting to feeling homesick a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topologically
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a ''topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ''ABCD'' would be denoted . Characterizations A convex quadrilateral is a square if and only if it is any one of the following: * A rectangle with two adjacent equal sides * A rhombus with a right vertex angle * A rhombus with all angles equal * A parallelogram with one right vertex angle and two adjacent equal sides * A quadrilateral with four equal sides and four right angles * A quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other (i.e., a rhombus with equal diagonals) * A convex quadrilateral with successiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectangle
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containing a right angle. A rectangle with four sides of equal length is a ''square''. The term "oblong" is occasionally used to refer to a non-square rectangle. A rectangle with vertices ''ABCD'' would be denoted as . The word rectangle comes from the Latin ''rectangulus'', which is a combination of ''rectus'' (as an adjective, right, proper) and ''angulus'' (angle). A crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals (therefore only two sides are parallel). It is a special case of an antiparallelogram, and its angles are not right angles and not all equal, though opposite angles are equal. Other geometries, such as spherical, elliptic, and hyperboli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icosidodecahedron
In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly, a quasiregular polyhedron. Geometry An icosidodecahedron has icosahedral symmetry, and its first stellation is the compound of a dodecahedron and its dual icosahedron, with the vertices of the icosidodecahedron located at the midpoints of the edges of either. Its dual polyhedron is the rhombic triacontahedron. An icosidodecahedron can be split along any of six planes to form a pair of pentagonal rotundae, which belong among the Johnson solids. The icosidodecahedron can be considered a ''pentagonal gyrobirotunda'', as a combination of two rotundae (compare pentagonal orthobirotunda, one of the Johnson solids) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncation (geometry)
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new Facet (geometry), facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids. Uniform truncation In general any polyhedron (or polytope) can also be truncated with a degree of freedom as to how deep the cut is, as shown in Conway polyhedron notation truncation operation. A special kind of truncation, usually implied, is a uniform truncation, a truncation operator applied to a regular polyhedron (or regular polytope) which creates a resulting uniform polyhedron (uniform polytope) with equal edge lengths. There are no degrees of freedom, and it represents a fixed geometric, just like the regular polyhedra. In general all single ringed uniform polytopes have a uniform truncation. For example, the icosidodecahedron, represented as Schläfli symbols r or \begin 5 \\ 3 \end, and Coxeter-Dynkin diagram or has a uniform truncation, the truncate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johannes Kepler
Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws of planetary motion, and his books ''Astronomia nova'', ''Harmonice Mundi'', and ''Epitome Astronomiae Copernicanae''. These works also provided one of the foundations for Newton's theory of universal gravitation. Kepler was a mathematics teacher at a seminary school in Graz, where he became an associate of Prince Hans Ulrich von Eggenberg. Later he became an assistant to the astronomer Tycho Brahe in Prague, and eventually the imperial mathematician to Emperor Rudolf II and his two successors Matthias and Ferdinand II. He also taught mathematics in Linz, and was an adviser to General Wallenstein. Additionally, he did fundamental work in the field of optics, invented an improved version of the refracting (or Keplerian) telescope, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
