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Truncated Small Rhombicuboctahedron
The truncated rhombicuboctahedron is a polyhedron, constructed as a truncation (geometry), truncation of the rhombicuboctahedron. It has 50 faces consisting of 18 octagons, 8 hexagons, and 24 squares. It can fill space with the truncated cube, truncated tetrahedron and triangular prism as a truncated runcic cubic honeycomb. Other names *Truncated small rhombicuboctahedron *Beveled cuboctahedron Zonohedron As a zonohedron, it can be constructed with all but 12 octagons as regular polygons. It has two sets of 48 vertices existing on two distances from its center. It represents the Minkowski sum of a cube, a truncated octahedron, and a rhombic dodecahedron. Excavated truncated rhombicuboctahedron The excavated truncated rhombicuboctahedron is a toroidal polyhedron, constructed from a truncated rhombicuboctahedron with its 12 irregular octagonal faces removed. It comprises a network of 6 square cupolae, 8 triangular cupolae, and 24 triangular prisms. It has 148 faces (8 tri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Rhombicuboctahedron2
Truncation is the term used for limiting the number of digits right of the decimal point by discarding the least significant ones. Truncation may also refer to: Mathematics * Truncation (statistics) refers to measurements which have been cut off at some value * Truncation error, Truncation (numerical analysis) refers to truncating an infinite sum by a finite one * Truncation (geometry) is the removal of one or more parts, as for example in truncated cube * Propositional truncation, a type former which truncates a type down to a mere proposition Computer science * Data truncation, an event that occurs when a file or other data is stored in a location too small to accommodate its entire length * Truncate (SQL), a command in the SQL data manipulation language to quickly remove all data from a table Biology * Truncate, a leaf shape * Truncated protein, a protein shortened by a mutation which specifically induces premature termination of messenger RNA translation Other uses * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski Sum
In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set : A + B = \. Analogously, the Minkowski difference (or geometric difference) is defined using the complement operation as : A - B = \left(A^c + (-B)\right)^c In general A - B \ne A + (-B). For instance, in a one-dimensional case A = 2, 2/math> and B = 1, 1/math> the Minkowski difference A - B = 1, 1/math>, whereas A + (-B) = A + B = 3, 3 In a two-dimensional case, Minkowski difference is closely related to erosion (morphology) in image processing. The concept is named for Hermann Minkowski. Example For example, if we have two sets ''A'' and ''B'', each consisting of three position vectors (informally, three points), representing the vertices of two triangles in \mathbb^2, with coordinates :A = \ and :B = \ then their Minkowski sum is :A + B = \ which comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points around the face. When done, these lines form a complete circuit, i.e. a polygon, around the vertex. This polygon is the vertex figure. More precise formal definitions can vary quite widely, according to circumstance. For example Coxeter (e.g. 1948, 1954) varies his definition as convenient for the current area of discussion. Most of the following definitions of a vertex figure apply equally well to infinite tessellation, tilings or, by extension, to Honeycomb (geometry), space-filling tessellation with polytope Cell (geometry), cells and other higher-dimensional polytopes. As a flat slice Make a slice through the corner of the polyhedron, cutting through all the edges ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Cuboctahedron
In geometry, the truncated cuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated cuboctahedron is a 9-zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism. Names There is a nonconvex uniform polyhedron with a similar name: the nonconvex great rhombicuboctahedron. Cartesian coordinates The Cartesian coordinates for the vertices of a truncated cuboctahedron having edge length 2 and centered at the origin are all the permutations of: :(±1, ±(1 + ), ±(1 + 2)). Area and volume The area ''A'' and the volume ''V'' of the truncated cuboctahedron of edge length ''a'' are: :\begin A &= 12\left(2+\sqrt+\sqrt\right) a^2 &&\approx 61.755\,1724~a^2, \\ V &= \left(22+14\sqrt\right) a^3 &&\approx 41. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Cuboctahedron
In geometry, the truncated cuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated cuboctahedron is a 9-zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism. Names There is a nonconvex uniform polyhedron with a similar name: the nonconvex great rhombicuboctahedron. Cartesian coordinates The Cartesian coordinates for the vertices of a truncated cuboctahedron having edge length 2 and centered at the origin are all the permutations of: :(±1, ±(1 + ), ±(1 + 2)). Area and volume The area ''A'' and the volume ''V'' of the truncated cuboctahedron of edge length ''a'' are: :\begin A &= 12\left(2+\sqrt+\sqrt\right) a^2 &&\approx 61.755\,1724~a^2, \\ V &= \left(22+14\sqrt\right) a^3 &&\approx 41. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Excavated Truncated Cuboctahedron '', a spider
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Excavation may refer to: * Excavation (archaeology) * Excavation (medicine) * ''Excavation'' (The Haxan Cloak album), 2013 * ''Excavation'' (Ben Monder album), 2000 * ''Excavation'' (novel), a 2000 novel by James Rollins * '' Excavation: A Memoir'', a 2014 memoir by Wendy C. Ortiz * ''Excavation'' (video game), a 2003 video game by WildTangent See also *Excavate (other) *Excavator (other) * Excavata, a taxonomic grouping of eukaryotic unicellular organisms *''Celaenia excavata ''Celaenia excavata'', the bird dropping spider of Australia and New Zealand, derives its name from mimicking bird droppings to avoid predators, mainly birds. However, there are other species of spider that resemble bird droppings, for example s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Excavated Truncated Rhombicuboctahedron '', a spider
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{{disambig ...
