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Pentiruncinated 6-cube
In six-dimensional geometry, a pentellated 6-orthoplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-orthoplex. There are unique 16 degrees of pentellations of the 6-orthoplex with permutations of truncations, cantellations, runcinations, and sterications. Ten are shown, with the other 6 more easily constructed as a pentellated 6-cube. The simple pentellated 6-orthoplex (Same as pentellated 5-cube) is also called an expanded 6-orthoplex, constructed by an expansion operation applied to the regular 6-orthoplex. The highest form, the ''pentisteriruncicantitruncated 6-orthoplex'', is called an ''omnitruncated 6-orthoplex'' with all of the nodes ringed. Pentitruncated 6-orthoplex Alternate names * Teritruncated hexacontatetrapeton (Acronym: tacox) (Jonathan Bowers) Images Penticantellated 6-orthoplex Alternate names * Terirhombated hexacontitetrapeton (Acronym: tapox) (Jonathan Bowers) Images Penticantitruncated 6-orthoplex Altern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Projection
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it were applied once (i.e. P is idempotent). It leaves its image unchanged. This definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object. Definitions A projection on a vector space V is a linear operator P : V \to V such that P^2 = P. When V has an inner product and is complete (i.e. when V is a Hilbert space) the concept of orthogonality can be used. A projection P on a Hilbert space V is called an orthogonal projection if it satisfies \langle P \mathbf x, \mathbf y \rangle = \langle \mathbf x, P \mathbf y \rangle for all \mathbf x, \mathbf y \in V. A projection ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6-cube T01345
In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces. It has Schläfli symbol , being composed of 3 5-cubes around each 4-face. It can be called a hexeract, a portmanteau of tesseract (the ''4-cube'') with ''hex'' for six (dimensions) in Greek. It can also be called a regular dodeca-6-tope or dodecapeton, being a 6-dimensional polytope constructed from 12 regular facets. Related polytopes It is a part of an infinite family of polytopes, called hypercubes. The dual of a 6-cube can be called a 6-orthoplex, and is a part of the infinite family of cross-polytopes. Applying an '' alternation'' operation, deleting alternating vertices of the 6-cube, creates another uniform polytope, called a 6-demicube, (part of an infinite family called demihypercubes), which has 12 5-demicube and 32 5-simplex facets. As a configuration This configuration matrix represents the 6-cube. ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others''Mathematical Programming'', by Melvyn W. Jeter (1986) p. 68/ref> (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In the influential textbooks of Grünbaum and Ziegler on the subject, as well as in many other texts in discrete geometry, convex polytopes are often simply called "polytopes". Grünbaum points out that this is solely to avo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups , and finite Coxeter groups were classified in 1935 . Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points around the face. When done, these lines form a complete circuit, i.e. a polygon, around the vertex. This polygon is the vertex figure. More precise formal definitions can vary quite widely, according to circumstance. For example Coxeter (e.g. 1948, 1954) varies his definition as convenient for the current area of discussion. Most of the following definitions of a vertex figure apply equally well to infinite tilings or, by extension, to space-filling tessellation with polytope cells and other higher-dimensional polytopes. As a flat slice Make a slice through the corner of the polyhedron, cutting through all the edges connected to the vertex. The cut surface is the vertex figure. This i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schläfli Symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more than three dimensions and discovered all their convex regular polytopes, including the six that occur in four dimensions. Definition The Schläfli symbol is a recursive description, starting with for a ''p''-sided regular polygon that is convex. For example, is an equilateral triangle, is a square, a convex regular pentagon, etc. Regular star polygons are not convex, and their Schläfli symbols contain irreducible fractions ''p''/''q'', where ''p'' is the number of vertices, and ''q'' is their turning number. Equivalently, is created from the vertices of , connected every ''q''. For example, is a pentagram; is a pentagon. A regular polyhedron that has ''q'' regular ''p''-sided polygon faces around each vertex is repr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expansion (geometry)
In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements ( vertices, edges, etc.). Equivalently this operation can be imagined by keeping facets in the same position but reducing their size. The expansion of a regular polytope creates a uniform polytope, but the operation can be applied to any convex polytope, as demonstrated for polyhedra in Conway polyhedron notation (which represents expansion with the letter ). For polyhedra, an expanded polyhedron has all the faces of the original polyhedron, all the faces of the dual polyhedron, and new square faces in place of the original edges. Expansion of regular polytopes According to Coxeter, this multidimensional term was defined by Alicia Boole StottCoxeter, ''Regular Polytopes'' (1973), p. 123. p.210 for creating new polytopes, specifically starting from regular polytopes to construct new uniform polytopes. The ''expansion'' ope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentellated 6-cube
In six-dimensional geometry, a pentellated 6-cube is a convex uniform 6-polytope with 5th order truncations of the regular 6-cube. There are unique 16 degrees of pentellations of the 6-cube with permutations of truncations, cantellations, runcinations, and sterications. The simple pentellated 6-cube is also called an expanded 6-cube, constructed by an expansion operation applied to the regular 6-cube. The highest form, the ''pentisteriruncicantitruncated 6-cube'', is called an ''omnitruncated 6-cube'' with all of the nodes ringed. Six of them are better constructed from the 6-orthoplex given at pentellated 6-orthoplex. Pentellated 6-cube Alternate names * Pentellated 6-orthoplex * Expanded 6-cube, expanded 6-orthoplex * Small teri-hexeractihexacontitetrapeton (Acronym: stoxog) (Jonathan Bowers) Images Pentitruncated 6-cube Alternate names * Teritruncated hexeract (Acronym: tacog) (Jonathan Bowers) Images Penticantellated 6-cube Alternate names * Terirhomba ... [...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 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 truncated icosidodecahedr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform 6-polytope
In six-dimensional geometry, a uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes. The complete set of convex uniform 6-polytopes has not been determined, but most can be made as Wythoff constructions from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter-Dynkin diagrams. Each combination of at least one ring on every connected group of nodes in the diagram produces a uniform 6-polytope. The simplest uniform polypeta are regular polytopes: the 6-simplex , the 6-cube (hexeract) , and the 6-orthoplex (hexacross) . History of discovery * Regular polytopes: (convex faces) ** 1852: Ludwig Schläfli proved in his manuscript ''Theorie der vielfachen Kontinuität'' that there are exactly 3 regular polytopes in 5 or more dimensions. * Convex semiregular polytopes: (Various definitions before Coxeter's uniform category) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
