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Great Icosidodecahedron
In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U54. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 30 vertices. It is given a Schläfli symbol r. It is the rectification of the great stellated dodecahedron and the great icosahedron. It was discovered independently by , and . Related polyhedra The name is constructed analogously as how a cube-octahedron creates a cuboctahedron, and how a dodecahedron-icosahedron creates a (small) icosidodecahedron. It shares the same vertex arrangement with the icosidodecahedron, its convex hull. Unlike the great icosahedron and great dodecahedron, the great icosidodecahedron is not a stellation of the icosidodecahedron, but a faceting of it instead. It also shares its edge arrangement with the great icosihemidodecahedron (having the triangular faces in common), and with the great dodecahemidodecahedron (having the pentagrammic faces in common). This polyhedron can be considered a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Icosidodecahedron
In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U54. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 30 vertices. It is given a Schläfli symbol r. It is the rectification of the great stellated dodecahedron and the great icosahedron. It was discovered independently by , and . Related polyhedra The name is constructed analogously as how a cube-octahedron creates a cuboctahedron, and how a dodecahedron-icosahedron creates a (small) icosidodecahedron. It shares the same vertex arrangement with the icosidodecahedron, its convex hull. Unlike the great icosahedron and great dodecahedron, the great icosidodecahedron is not a stellation of the icosidodecahedron, but a faceting of it instead. It also shares its edge arrangement with the great icosihemidodecahedron (having the triangular faces in common), and with the great dodecahemidodecahedron (having the pentagrammic faces in common). This polyhedron can be considered a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Dodecahemidodecahedron
In geometry, the great dodecahemidodecahedron is a nonconvex uniform polyhedron, indexed as U70. It has 18 faces (12 Pentagram, pentagrams and 6 Decagram (geometry), decagrams), 60 edges, and 30 vertices. Its vertex figure is a quadrilateral#More quadrilaterals, crossed quadrilateral. Aside from the regular small stellated dodecahedron and great stellated dodecahedron , it is the only nonconvex uniform polyhedron whose faces are all non-convex regular polygons (star polygons), namely the star polygons pentagram, and decagram (geometry), . It is a hemipolyhedron with 6 Decagram (geometry), decagrammic faces passing through the model center. Related polyhedra Its convex hull is the icosidodecahedron. It also shares its edge arrangement with the great icosidodecahedron (having the pentagrammic faces in common) and the great icosihemidodecahedron (having the decagrammic faces in common). Gallery See also * List of uniform polyhedra References External links * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Rhombic Triacontahedron
In geometry, the great rhombic triacontahedron is a nonconvex isohedral, isotoxal polyhedron. It is the dual of the great icosidodecahedron (U54). Like the convex rhombic triacontahedron it has 30 rhombic faces, 60 edges and 32 vertices (also 20 on 3-fold and 12 on 5-fold axes). It can be constructed from the convex solid by expanding the faces by factor of \varphi^3 \approx 4.236, where \varphi\! is the golden ratio. This solid is to the compound of great icosahedron and great stellated dodecahedron what the convex one is to the compound of dodecahedron and icosahedron: The crossing edges in the dual compound are the diagonals of the rhombs. What resembles an "excavated" rhombic triacontahedron (compare excavated dodecahedron and excavated icosahedron) can be seen within the middle of this compound. The rest of the polyhedron strikingly resembles a rhombic hexecontahedron. The rhombs have two angles of \arccos(\frac\sqrt)\approx 63.434\,948\,822\,92^, and two of \arccos(-\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Truncated Icosahedron
In geometry, the truncated great icosahedron (or great truncated icosahedron) is a nonconvex uniform polyhedron, indexed as U55. It has 32 faces (12 pentagrams and 20 hexagons), 90 edges, and 60 vertices. It is given a Schläfli symbol t or t0,1 as a truncated great icosahedron. Cartesian coordinates Cartesian coordinates for the vertices of a ''truncated great icosahedron'' centered at the origin are all the even permutations of : (±1, 0, ±3/τ) : (±2, ±1/τ, ±1/τ3) : (±(1+1/τ2), ±1, ±2/τ) where τ = (1+√5)/2 is the golden ratio (sometimes written φ). Using 1/τ2 = 1 − 1/τ one verifies that all vertices are on a sphere, centered at the origin, with the radius squared equal to 10−9/τ. The edges have length 2. Related polyhedra This polyhedron is the truncation of the great icosahedron: The truncated ''great stellated dodecahedron'' is a degenerate polyhedron, with 20 triangular faces from the truncated vertices, and 12 (hidden) pentagonal faces as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . 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 , con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Diagram
