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Regular Polyhedra
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is identified by its Schläfli symbol of the form , where ''n'' is the number of sides of each face and ''m'' the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra (the Platonic solids), and four regular star polyhedra (the Kepler–Poinsot polyhedra), making nine regular polyhedra in all. In addition, there are five regular compounds of the regular polyhedra. The regular polyhedra There are five convex regular polyhedra, known as the Platonic solids; four regular star polyhedra, the Kepler–Poinsot polyhedra; and five regular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Polyhedron
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surface (mathematics), surface. The terms solid polyhedron and polyhedral surface are commonly used to distinguish the two concepts. Also, the term ''polyhedron'' is often used to refer implicitly to the whole structure (mathematics), structure formed by a solid polyhedron, its polyhedral surface, its faces, its edges, and its vertices. There are many definitions of polyhedron. Nevertheless, the polyhedron is typically understood as a generalization of a two-dimensional polygon and a three-dimensional specialization of a polytope, a more general concept in any number of dimensions. Polyhedra have several general characteristics that include the number of faces, topological classification by Eule ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Hexahedron
A hexahedron (: hexahedra or hexahedrons) or sexahedron (: sexahedra or sexahedrons) is any polyhedron with six faces. A cube, for example, is a regular hexahedron with all its faces square, and three squares around each vertex. There are seven topologically distinct ''convex'' hexahedra, one of which exists in two mirror image forms. Additional non-convex hexahedra exist, with their number depending on how polyhedra are defined. Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces. Convex Cuboid A hexahedron that is combinatorially equivalent to a cube may be called a cuboid, although this term is often used more specifically to mean a rectangular cuboid, a hexahedron with six rectangular sides. Different types of cuboids include the ones depicted and linked below. Others There a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Great Dodecahedron
In geometry, the great dodecahedron is one of four Kepler–Poinsot polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagons), intersecting each other making a pentagrammic path, with five pentagons meeting at each vertex. Construction One way to construct a great dodecahedron is by faceting the regular icosahedron. In other words, it is constructed from the regular icosahedron by removing its polygonal faces without changing or creating new vertices. For each vertex of the icosahedron, the five neighboring vertices become those of a regular pentagon face of the great dodecahedron. The resulting shape has a pentagram as its vertex figure, so its Schläfli symbol is \ . The great dodecahedron may also be interpreted as the ''second stellation of dodecahedron''. The construction started from a regular dodecahedron by attaching 12 pentagonal pyramids onto each of its faces, known as the ''first stellation''. The second stellation appears when 30 wed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Small Stellated Dodecahedron
In geometry, the small stellated dodecahedron is a Kepler–Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol . It is one of four nonconvex List of regular polytopes#Non-convex 2, regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex. It shares the same vertex arrangement as the convex regular icosahedron. It also shares the same edge arrangement with the great icosahedron, with which it forms Great complex icosidodecahedron, a degenerate uniform compound figure. It is the List of Wenninger polyhedron models#Stellations of dodecahedron, second of four stellations of the dodecahedron (including the original dodecahedron itself). The small stellated dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the edges (1-faces) of the core polytope until a point is reached where they intersect. Construction and properties The small stellated dodecahedron is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Euler Characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by \chi (Greek alphabet, Greek lower-case letter chi (letter), chi). The Euler characteristic was originally defined for polyhedron, polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In modern mathematics, the Euler characteristic arises from homology (mathematics), homology and, more abstractly, homological algebra. Polyhedra The Euler characteristic was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Regular Icosahedron
The regular icosahedron (or simply ''icosahedron'') is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with Regular polygon, regular faces to each of its pentagonal faces, or by putting points onto the cube. The resulting polyhedron has 20 equilateral triangles as its faces, 30 edges, and 12 vertices. It is an example of a Platonic solid and of a deltahedron. The icosahedral graph represents the Skeleton (topology), skeleton of a regular icosahedron. Many polyhedra are constructed from the regular icosahedron. A notable example is the stellation of regular icosahedron, which consists of 59 polyhedrons. The great dodecahedron, one of the Kepler–Poinsot polyhedra, is constructed by either stellation or faceting. Some of the Johnson solids can be constructed by removing the pentagonal pyramids. The regular icosahedron's dual polyhedron is the regular dodecahedron, and their relation has a historical background on the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Regular Dodecahedron
A regular dodecahedron or pentagonal dodecahedronStrictly speaking, a pentagonal dodecahedron need not be composed of regular pentagons. The name "pentagonal dodecahedron" therefore covers a wider class of solids than just the Platonic solid, the regular dodecahedron. is a dodecahedron composed of regular polygon, regular pentagonal faces, three meeting at each Vertex (geometry), vertex. It is an example of Platonic solids, described as cosmic stellation by Plato in his dialogues, and it was used as part of Solar System proposed by Johannes Kepler. However, the regular dodecahedron, including the other Platonic solids, has already been described by other philosophers since antiquity. The regular dodecahedron is a truncated trapezohedron because it is the result of Truncation (geometry), truncating axial vertices of a pentagonal trapezohedron. It is also a Goldberg polyhedron because it is the initial polyhedron to construct new polyhedrons by the process of chamfering. It has a re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It is a type of parallelepiped, with pairs of parallel opposite faces, and more specifically a rhombohedron, with congruent edges, and a rectangular cuboid, with right angles between pairs of intersecting faces and pairs of intersecting edges. It is an example of many classes of polyhedra: Platonic solid, regular polyhedron, parallelohedron, zonohedron, and plesiohedron. The dual polyhedron of a cube is the regular octahedron. The cube can be represented in many ways, one of which is the graph known as the cubical graph. It can be constructed by using the Cartesian product of graphs. The cube is the three-dimensional hypercube, a family of polytopes also including the two-dimensional square and four-dimensional tesseract. A cube with 1, unit s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |