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Stericated 6-demicube
In six-dimensional geometry, a pentic 6-cube is a convex uniform 6-polytope. There are 8 pentic forms of the 6-cube. Pentic 6-cube The ''pentic 6-cube'', , has half of the vertices of a pentellated 6-cube, . Alternate names * Stericated 6-demicube/demihexeract * Small cellated hemihexeract (Acronym: sochax) (Jonathan Bowers) Cartesian coordinates The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: : (±1,±1,±1,±1,±1,±3) with an odd number of plus signs. Images Penticantic 6-cube The ''penticantic 6-cube'', , has half of the vertices of a penticantellated 6-cube, . Alternate names * Steritruncated 6-demicube/demihexeract * cellitruncated hemihexeract (Acronym: cathix) (Jonathan Bowers) Cartesian coordinates The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: : (±1,±1,±3,±3,±3,±5) with an odd number of plus signs. Images Pentiruncic 6-cube The ''pentiruncic 6-cub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6-demicube T0 D5
In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a ''6-cube'' (hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM6 for a 6-dimensional ''half measure'' polytope. Coxeter named this polytope as 131 from its Coxeter diagram, with a ring on one of the 1-length branches, . It can named similarly by a 3-dimensional exponential Schläfli symbol \left\ or . Cartesian coordinates Cartesian coordinates for the vertices of a demihexeract centered at the origin are alternate halves of the hexeract: : (±1,±1,±1,±1,±1,±1) with an odd number of plus signs. As a configuration This configuration matrix represents the 6-demicube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-demicub ... [...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] |
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] |
Pentisteriruncicantellated 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] |
Pentiruncicantellated 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] |
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] |
Penticantellated 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] |
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 ... [...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] |
