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Cantic 5-cube
In geometry of five dimensions or higher, a cantic 5-cube, cantihalf 5-cube, truncated 5-demicube is a uniform 5-polytope, being a truncation of the 5-demicube. It has half the vertices of a cantellated 5-cube. Cartesian coordinates The Cartesian coordinates for the 160 vertices of a cantic 5-cube centered at the origin and edge length 6 are coordinate permutations: : (±1,±1,±3,±3,±3) with an odd number of plus signs. Alternate names * Cantic penteract, truncated demipenteract * Truncated hemipenteract (thin) (Jonathan Bowers)Klitzing, (x3x3o *b3o3o - thin) Images Related polytopes It has half the vertices of the cantellated 5-cube, as compared here in the B5 Coxeter plane projections: This polytope is based on the 5-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family. There are 23 uniform 5-polytope that can be constructed from the D5 symmetry of the 5-demicube, of which are uniqu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated 5-demicube D5
Truncation is the term used for limiting the number of digits right of the decimal point by discarding the least significant ones. Truncation may also refer to: Mathematics * Truncation (statistics) refers to measurements which have been cut off at some value * Truncation (numerical analysis) refers to truncating an infinite sum by a finite one * Truncation (geometry) is the removal of one or more parts, as for example in truncated cube * Propositional truncation, a type former which truncates a type down to a mere proposition Computer science * Data truncation, an event that occurs when a file or other data is stored in a location too small to accommodate its entire length * Truncate (SQL), a command in the SQL data manipulation language to quickly remove all data from a table Biology * Truncate, a leaf shape * Truncated protein, a protein shortened by a mutation which specifically induces premature termination of messenger RNA translation Other uses * Cheque truncatio ... [...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] |
Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to Harold Samuel Coxeter and Lucy (). His father had taken over the family business of Coxeter & Son, manufacturers of surgical instruments and compressed gases (including a mechanism for anaesthetising surgical patients with nitrous oxide), but was able to retire early and focus on sculpting and baritone singing; Lucy Coxeter was a portrait and landscape painter who had attended the Royal Academy of Arts. A maternal cousin was the architect Sir Giles Gilbert Scott. In his youth, Coxeter composed music and was an accomplished pianist at the age of 10. Roberts, Siobhan, ''King of Infinite Space: Donald Coxeter, The Man Who Saved Geometry'', Walker & Company, 2006, He felt that mathematics and music were intimately related, outlining his i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-cube
In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces. It is represented by Schläfli symbol or , constructed as 3 tesseracts, , around each cubic ridge. It can be called a penteract, a portmanteau of the Greek word , for 'five' (dimensions), and the word ''tesseract'' (the 4-cube). It can also be called a regular deca-5-tope or decateron, being a 5-dimensional polytope constructed from 10 regular facets. Related polytopes It is a part of an infinite hypercube family. The dual of a 5-cube is the 5-orthoplex, of the infinite family of orthoplexes. Applying an '' alternation'' operation, deleting alternating vertices of the 5-cube, creates another uniform 5-polytope, called a 5-demicube, which is also part of an infinite family called the demihypercubes. The 5-cube can be seen as an ''order-3 tesseractic honeycomb'' on a 4-sphere. It is related to the Euclidea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in ''n'' dimensions is equal to \sqrt. An ''n''-dimensional hypercube is more commonly referred to as an ''n''-cube or sometimes as an ''n''-dimensional cube. The term measure polytope (originally from Elte, 1912) is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γn polytopes. The hypercube is the special case of a hyperrectangle (also called an ''n-orthotope''). A ''unit hypercube'' is a hypercube whose side has length one unit. Often, the hypercube whose corners (or ''vertices'') are the 2''n'' points in R''n'' with each coordinate equal to 0 or 1 is called ''the'' unit hypercube. Construction A hyp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternation (geometry)
In geometry, an alternation or ''partial truncation'', is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.Coxeter, Regular polytopes, pp. 154–156 8.6 Partial truncation, or alternation Coxeter labels an ''alternation'' by a prefixed ''h'', standing for ''hemi'' or ''half''. Because alternation reduces all polygon faces to half as many sides, it can only be applied to polytopes with all even-sided faces. An alternated square face becomes a digon, and being degenerate, is usually reduced to a single edge. More generally any vertex-uniform polyhedron or tiling with a vertex configuration consisting of all even-numbered elements can be ''alternated''. For example, the alternation of a vertex figure with ''2a.2b.2c'' is ''a.3.b.3.c.3'' where the three is the number of elements in this vertex figure. A special case is square faces whose order divides in half into degenerate digons. So for example, the cube ''4.4.4'' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Demihypercube
In geometry, demihypercubes (also called ''n-demicubes'', ''n-hemicubes'', and ''half measure polytopes'') are a class of ''n''-polytopes constructed from alternation of an ''n''-hypercube, labeled as ''hγn'' for being ''half'' of the hypercube family, ''γn''. Half of the vertices are deleted and new facets are formed. The 2''n'' facets become 2''n'' (''n''−1)-demicubes, and 2''n'' (''n''−1)-simplex facets are formed in place of the deleted vertices. They have been named with a ''demi-'' prefix to each hypercube name: demicube, demitesseract, etc. The demicube is identical to the regular tetrahedron, and the demitesseract is identical to the regular 16-cell. The demipenteract is considered ''semiregular'' for having only regular facets. Higher forms don't have all regular facets but are all uniform polytopes. The vertices and edges of a demihypercube form two copies of the halved cube graph. An ''n''-demicube has inversion symmetry if ''n'' is even. Discovery ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vertex-transitive even-sided polygons that alternate two different lengths of edges). This is a generalization of the older category of ''semiregular'' polytopes, but also includes the regular polytopes. Further, star regular faces and vertex figures ( star polygons) are allowed, which greatly expand the possible solutions. A strict definition requires uniform polytopes to be finite, while a more expansive definition allows uniform honeycombs (2-dimensional tilings and higher dimensional honeycombs) of Euclidean and hyperbolic space to be considered polytopes as well. Operations Nearly every uniform polytope can be generated by a Wythoff construction, and represented by a Coxeter diagram. Notable exceptions include the great dirhombic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-cube T02
In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces. It is represented by Schläfli symbol or , constructed as 3 tesseracts, , around each cubic ridge. It can be called a penteract, a portmanteau of the Greek word , for 'five' (dimensions), and the word ''tesseract'' (the 4-cube). It can also be called a regular deca-5-tope or decateron, being a 5-dimensional polytope constructed from 10 regular facets. Related polytopes It is a part of an infinite hypercube family. The dual of a 5-cube is the 5-orthoplex, of the infinite family of orthoplexes. Applying an '' alternation'' operation, deleting alternating vertices of the 5-cube, creates another uniform 5-polytope, called a 5-demicube, which is also part of an infinite family called the demihypercubes. The 5-cube can be seen as an ''order-3 tesseractic honeycomb'' on a 4-sphere. It is related to the Euclide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-demicube T01 B5
In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a ''5-hypercube'' (penteract) with alternated vertices removed. It was discovered by Thorold Gosset. Since it was the only semiregular 5-polytope (made of more than one type of regular facets), he called it a 5-ic semi-regular. E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM5 for a 5-dimensional ''half measure'' polytope. Coxeter named this polytope as 121 from its Coxeter diagram, which has branches of length 2, 1 and 1 with a ringed node on one of the short branches, and Schläfli symbol \left\ or . It exists in the k21 polytope family as 121 with the Gosset polytopes: 221, 321, and 421. The graph formed by the vertices and edges of the demipenteract is sometimes called the Clebsch graph, though that name sometimes refers to the folded cube graph of order five instead. Cartesian coordinates Cartesian coordinates for the vertices ... [...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] |
Cantellated 5-cube
In six-dimensional geometry, a cantellated 5-cube is a convex uniform 5-polytope, being a cantellation of the regular 5-cube. There are 6 unique cantellation for the 5-cube, including truncations. Half of them are more easily constructed from the dual 5-orthoplex Cantellated 5-cube Alternate names * Small rhombated penteract (Acronym: sirn) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of a ''cantellated 5-cube'' having edge length 2 are all permutations of: :\left(\pm1,\ \pm1,\ \pm(1+\sqrt),\ \pm(1+\sqrt),\ \pm(1+\sqrt)\right) Images Bicantellated 5-cube In five-dimensional geometry, a bicantellated 5-cube is a uniform 5-polytope. Alternate names * Bicantellated penteract, bicantellated 5-orthoplex, or bicantellated pentacross * Small birhombated penteractitriacontiditeron (Acronym: sibrant) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of a ''bicantellated 5-cube'' having edge length 2 are a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