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 ide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Icosahedron
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. Images As a snub The ''great icosahedron'' can be constructed a uniform snub, with different colored faces and only tetrahedral symmetry: . This construction can be called a ''retrosnub tetrahedron'' or ''retrosnub tetratetrahedron'', similar to the snub tetrahedron symmetry of the icosahedron, as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Great Icosahedron
In geometry, the truncated great icosahedron (or great truncated icosahedron) is a nonconvex uniform polyhedron, indexed as U55. It has 32 faces (12 pentagrams and 20 hexagons), 90 edges, and 60 vertices. It is given a Schläfli symbol t or t0,1 as a truncated great icosahedron. Cartesian coordinates Cartesian coordinates for the vertices of a ''truncated great icosahedron'' centered at the origin are all the even permutations of : (±1, 0, ±3/τ) : (±2, ±1/τ, ±1/τ3) : (±(1+1/τ2), ±1, ±2/τ) where τ = (1+√5)/2 is the golden ratio (sometimes written φ). Using 1/τ2 = 1 − 1/τ one verifies that all vertices are on a sphere, centered at the origin, with the radius squared equal to 10−9/τ. The edges have length 2. Related polyhedra This polyhedron is the truncation of the great icosahedron: The truncated ''great stellated dodecahedron'' is a degenerate polyhedron, with 20 triangular faces from the truncated vertices, and 12 (hidden) pentagonal faces as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Great Stellated Dodecahedron
In geometry, the small complex icosidodecahedron is a degenerate uniform star polyhedron. Its edges are doubled, making it degenerate. The star has 32 faces (20 triangles and 12 pentagons), 60 (doubled) edges and 12 vertices and 4 sharing faces. The faces in it are considered as two overlapping edges as topological polyhedron. A small complex icosidodecahedron can be constructed from a number of different vertex figures. A very similar figure emerges as a geometrical truncation of the great stellated dodecahedron, where the pentagram faces become doubly-wound pentagons ( --> ), making the internal pentagonal planes, and the three meeting at each vertex become triangles, making the external triangular planes. As a compound The small complex icosidodecahedron can be seen as a compound of the icosahedron and the great dodecahedron where all vertices are precise and edges coincide. The small complex icosidodecahedron resembles an icosahedron, because the great dodecahedron is c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Stellated Dodecahedron
In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each vertex. It shares its vertex arrangement, although not its vertex figure or vertex configuration, with the regular dodecahedron, as well as being a stellation of a (smaller) dodecahedron. It is the only dodecahedral stellation with this property, apart from the dodecahedron itself. Its dual, the great icosahedron, is related in a similar fashion to the icosahedron. Shaving the triangular pyramids off results in an icosahedron. If the pentagrammic faces are broken into triangles, it is topologically related to the triakis icosahedron, with the same face connectivity, but much taller isosceles triangle faces. If the triangles are instead made to invert themselves and excavate the central icosahedron, the result is a great dodecahedron. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degeneracy (mathematics)
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case. The definitions of many classes of composite or structured objects often implicitly include inequalities. For example, the angles and the side lengths of a triangle are supposed to be positive. The limiting cases, where one or several of these inequalities become equalities, are degeneracies. In the case of triangles, one has a ''degenerate triangle'' if at least one side length or angle is zero. Equivalently, it becomes a "line segment". Often, the degenerate cases are the exceptional cases where changes to the usual dimension or the cardinality of the object (or of some part of it) occur. For example, a triangle is an object of dimension two, and a degenerate triangle is contained in a line, which makes its dimension one. This is similar ... [...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 (geometry), 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 truncate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
