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Hexagonal Dipyramid
A hexagonal bipyramid is a polyhedron formed from two hexagonal pyramids joined at their bases. The resulting solid has 12 triangular faces, 8 vertices and 18 edges. The 12 faces are identical isosceles triangles. Although it is face-transitive, it is not a Platonic solid because some vertices have four faces meeting and others have six faces, and it is not a Johnson solid because its faces cannot be equilateral triangles; 6 equilateral triangles would make a flat vertex. It is one of an infinite set of bipyramids. Having twelve faces, it is a type of dodecahedron, although that name is usually associated with the regular polyhedral form with pentagonal faces. The hexagonal bipyramid has a plane of symmetry (which is horizontal in the figure to the right) where the bases of the two pyramids are joined. This plane is a regular hexagon. There are also six planes of symmetry crossing through the two apices. These planes are rhombic and lie at 30° angles to each other, perp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipyramid
A (symmetric) -gonal bipyramid or dipyramid is a polyhedron formed by joining an -gonal pyramid and its mirror image base-to-base. An -gonal bipyramid has triangle faces, edges, and vertices. The "-gonal" in the name of a bipyramid does not refer to a face but to the internal polygon base, lying in the mirror plane that connects the two pyramid halves. (If it were a face, then each of its edges would connect three faces instead of two.) "Regular", right bipyramids A ''"regular"'' bipyramid has a ''regular'' polygon base. It is usually implied to be also a ''right'' bipyramid. A ''right'' bipyramid has its two apices ''right'' above and ''right'' below the center or the '' centroid'' of its polygon base. A "regular" right (symmetric) -gonal bipyramid has Schläfli symbol . A right (symmetric) bipyramid has Schläfli symbol , for polygon base . The "regular" right (thus face-transitive) -gonal bipyramid with regular vertices is the dual of the -gonal uniform (thus ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has Schläfli symbol and can also be constructed as a truncated equilateral triangle, t, which alternates two types of edges. A regular hexagon is defined as a hexagon that is both equilateral and equiangular. It is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle). The common length of the sides equals the radius of the circumscribed circle or circumcircle, which equals \tfrac times the apothem (radius of the inscribed circle). All internal angles are 120 degrees. A regular hexagon has six rotational symmetries (''rotational symmetry of order six'') and six reflection symmetries (''six lines of symmetry''), making up the dihedral group D6. The longest diagonals of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetry Group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object ''X'' is ''G'' = Sym(''X''). For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space. This article mainly considers symmetry groups in Euclidean geometry, but the concept may also be studied for more general types of geometric structure. Introduction We consider the "objects" possessing symmetry to be geometric figures, images, and patterns, such as a wallpaper pattern. For symmetry of physical objects, one may also take their physical composition as part of the pattern. (A pattern may be specified formally as a scalar fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Face Configuration
In geometry, a vertex configurationCrystallography of Quasicrystals: Concepts, Methods and Structures by Walter Steurer, Sofia Deloudi, (2009) pp. 18–20 and 51–53Physical Metallurgy: 3-Volume Set, Volume 1 edited by David E. Laughlin, (2014) pp. 16–20 is a shorthand notation for representing the of a or [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snub Rectified Hexagonal Bipyramid Sequence
A snub, cut or slight is a refusal to recognise an acquaintance by ignoring them, avoiding them or pretending not to know them. For example, a failure to greet someone may be considered a snub. In Awards and Lists For awards, the term "snub" is usually used to refer to a work or person that fails to be nominated or win award, with whether or not a person or work was legitimately snubbed for an award has often been subject for public debate. The term Snub has also been used in relation to lists, such as the NBA 75th Anniversary Team. Many of the most notable people and works have failed to be nominated or win a major award for example Alfred Hitchcock, Stanley Kubrick, and Spike Lee never won best director at the Oscars despite being nominated five, four, and one time respectively, Glenn Close, Peter O'Toole, and Cicely Tyson were also notable for having never won an Oscar related to acting despite each of them having multiple nominations. Among films, '' Citizen Kane'', '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectification (geometry)
In Euclidean geometry, rectification, also known as critical truncation or complete-truncation, is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope. A rectification operator is sometimes denoted by the letter with a Schläfli symbol. For example, is the rectified cube, also called a cuboctahedron, and also represented as \begin 4 \\ 3 \end. And a rectified cuboctahedron is a rhombicuboctahedron, and also represented as r\begin 4 \\ 3 \end. Conway polyhedron notation uses for ambo as this operator. In graph theory this operation creates a medial graph. The rectification of any regular self-dual polyhedron or tiling will result in another regular polyhedron or tiling with a tiling order of 4, for example the tetrahedron becoming an octahedron As a special case, a square tili ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snub Hexagonal Bipyramid Sequence
A snub, cut or slight is a refusal to recognise an acquaintance by ignoring them, avoiding them or pretending not to know them. For example, a failure to greet someone may be considered a snub. In Awards and Lists For awards, the term "snub" is usually used to refer to a work or person that fails to be nominated or win award, with whether or not a person or work was legitimately snubbed for an award has often been subject for public debate. The term Snub has also been used in relation to lists, such as the NBA 75th Anniversary Team. Many of the most notable people and works have failed to be nominated or win a major award for example Alfred Hitchcock, Stanley Kubrick, and Spike Lee never won best director at the Oscars despite being nominated five, four, and one time respectively, Glenn Close, Peter O'Toole, and Cicely Tyson were also notable for having never won an Oscar related to acting despite each of them having multiple nominations. Among films, '' Citizen Kane'', '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snub (geometry)
In geometry, a snub is an operation applied to a polyhedron. The term originates from Kepler's names of two Archimedean solids, for the snub cube () and snub dodecahedron (). In general, snubs have chiral symmetry with two forms: with clockwise or counterclockwise orientation. By Kepler's names, a snub can be seen as an expansion of a regular polyhedron: moving the faces apart, twisting them about their centers, adding new polygons centered on the original vertices, and adding pairs of triangles fitting between the original edges. The terminology was generalized by Coxeter, with a slightly different definition, for a wider set of uniform polytopes. Conway snubs John Conway explored generalized polyhedron operators, defining what is now called Conway polyhedron notation, which can be applied to polyhedra and tilings. Conway calls Coxeter's operation a ''semi-snub''. In this notation, snub is defined by the dual and gyro operators, as ''s'' = ''dg'', and it is e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
