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Great Icosahedron
In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra (nonconvex List of regular polytopes#Non-convex 2, regular polyhedra), with Schläfli symbol and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence. The great icosahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the -dimensional simplex faces of the core -polytope (equilateral triangles for the great icosahedron, and line segments for the pentagram) until the figure regains regular faces. The grand 600-cell can be seen as its four-dimensional analogue using the same process. Construction The edge length of a great icosahedron is \frac times that of the original icosahedron. Images Formulas For a great icosahedron with edge length E (the edge of its dodecahedron core), \text = \frac \text = \frac \text = \frac \text = 3\sqrt(5+4 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stellation
In geometry, stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original. The word ''stellation'' comes from the Latin ''stellātus'', "starred", which in turn comes from the Latin ''stella'', "star". Stellation is the reciprocal or dual process to '' faceting''. Kepler's definition In 1619 Kepler defined stellation for polygons and polyhedra as the process of extending edges or faces until they meet to form a new polygon or polyhedron. He stellated the regular dodecahedron to obtain two regular star polyhedra, the small stellated dodecahedron and the great stellated dodecahedron. He also stellated the regular oct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Retrosnub Tetrahedron
In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra ( nonconvex regular polyhedra), with Schläfli symbol and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence. The great icosahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the -dimensional simplex faces of the core -polytope (equilateral triangles for the great icosahedron, and line segments for the pentagram) until the figure regains regular faces. The grand 600-cell can be seen as its four-dimensional analogue using the same process. Construction The edge length of a great icosahedron is \frac times that of the original icosahedron. Images Formulas For a great icosahedron with edge length E (the edge of its dodecahedron core), \text = \frac \text = \frac \text = \frac \text = 3\sqrt(5+4\sqrt)\text^2 \text = \text^3 As ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pyritohedral Symmetry
150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation. The group of all (not necessarily orientation preserving) symmetries is isomorphic to the group S4, the symmetric group of permutations of four objects, since there is exactly one such symmetry for each permutation of the vertices of the tetrahedron. The set of orientation-preserving symmetries forms a group referred to as the alternating subgroup A4 of S4. Details Chiral and full (or achiral tetrahedral symmetry and pyritohedral symmetry) are discrete point symmetries (or equivalently, symmetries on the sphere). They are among the crystallographic point groups of the cubic crystal system. Seen in stereographic projection the edges of the tetrakis hexahedron form 6 circles (or centrally radial lines) in the plane. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Omnitruncation
In geometry, an omnitruncation of a convex polytope is a simple polytope of the same dimension, having a vertex for each flag of the original polytope and a facet for each face of any dimension of the original polytope. Omnitruncation is the dual operation to barycentric subdivision. Because the barycentric subdivision of any polytope can be realized as another polytope, the same is true for the omnitruncation of any polytope. When omnitruncation is applied to a regular polytope (or honeycomb) it can be described geometrically as a Wythoff construction that creates a maximum number of facets. It is represented in a Coxeter–Dynkin diagram with all nodes ringed. It is a ''shortcut'' term which has a different meaning in progressively-higher-dimensional polytopes: * Uniform polytope truncation operators ** For regular polygons: An ordinary truncation, t_\ = t\ = \. *** Coxeter-Dynkin diagram ** For uniform polyhedra (3-polytopes): A cantitruncation, t_\ = tr\. (Application ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Octahedron
In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Square (geometry), squares), 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a 6-zonohedron. It is also the Goldberg polyhedron GIV(1,1), containing square and hexagonal faces. Like the cube, it can tessellate (or "pack") 3-dimensional space, as a permutohedron. The truncated octahedron was called the "mecon" by Buckminster Fuller. Its dual polyhedron is the tetrakis hexahedron. If the original truncated octahedron has unit edge length, its dual tetrakis hexahedron has edge lengths and . Classifications As an Archimedean solid A truncated octahedron is constructed from a regular octahedron by cutting off all vertices. This resulting polyhedron has six squares and eight hexagons, leaving ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical than others. The best known is the ( convex, non- stellated) regular icosahedron—one of the Platonic solids—whose faces are 20 equilateral triangles. Regular icosahedra There are two objects, one convex and one nonconvex, that can both be called regular icosahedra. Each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. Both have icosahedral symmetry. The term "regular icosahedron" generally refers to the convex variety, while the nonconvex form is called a ''great icosahedron''. Convex regular icosahedron The convex regular icosahedron is usually referred to simply as the ''regular icosahedron'', one of the five regular Platonic solids, and is represented by its Schläfli symbol , contai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedral Symmetry
image:tetrahedron.svg, 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation. The group of all (not necessarily orientation preserving) symmetries is isomorphic to the group S4, the symmetric group of permutations of four objects, since there is exactly one such symmetry for each permutation of the vertices of the tetrahedron. The set of orientation-preserving symmetries forms a group referred to as the alternating group, alternating subgroup A4 of S4. Details Chiral and full (or achiral tetrahedral symmetry and pyritohedral symmetry) are Point groups in three dimensions, discrete point symmetries (or equivalently, List of spherical symmetry groups, symmetries on the sphere). They are among the Crystal system#Overview of point groups by crystal system, crystallographic point gro ... [...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 Chirality, chiral symmetry with two forms: with clockwise or counterclockwise orientation. By Kepler's names, a snub can be seen as an Expansion (geometry), 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 Horton Conway, 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, Conway snub operator, snub is defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Icosahedron Tiling
Great may refer to: Descriptions or measurements * Great, a relative measurement in physical space, see Size * Greatness, being divine, majestic, superior, majestic, or transcendent People * List of people known as "the Great" * Artel Great (born 1981), American actor * Great Osobor (born 2002), Spanish-born British basketball player Other uses * ''Great'' (1975 film), a British animated short about Isambard Kingdom Brunel * ''Great'' (2013 film), a German short film * Great (supermarket), a supermarket in Hong Kong * GReAT, Graph Rewriting and Transformation, a Model Transformation Language * Gang Resistance Education and Training Gang Resistance Education And Training, abbreviated G.R.E.A.T., provides a school-based, police officer-instructed program in America that includes classroom instruction and a variety of learning activities. The program was originally adminis ..., or GREAT, a school-based and police officer-instructed program * Global Research and Analysis Te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Tiling
In geometry, a spherical polyhedron or spherical tiling is a tessellation, tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called ''spherical polygons''. A polyhedron whose vertices are equidistant from its center can be conveniently studied by projecting its edges onto the sphere to obtain a corresponding spherical polyhedron. The most familiar spherical polyhedron is the Ball (association football), soccer ball, thought of as a spherical truncated icosahedron. The next most popular spherical polyhedron is the beach ball, thought of as a hosohedron. Some #Improper_cases, "improper" polyhedra, such as hosohedron, hosohedra and their dual polyhedron, duals, dihedron, dihedra, exist as spherical polyhedra, but their flat-faced analogs are Degeneracy (mathematics), degenerate. The example hexagonal beach ball, is a hosohedron, and is its dual dihedron. History During the 10th Century, the Islamic scholar Abū al-Wafā' Būz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
