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Snub Polyhedra
In geometry, a snub polyhedron is a polyhedron obtained by performing a snub operation: alternating a corresponding omnitruncated or truncated polyhedron, depending on the definition. Some, but not all, authors include antiprisms as snub polyhedra, as they are obtained by this construction from a degenerate "polyhedron" with only two faces (a dihedron). Chiral snub polyhedra do not always have reflection symmetry and hence sometimes have two ''enantiomorphous'' (left- and right-handed) forms which are reflections of each other. Their symmetry groups are all point groups. For example, the snub cube: Snub polyhedra have Wythoff symbol and by extension, vertex configuration . Retrosnub polyhedra (a subset of the snub polyhedron, containing the great icosahedron, small retrosnub icosicosidodecahedron, and great retrosnub icosidodecahedron) still have this form of Wythoff symbol, but their vertex configurations are instead List of snub polyhedra Uniform There are 12 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...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 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 . Construction A truncated octahedron is constructed from a regular octahedron with side length 3''a'' by the removal of six right square pyramids, one from each point. These pyramids have both base side len ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snub Tetrahedron
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] |
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] |
Great Snub Dodecicosidodecahedron
In geometry, the great snub dodecicosidodecahedron (or great snub dodekicosidodecahedron) is a nonconvex uniform polyhedron, indexed as U64. It has 104 faces (80 triangles and 24 pentagrams), 180 edges, and 60 vertices. It has Coxeter diagram, . It has the unusual feature that its 24 pentagram faces occur in 12 coplanar pairs. Related polyhedra It shares its vertices and edges, as well as 20 of its triangular faces and all its pentagrammic faces, with the great dirhombicosidodecahedron, (although the latter has 60 edges not contained in the great snub dodecicosidodecahedron). It shares its other 60 triangular faces (and its pentagrammic faces again) with the great disnub dirhombidodecahedron. The edges and triangular faces also occur in the compound of twenty octahedra. In addition, 20 of the triangular faces occur in one enantiomer of the compound of twenty tetrahemihexahedra, and the other 60 triangular faces occur in the other enantiomer. Gallery See also * List ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snub Icosidodecadodecahedron
In geometry, the snub icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U46. It has 104 faces (80 triangles, 12 pentagons, and 12 pentagrams), 180 edges, and 60 vertices. As the name indicates, it belongs to the family of snub polyhedra. The circumradius of the snub icosidodecadodecahedron with unit edge length is :\frac\sqrt, where ρ is the plastic constant, or the unique real root of . Related polyhedra Medial hexagonal hexecontahedron The medial hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform snub icosidodecadodecahedron. See also * List of uniform polyhedra In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive ( transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are c ... References * External links * Uniform polyhedra {{Polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Snub Icosicosidodecahedron
In geometry, the small snub icosicosidodecahedron or snub disicosidodecahedron is a uniform star polyhedron, indexed as U32. It has 112 faces (100 triangles and 12 pentagrams), 180 edges, and 60 vertices. Its stellation core is a truncated pentakis dodecahedron. It also called a holosnub icosahedron, ß. The 40 non-snub triangular faces form 20 coplanar pairs, forming star hexagons that are not quite regular. Unlike most snub polyhedra, it has reflection symmetries. Convex hull Its convex hull is a nonuniform truncated icosahedron. Cartesian coordinates Cartesian coordinates for the vertices of a small snub icosicosidodecahedron are all the even permutations of : (±(1-ϕ+α), 0, ±(3+ϕα)) : (±(ϕ-1+α), ±2, ±(2ϕ-1+ϕα)) : (±(ϕ+1+α), ±2(ϕ-1), ±(1+ϕα)) where ϕ = (1+)/2 is the golden ratio and α = . See also * List of uniform polyhedra In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive ( t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isosceles Triangle
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids. The mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics. Isosceles triangles have been used as decoration from even earlier times, and appear frequently in architecture and design, for instance in the pediments and gables of buildings. The two equal sides are called the legs and the third side is called the base of the triangle. The other dimensions of the triangle, such as its height, area, and perimeter, can be calculated by simple formulas from the lengths of the legs an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwarz Triangle
In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere (spherical tiling), possibly overlapping, through reflections in its edges. They were classified in . These can be defined more generally as tessellations of the sphere, the Euclidean plane, or the hyperbolic plane. Each Schwarz triangle on a sphere defines a finite group, while on the Euclidean or hyperbolic plane they define an infinite group. A Schwarz triangle is represented by three rational numbers each representing the angle at a vertex. The value means the vertex angle is of the half-circle. "2" means a right triangle. When these are whole numbers, the triangle is called a Möbius triangle, and corresponds to a ''non''-overlapping tiling, and the symmetry group is called a triangle group. In the sphere there are three Möbius triangles plus one one-parameter family; in the plane there are three Möbius triangles, while in hyperbolic space there is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Disnub Dirhombidodecahedron
In geometry, the great disnub dirhombidodecahedron, also called ''Skilling's figure'', is a degenerate uniform star polyhedron. It was proven in 1970 that there are only 75 uniform polyhedra other than the infinite families of prisms and antiprisms. John Skilling discovered another degenerate example, the great disnub dirhombidodecahedron, by relaxing the condition that edges must be single. More precisely, he allowed any even amount of faces to meet at each edge, as long as the set of faces couldn't be separated into two connected sets (Skilling, 1975). Due to its geometric realization having some double edges where 4 faces meet, it is considered a degenerate uniform polyhedron but not strictly a uniform polyhedron. The number of edges is ambiguous, because the underlying abstract polyhedron has 360 edges, but 120 pairs of these have the same image in the geometric realization, so that the geometric realization has 120 single edges and 120 double edges where 4 faces meet, for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets. For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sphere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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