Excavation may refer to: * Excavation (archaeology) * Excavation (medicine) * ''Excavation'' (The Haxan Cloak album), 2013 * ''Excavation'' (Ben Monder album), 2000 * ''Excavation'' (novel), a 2000 novel by James Rollins * '' Excavation: A Memoir'', a 2014 memoir by Wendy C. Ortiz * ''Excavation'' (video game), a 2003 video game by WildTangent See also *Excavate (other) *Excavator (other) * Excavata, a taxonomic grouping of eukaryotic unicellular organisms *''Celaenia excavata ''Celaenia excavata'', the bird dropping spider of Australia and New Zealand, derives its name from mimicking bird droppings to avoid predators, mainly birds. However, there are other species of spider that resemble bird droppings, for example s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Cupola
In geometry, the triangular cupola is one of the Johnson solids (). It can be seen as half a cuboctahedron. Formulae The following formulae for the volume (V), the surface area (A) and the height (H) can be used if all faces are regular, with edge length ''a'': :V=\left(\frac\right) a^3\approx1.17851...a^3 :A=\left(3+\frac \right) a^2\approx7.33013...a^2 :H = \frac a\approx 0.816496...a Dual polyhedron The dual of the triangular cupola has 6 triangular and 3 kite faces: Related polyhedra and honeycombs The triangular cupola can be augmented by 3 square pyramids, leaving adjacent coplanar faces. This isn't a Johnson solid because of its coplanar faces. Merging those coplanar triangles into larger ones, topologically this is another triangular cupola with isosceles trapezoidal side faces. If all the triangles are retained and the base hexagon is replaced by 6 triangles, it generates a coplanar deltahedron with 22 faces. : The triangular cupola can form a tessel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Cupola
In geometry, the square cupola, sometimes called lesser dome, is one of the Johnson solids (). It can be obtained as a slice of the rhombicuboctahedron. As in all cupolae, the base polygon has twice as many edges and vertices as the top; in this case the base polygon is an octagon. Formulae The following formulae for the circumradius, surface area, volume, and height can be used if all faces are regular, with edge length ''a'': :C=\left(\frac\sqrt\right)a\approx1.39897a, :A=\left(7+2\sqrt+\sqrt\right)a^2\approx11.56048a^2, :V=\left(1+\frac\right)a^3\approx1.94281a^3. :h = \fraca \approx 0.70711a Related polyhedra and honeycombs Other convex cupolae Dual polyhedron The dual of the square cupola has 8 triangular and 4 kite faces: Crossed square cupola The crossed square cupola is one of the nonconvex Johnson solid isomorphs, being topologically identical to the convex square cupola. It can be obtained as a slice of the nonconvex great rhombicuboctahedron or qu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toroidal Polyhedron
In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a -holed torus), having a topological genus () of 1 or greater. Notable examples include the Császár and Szilassi polyhedra. Variations in definition Toroidal polyhedra are defined as collections of polygons that meet at their edges and vertices, forming a manifold as they do. That is, each edge should be shared by exactly two polygons, and at each vertex the edges and faces that meet at the vertex should be linked together in a single cycle of alternating edges and faces, the link of the vertex. For toroidal polyhedra, this manifold is an orientable surface. Some authors restrict the phrase "toroidal polyhedra" to mean more specifically polyhedra topologically equivalent to the (genus 1) torus. In this area, it is important to distinguish embedded toroidal polyhedra, whose faces are flat polygons in three-dimensional Euclidean space that do not cross themselves or each other, from abstract pol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Genus (mathematics)
In mathematics, genus (plural genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1. Topology Orientable surfaces The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It is equal to the number of handles on it. Alternatively, it can be defined in terms of the Euler characteristic ''χ'', via the relationship ''χ'' = 2 − 2''g'' for closed surfaces, where ''g'' is the genus. For surfaces with ''b'' boundary components, the equation reads ''χ'' = 2 − 2''g'' − ''b''. In layman's terms, it's the number of "holes" an object has ("holes" interpreted in the sense of doughnut holes; a hollow sphere would be considered as having zero holes in this sense). A torus has 1 such h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 lower-case letter chi). The Euler characteristic was originally defined for 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 and, more abstractly, homological algebra. Polyhedra The Euler characteristic \chi was classically defined for the surfaces of polyhedra, acc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
