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Simple Polytope
In geometry, a -dimensional simple polytope is a -dimensional polytope each of whose vertices are adjacent to exactly edges (also facets). The vertex figure of a simple -polytope is a -simplex. Simple polytopes are topologically dual to simplicial polytopes. The family of polytopes which are both simple and simplicial are simplices or two-dimensional polygons. A ''simple polyhedron'' is a three-dimensional polyhedron whose vertices are adjacent to three edges and three faces. The dual to a simple polyhedron is a ''simplicial polyhedron'', in which all faces are triangles. Examples Three-dimensional simple polyhedra include the prisms (including the cube), the regular tetrahedron and dodecahedron, and, among the Archimedean solids, the truncated tetrahedron, truncated cube, truncated octahedron, truncated cuboctahedron, truncated dodecahedron, truncated icosahedron, and truncated icosidodecahedron. They also include the Goldberg polyhedra and fullerenes, including the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Associahedron K5
In mathematics, an associahedron is an -dimensional convex polytope in which each vertex (geometry), vertex corresponds to a way of correctly inserting opening and closing parentheses in a string of letters, and the Edge (geometry), edges correspond to single application of the associativity rule. Equivalently, the vertices of an associahedron correspond to the polygon triangulation, triangulations of a regular polygon with sides and the edges correspond to edge flips in which a single diagonal is removed from a triangulation and replaced by a different diagonal. Associahedra are also called Stasheff polytopes after the work of Jim Stasheff, who rediscovered them in the early 1960s after earlier work on them by Dov Tamari (mathematician), Dov Tamari. Examples The one-dimensional associahedron ''K''3 represents the two parenthesizations ((''xy'')''z'') and (''x''(''yz'')) of three symbols, or the two triangulations of a square. It is itself a line segment. The two-dimensio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Truncated Tetrahedron
In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncation (geometry), truncating all 4 vertices of a regular tetrahedron. Construction The truncated tetrahedron can be constructed from a regular tetrahedron by cutting all of its vertices off, a process known as Truncation (geometry), truncation. The resulting polyhedron has 4 equilateral triangles and 4 regular hexagons, 18 edges, and 12 vertices. With edge length 1, the Cartesian coordinates of the 12 vertices are points \bigl( , \pm\tfrac, \pm\tfrac \bigr) that have an even number of minus signs. Properties Given the edge length a . The surface area of a truncated tetrahedron A is the sum of 4 regular hexagons and 4 equilateral triangles' area, and its volume V is: \begin A &= 7\sqrta^2 &&\approx 12.124a^2, \\ V &= \tfrac\sqrta^3 &&\approx 2.711a^3. \end The dihedral ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Truncated Trapezohedron
In geometry, an truncated trapezohedron is a polyhedron formed by a trapezohedron with Pyramid (geometry), pyramids Truncation (geometry), truncated from its two polar axis Vertex (geometry), vertices. The vertices exist as 4 in four parallel planes, with alternating orientation in the middle creating the pentagons. The regular dodecahedron is the most common polyhedron in this class, being a Platonic solid, with 12 Congruence (geometry), congruent pentagonal faces. A truncated trapezohedron has all vertices with 3 faces. This means that the dual polyhedra, the set of gyroelongated dipyramids, have all triangular faces. For example, the icosahedron is the dual of the dodecahedron. Forms *Triangular truncated trapezohedron (Dürer's solid) – 6 pentagons, 2 triangles, dual gyroelongated triangular dipyramid *Truncated square trapezohedron – 8 pentagons, 2 squares, dual gyroelongated square dipyramid *''Truncated pentagonal trapezohedron'' or regular dodecahedron – 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Chamfered Dodecahedron
In geometry, the chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer (edge-truncation) of a regular dodecahedron. The pentagons are reduced in size and new hexagonal faces are added in place of all the original edges. Its dual is the pentakis icosidodecahedron. It is also called a truncated rhombic triacontahedron, constructed as a truncation of the rhombic triacontahedron. It can more accurately be called an order-5 truncated rhombic triacontahedron because only the order-5 vertices are truncated. Structure These 12 order-5 vertices can be truncated such that all edges are equal length. The original 30 rhombic faces become non-regular hexagons, and the truncated vertices become regular pentagons. The hexagon faces can be equilateral but not regular with D symmetry. The angles at the two vertices with vertex configuration are \arccos\left(\frac\right) \approx 116.565^ and at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Chamfered Cube
In geometry, a chamfer or edge-truncation is a topological operator that modifies one polyhedron into another. It separates the Face (geometry), faces by reducing them, and adds a new face between each two adjacent faces (moving the vertices inward). Oppositely, similar to Expansion (geometry), expansion, it moves the faces apart outward, and adds a new face between each two adjacent faces; but contrary to expansion, it maintains the original Vertex (geometry), vertices. For a polyhedron, this operation adds a new hexagonal face in place of each original Edge (geometry), edge. In Conway polyhedron notation, ''chamfering'' is represented by the letter "c". A polyhedron with edges will have a chamfered form containing new vertices, new edges, and new hexagonal faces. Chamfered Platonic solids Chamfers of five Platonic solids are described in detail below. Each is shown in an Equilateral polygon, equilateral version where all edges have the same length, and in a canonic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Chamfered Tetrahedron
