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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 . Classifications As an Archimedean solid A truncated octahedron is constructed from a regular octahedron by cutting off all vertices. This resulting polyhedron has six squares and eight hexagons, leaving ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallelohedron
In geometry, a parallelohedron or Fedorov polyhedron is a convex polyhedron that can be Translation (geometry), translated without rotations to fill Euclidean space, producing a Honeycomb (geometry), honeycomb in which all copies of the polyhedron meet face-to-face. Evgraf Fedorov identified the five types of parallelohedron in 1885 in his studies of crystallographic systems. They are the cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, and truncated octahedron. Each parallelohedron is centrally symmetric with symmetric faces, making it a special case of a zonohedron. Each parallelohedron is also a stereohedron, a polyhedron that tiles space so that isohedral tiling, all tiles are symmetric. The centers of the tiles in a tiling of space by parallelohedra form a Bravais lattice, and every Bravais lattice can be formed in this way. Adjusting the lengths of parallel edges in a parallelohedron, or performing an affine transformation of the parallelohedron, resul ... [...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 . Classifications As an Archimedean solid A truncated octahedron is constructed from a regular octahedron by cutting off all vertices. This resulting polyhedron has six squares and eight hexagons, leaving ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedean Solid
The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygon and are vertex-transitive, although they aren't face-transitive. The solids were named after Archimedes, although he did not claim credit for them. They belong to the class of uniform polyhedra, the polyhedra with regular faces and symmetric vertices. Some Archimedean solids were portrayed in the works of artists and mathematicians during the Renaissance. The elongated square gyrobicupola or ' is an extra polyhedron with regular faces and congruent vertices, but it is not generally counted as an Archimedean solid because it is not vertex-transitive. The solids The Archimedean solids have a single vertex configuration and highly symmetric properties. A vertex configuration indicates which regular polygons meet at each vertex. For instance, the configuration 3 \cdot 5 \cdot 3 \cdot 5 indicates a polyhedron in which each vertex is met by alternating two triangles and two pentagons. Highl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Translation (geometry)
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same Distance geometry, distance in a given direction (geometry), direction. A translation can also be interpreted as the addition of a constant vector space, vector to every point, or as shifting the Origin (mathematics), origin of the coordinate system. In a Euclidean space, any translation is an isometry. As a function If \mathbf is a fixed vector, known as the ''translation vector'', and \mathbf is the initial position of some object, then the translation function T_ will work as T_(\mathbf)=\mathbf+\mathbf. If T is a translation, then the image (mathematics), image of a subset A under the function (mathematics), function T is the translate of A by T . The translate of A by T_ is often written as A+\mathbf . Application in classical physics In classical physics, translational motion is movement that changes the Position (geometry), positio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Delone Set
In the mathematical theory of metric spaces, -nets, -packings, -coverings, uniformly discrete sets, relatively dense sets, and Delone sets (named after Boris Delone) are several closely related definitions of well-spaced sets of points, and the packing radius and covering radius of these sets measure how well-spaced they are. These sets have applications in coding theory, approximation algorithms, and the theory of quasicrystals. Definitions If is a metric space, and is a subset of , then the packing radius, , of is half of the smallest distance between distinct members of . Open balls of radius centered at the points of will all be disjoint from each other. The covering radius, , of is the smallest distance such that every point of is within distance of at least one point in ; that is, is the smallest radius such that closed balls of that radius centered at the points of have all of as their union. An -packing is a set of packing radius (equivalently, minimu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Voronoi Cell
In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. It can be classified also as a tessellation. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed there is a corresponding region, called a Voronoi cell, consisting of all points of the plane closer to that seed than to any other. The Voronoi diagram of a set of points is dual to that set's Delaunay triangulation. The Voronoi diagram is named after mathematician Georgy Voronoy, and is also called a Voronoi tessellation, a Voronoi decomposition, a Voronoi partition, or a Dirichlet tessellation (after Peter Gustav Lejeune Dirichlet). Voronoi cells are also known as Thiessen polygons, after Alfred H. Thiessen. Voronoi diagrams have practical and theoretical applications in many fields, mainly in science and technology, but also in visual art. Simplest case In the simplest case, shown in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plesiohedron
In geometry, a plesiohedron is a special kind of space-filling polyhedron, defined as the Voronoi cell of a symmetric Delone set. Three-dimensional Euclidean space can be completely filled by copies of any one of these shapes, with no overlaps. The resulting honeycomb will have symmetries that take any copy of the plesiohedron to any other copy. The plesiohedra include such well-known shapes as the cube, hexagonal prism, rhombic dodecahedron, and truncated octahedron. The largest number of faces that a plesiohedron can have is 38. Definition A set S of points in Euclidean space is a Delone set if there exists a number \varepsilon>0 such that every two points of S are at least at distance \varepsilon apart from each other and such that every point of space is within distance 1/\varepsilon of at least one point in S. So S fills space, but its points never come too close to each other. For this to be true, S must be infinite. Additionally, the set S is symmetric (in the sense needed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \mathrm_n defined over a finite set of n symbols consists of the permutations that can be performed on the n symbols. Since there are n! (n factorial) such permutation operations, the order (number of elements) of the symmetric group \mathrm_n is n!. Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley Graph
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, is a Graph (discrete mathematics), graph that encodes the abstract structure of a group (mathematics), group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley), and uses a specified generating set of a group, set of generators for the group. It is a central tool in combinatorial group theory, combinatorial and geometric group theory. The structure and symmetry of Cayley graphs make them particularly good candidates for constructing expander graphs. Definition Let G be a group (mathematics), group and S be a generating set of a group, generating set of G. The Cayley graph \Gamma = \Gamma(G,S) is an Edge coloring, edge-colored directed graph constructed as follows: In his Collected Mathematical Papers 10: 403–405. * Each element g of G is assigned a vertex: the vertex set of \Gamma is identified with G. * Each element s of S is assigned a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Coordinates
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular oriented lines, called '' coordinate lines'', ''coordinate axes'' or just ''axes'' (plural of ''axis'') of the system. The point where the axes meet is called the '' origin'' and has as coordinates. The axes directions represent an orthogonal basis. The combination of origin and basis forms a coordinate frame called the Cartesian frame. Similarly, the position of any point in three-dimensional space can be specified by three ''Cartesian coordinates'', which are the signed distances from the point to three mutually perpendicular planes. More generally, Cartesian coordinates specify the point in an -dimensional Euclidean space for any dimension . These coordinates are the signed distances from the point to mutually perpendicular fixed h ... [...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 general -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 conn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
