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Diminishment (geometry)
In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two solids with such a property: the first solids are the pyramids, cupolas, and a rotunda; some of the solids may be constructed by attaching with those previous solids, whereas others may not. Definition and background A Johnson solid is a convex polyhedron whose faces are all regular polygons. The convex polyhedron means as bounded intersections of finitely many half-spaces, or as the convex hull of finitely many points. Although there is no restriction that any given regular polygon cannot be a face of a Johnson solid, some authors required that Johnson solids are not uniform. This means that a Johnson solid is not a Platonic solid, Archimedean solid, prism, or antiprism. A convex polyhedron in which all faces are nearly regular, but some are not pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedean Solid
The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygon and are vertex-transitive, although they aren't face-transitive. The solids were named after Archimedes, although he did not claim credit for them. They belong to the class of uniform polyhedra, the polyhedra with regular faces and symmetric vertices. Some Archimedean solids were portrayed in the works of artists and mathematicians during the Renaissance. The elongated square gyrobicupola or ' is an extra polyhedron with regular faces and congruent vertices, but it is not generally counted as an Archimedean solid because it is not vertex-transitive. The solids The Archimedean solids have a single vertex configuration and highly symmetric properties. A vertex configuration indicates which regular polygons meet at each vertex. For instance, the configuration 3 \cdot 5 \cdot 3 \cdot 5 indicates a polyhedron in which each vertex is met by alternating two triangles and two pentagons. Highl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gyration
In geometry, a gyration is a rotation in a discrete subgroup of symmetries of the Euclidean plane such that the subgroup does not also contain a reflection symmetry whose axis passes through the center of rotational symmetry. In the orbifold corresponding to the subgroup, a gyration corresponds to a rotation point that does not lie on a mirror A mirror, also known as a looking glass, is an object that Reflection (physics), reflects an image. Light that bounces off a mirror forms an image of whatever is in front of it, which is then focused through the lens of the eye or a camera ..., called a gyration point. For example, having a sphere rotating about any point that is ''not'' the center of the sphere, the sphere is gyrating. If it was rotating about its center, the rotation would be symmetrical and it would not be considered gyration. References External links * Euclidean geometry {{geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pyramid (geometry)
In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex (geometry), apex. Each base edge (geometry), edge and apex form a triangle, called a lateral face. A pyramid is a cone, conic solid with a polygonal base. Many types of pyramids can be found by determining the shape of bases, either by based on a regular polygon (regular pyramids) or by cutting off the apex (truncated pyramid). It can be generalized into higher dimensions, known as hyperpyramid. All pyramids are Self-dual polyhedron, self-dual. Etymology The word "pyramid" derives from the ancient Greek term "πυραμίς" (pyramis), which referred to a pyramid-shaped structure and a type of wheat cake. The term is rooted in the Greek "πυρ" (pyr, 'fire') and "άμις" (amis, 'vessel'), highlighting the shape's pointed, flame-like appearance. In Byzantine Greek, the term evolved to "πυραμίδα" (pyramída), continuing to denote pyramid structures. The Greek term " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Orthobicupola
In geometry, the triangular orthobicupola is one of the Johnson solids (). As the name suggests, it can be constructed by attaching two triangular cupolas () along their bases. It has an equal number of squares and triangles at each vertex; however, it is not vertex-transitive. It is also called an ''anticuboctahedron'', ''twisted cuboctahedron'' or ''disheptahedron''. It is also a canonical polyhedron. The ''triangular orthobicupola'' is the first in an infinite set of orthobicupolae. Construction The ''triangular orthobicupola'' can be constructed by attaching two triangular cupolas onto their bases. Similar to the cuboctahedron, which would be known as the ''triangular gyrobicupola'', the difference is that the two triangular cupolas that make up the triangular orthobicupola are joined so that pairs of matching sides abut (hence, "ortho"); the cuboctahedron is joined so that triangles abut squares and vice versa. Given a triangular orthobicupola, a 60-degree rotation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagonal Bipyramid
The pentagonal bipyramid (or pentagonal dipyramid) is a polyhedron with ten triangular faces. It is constructed by attaching two pentagonal pyramids to each of their bases. If the triangular faces are equilateral, the pentagonal bipyramid is an example of deltahedra, composite polyhedron, and Johnson solid. The pentagonal bipyramid may be represented as four-connected well-covered graph. This polyhedron may be used in the chemical compound as the description of an atom cluster known as pentagonal bipyramidal molecular geometry, as a solution in Thomson problem, as well as in decahedral nanoparticles. Special cases As a right bipyramid Like other bipyramids, the pentagonal bipyramid can be constructed by attaching the base of two pentagonal pyramids. These pyramids cover their pentagonal base, such that the resulting polyhedron has ten triangles as its faces, fifteen edges, and seven vertices. The pentagonal bipyramid is said to be right if the pyramids are symmetri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prism (geometry)
In geometry, a prism is a polyhedron comprising an polygon Base (geometry), base, a second base which is a Translation (geometry), translated copy (rigidly moved without rotation) of the first, and other Face (geometry), faces, necessarily all parallelograms, joining corresponding sides of the two bases. All Cross section (geometry), cross-sections parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids. Like many basic geometric terms, the word ''prism'' () was first used in Euclid's Elements, Euclid's ''Elements''. Euclid defined the term in Book XI as "a solid figure contained by two opposite, equal and parallel planes, while the rest are parallelograms". However, this definition has been criticized for not being specific enough in regard to the nature of the bases (a cause of some confusion amongst generations of later geometry writers). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotunda (geometry)
In geometry, a rotunda is any member of a family of Dihedral group, dihedral-symmetric polyhedra. They are similar to a Cupola (geometry), cupola but instead of alternating squares and triangles, it alternates pentagons and triangles around an axis. The pentagonal rotunda is a Johnson solid. Other forms can be generated with dihedral symmetry and distorted equilateral pentagons. Examples Star-rotunda See also * Birotunda References *Norman Johnson (mathematician), Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others. * The first proof that there are only 92 Johnson solids. {{Polyhedron navigator Johnson solids ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cupola (geometry)
In geometry, a cupola is a Polyhedron, solid formed by joining two polygons, one (the base) with twice as many Edge (geometry), edges as the other, by an alternating band of isosceles triangles and rectangles. If the triangles are equilateral triangle, equilateral and the rectangles are Square (geometry), squares, while the base and its opposite face are regular polygons, the triangular cupola, triangular, square cupola, square, and pentagonal cupola, pentagonal cupolae all count among the Johnson solids, and can be formed by taking sections of the cuboctahedron, rhombicuboctahedron, and rhombicosidodecahedron, respectively. A cupola can be seen as a prism (geometry), prism where one of the polygons has been collapsed in half by merging alternate vertices. A cupola can be given an extended Schläfli symbol representing a regular polygon joined by a parallel of its Truncation (geometry)#Truncation of polygons, truncation, or Cupolae are a subclass of the prismatoids. Its d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triaugmented Triangular Prism (symmetric View)
The triaugmented triangular prism, in geometry, is a convex polyhedron with 14 equilateral triangles as its faces. It can be constructed from a triangular prism by attaching equilateral square pyramids to each of its three square faces. The same shape is also called the tetrakis triangular prism, tricapped trigonal prism, tetracaidecadeltahedron, or tetrakaidecadeltahedron; these last names mean a polyhedron with 14 triangular faces. It is an example of a deltahedron, composite polyhedron, and Johnson solid. The edges and vertices of the triaugmented triangular prism form a maximal planar graph with 9 vertices and 21 edges, called the Fritsch graph. It was used by Rudolf and Gerda Fritsch to show that Alfred Kempe's attempted proof of the four color theorem was incorrect. The Fritsch graph is one of only six graphs in which every neighborhood is a 4- or 5-vertex cycle. The dual polyhedron of the triaugmented triangular prism is an associahedron, a polyhedron with four quadril ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Victor Zalgaller
Victor (Viktor) Abramovich Zalgaller (; ; 25 December 1920 – 2 October 2020) was a Russian-Israeli mathematician in the fields of geometry and optimization. He is best known for the results he achieved on convex polyhedra, linear and dynamic programming, isoperimetry, and differential geometry. Biography Zalgaller was born in Parfino, Novgorod Governorate on 25 December 1920. In 1936, he was one of the winners of the Leningrad Mathematics Olympiads for high school students. He started his studies at the Leningrad State University, however, World War II intervened in 1941, and Zalgaller joined the Red Army. He took part in the defence of Leningrad, and in 1945 marched into Germany. He worked as a teacher at the Saint Petersburg Lyceum 239, and received his 1963 doctoral dissertation on polyhedra with the aid of his high school students who wrote the computer programs for the calculation. Zalgaller did his early work under direction of Aleksandr Aleksandrov and Leonid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norman Johnson (mathematician)
Norman Woodason Johnson (November 12, 1930 – July 13, 2017) was an American 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