In geometry, a chamfer or edge-truncation is a topological operator that modifies one polyhedron into another. It separates the faces by reducing them, and adds a new face between each two adjacent faces (moving the vertices inward). Oppositely, similar to expansion, it moves the faces apart outward, and adds a new face between each two adjacent faces; but contrary to expansion, it maintains the original vertices. For a polyhedron, this operation adds a new hexagonal face in place of each original edge. In Conway polyhedron notation, ''chamfering'' is represented by the letter "c". A polyhedron with edges will have a chamfered form containing new vertices, new edges, and new hexagonal faces. Chamfered Platonic solids Chamfers of five Platonic solids are described in detail below. Each is shown in an equilateral version where all edges have the same length, and in a canonical version where all edges touch the same midsphere. The shown dual polyhedra are dual to the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Fullerene
A fullerene is an allotropes of carbon, allotrope of carbon whose molecules consist of carbon atoms connected by single and double bonds so as to form a closed or partially closed mesh, with fused rings of five to six atoms. The molecules may have hollow sphere- and ellipsoid-like forms, cylinder (geometry), tubes, or other shapes. Fullerenes with a closed mesh topology are informally denoted by their empirical formula C''n'', often written C''n'', where ''n'' is the number of carbon atoms. However, for some values of ''n'' there may be more than one isomer. The family is named after buckminsterfullerene (C60), the most famous member, which in turn is named after Buckminster Fuller. The closed fullerenes, especially C60, are also informally called buckyballs for their resemblance to the standard ball (association football), ball of association football. Nested closed fullerenes have been named bucky onions. Cylindrical fullerenes are also called carbon nanotubes or buckytubes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Goldberg Polyhedron
In mathematics, and more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described in 1937 by Michael Goldberg (mathematician), Michael Goldberg (1902–1990). They are defined by three properties: each Face (geometry), face is either a pentagon or hexagon, exactly three faces meet at each Vertex (geometry), vertex, and they have rotational icosahedral symmetry. They are not necessarily Reflection symmetry, mirror-symmetric; e.g. and are enantiomorphs of each other. A Goldberg polyhedron is a dual polyhedron of a geodesic polyhedron. A consequence of Euler's polyhedron formula is that a Goldberg polyhedron always has exactly 12 pentagonal faces. Icosahedral symmetry ensures that the pentagons are always regular polygon, regular and that there are always 12 of them. If the vertices are not constrained to a sphere, the polyhedron can be constructed with planar equilateral (but not in general ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Truncated Icosidodecahedron
In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron,Wenninger Model Number 16 great rhombicosidodecahedron,Williams (Section 3-9, p. 94)Cromwell (p. 82) omnitruncated dodecahedron or omnitruncated icosahedronNorman Woodason Johnson, "The Theory of Uniform Polytopes and Honeycombs", 1966 is an Archimedean solid, one of thirteen Convex polytope, convex, Isogonal figure, isogonal, non-Prism (geometry), prismatic solids constructed by two or more types of regular polygon, regular polygon Face (geometry), faces. It has 62 faces: 30 square (geometry), squares, 20 regular hexagons, and 12 regular decagons. It has the most edges and vertices of all Platonic solid, Platonic and Archimedean solids, though the snub dodecahedron has more faces. Of all vertex-transitive polyhedra, it occupies the largest percentage (89.80%) of the volume of a Circumscribed sphere, sphere in which it is inscribed, very narrowly beating the snub dodecahedron (89.63%) and small rhombicos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Truncated Icosahedron
In geometry, the truncated icosahedron is a polyhedron that can be constructed by Truncation (geometry), truncating all of the regular icosahedron's vertices. Intuitively, it may be regarded as Ball (association football), footballs (or soccer balls) that are typically patterned with white hexagons and black pentagons. It can be found in the application of geodesic dome structures such as those whose architecture Buckminster Fuller pioneered are often based on this structure. It is an example of an Archimedean solid, as well as a Goldberg polyhedron. Construction The truncated icosahedron can be constructed from a regular icosahedron by cutting off all of its vertices, known as Truncation (geometry), truncation. Each of the 12 vertices at the one-third mark of each edge creates 12 pentagonal faces and transforms the original 20 triangle faces into regular hexagons. Therefore, the resulting polyhedron has 32 faces, 90 edges, and 60 vertices. A Goldberg polyhedron is one whose f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Truncated Dodecahedron
In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges. Construction The truncated dodecahedron is constructed from a regular dodecahedron by cutting all of its vertices off, a process known as truncation. Alternatively, the truncated dodecahedron can be constructed by expansion: pushing away the edges of a regular dodecahedron, forming the pentagonal faces into decagonal faces, as well as the vertices into triangles. Therefore, it has 32 faces, 90 edges, and 60 vertices. The truncated dodecahedron may also be constructed by using Cartesian coordinates. With an edge length 2\varphi - 2 centered at the origin, they are all even permutations of \left(0, \pm \frac, \pm (2 + \varphi) \right), \qquad \left(\pm \frac, \pm \varphi, \pm 2 \varphi \right), \qquad \left(\pm \varphi, \pm 2, \pm (\varphi + 1) \right), where \varphi = \frac is the golden ratio. Properties The surfac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
