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Infinite-order Tetrahedral Honeycomb
In the geometry of hyperbolic 3-space, the order-7 tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol . It has seven tetrahedra around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many tetrahedra existing around each vertex in an order-7 triangular tiling vertex arrangement. Images Related polytopes and honeycombs It is a part of a sequence of regular polychora and honeycombs with tetrahedral cells, . It is a part of a sequence of hyperbolic honeycombs with order-7 triangular tiling vertex figures, . It is a part of a sequence of hyperbolic honeycombs, . Order-8 tetrahedral honeycomb In the geometry of hyperbolic 3-space, the order-8 tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol . It has eight tetrahedra around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many tetrahedra exis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Regular Polytopes
This article lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces. The Schläfli symbol describes every regular tessellation of an ''n''-sphere, Euclidean and hyperbolic spaces. A Schläfli symbol describing an ''n''-polytope equivalently describes a tessellation of an (''n'' − 1)-sphere. In addition, the symmetry of a regular polytope or tessellation is expressed as a Coxeter group, which Coxeter expressed identically to the Schläfli symbol, except delimiting by square brackets, a notation that is called Coxeter notation. Another related symbol is the Coxeter-Dynkin diagram which represents a symmetry group with no rings, and the represents regular polytope or tessellation with a ring on the first node. For example, the cube has Schläfli symbol , and with its octahedral symmetry, ,3or , it is represented by Coxeter diagram . The regular polytopes are grouped by dimension and subgrouped by convex, nonconve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Honeycomb 3-3-7 Poincare Cc
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they manifest hyperbolas, not because something about them is exaggerated. * Hyperbolic angle, an unbounded variable referring to a hyperbola instead of a circle * Hyperbolic coordinates, location by geometric mean and hyperbolic angle in quadrant I *Hyperbolic distribution, a probability distribution characterized by the logarithm of the probability density function being a hyperbola * Hyperbolic equilibrium point, a fixed point that does not have any center manifolds * Hyperbolic function, an analog of an ordinary trigonometric or circular function * Hyperbolic geometric graph, a random network generated by connecting nearby points sprinkled in a hyperbolic space * Hyperbolic geometry, a non-Euclidean geometry * Hyperbolic group, a finitely ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-3-7 Heptagonal Honeycomb
In the geometry of hyperbolic 3-space, the order-3-7 heptagonal honeycomb a regular space-filling tessellation (or honeycomb) with Schläfli symbol . Geometry All vertices are ultra-ideal (existing beyond the ideal boundary) with seven heptagonal tilings existing around each edge and with an order-7 triangular tiling vertex figure. Related polytopes and honeycombs It a part of a sequence of regular polychora and honeycombs : Order-3-8 octagonal honeycomb In the geometry of hyperbolic 3-space, the order-3-8 octagonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol . It has eight octagonal tilings, , around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octagonal tilings existing around each vertex in an order-8 triangular tiling vertex arrangement. It has a second construction as a uniform honeycomb, Schläfli symbol , Coxeter diagram, , with alternating types or colors of cells. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-3-7 Hexagonal Honeycomb
In the geometry of Hyperbolic space, hyperbolic 3-space, the order-3-7 hexagonal honeycomb or (6,3,7 honeycomb) a regular space-filling tessellation (or honeycomb (geometry), honeycomb) with Schläfli symbol . Geometry All vertices are ultra-ideal (existing beyond the ideal boundary) with seven hexagonal tilings existing around each edge and with an order-7 triangular tiling vertex figure. Related polytopes and honeycombs It a part of a sequence of regular polychora and honeycombs with hexagonal tiling cell (geometry), cells. Order-3-8 hexagonal honeycomb In the geometry of Hyperbolic space, hyperbolic 3-space, the order-3-8 hexagonal honeycomb or (6,3,8 honeycomb) is a regular space-filling tessellation (or honeycomb (geometry), honeycomb) with Schläfli symbol . It has eight hexagonal tilings, , around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an order-8 triangular ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-7 Dodecahedral Honeycomb
In the geometry of Hyperbolic space, hyperbolic 3-space, the order-7 dodecahedral honeycomb a regular space-filling tessellation (or honeycomb (geometry), honeycomb). Geometry With Schläfli symbol , it has seven dodecahedra around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many dodecahedra existing around each vertex in an order-7 triangular tiling vertex arrangement. Related polytopes and honeycombs It a part of a sequence of regular polytopes and honeycombs with Dodecahedron, dodecahedral cell (geometry), cells, . It a part of a sequence of honeycombs . It a part of a sequence of honeycombs . Order-8 dodecahedral honeycomb In the geometry of Hyperbolic space, hyperbolic 3-space, the order-8 dodecahedral honeycomb a regular space-filling tessellation (or honeycomb (geometry), honeycomb). With Schläfli symbol , it has eight dodecahedra around each edge. All vertices are ultra-ideal (existing beyond the ideal boundar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-7 Cubic Honeycomb
In the geometry of Hyperbolic space, hyperbolic 3-space, the order-7 cubic honeycomb is a regular space-filling tessellation (or honeycomb (geometry), honeycomb). With Schläfli symbol , it has seven cubes around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many cubes existing around each vertex in an order-7 triangular tiling vertex arrangement. Images Related polytopes and honeycombs It is one of a series of regular polytopes and honeycombs with cubic cells: : It is a part of a sequence of hyperbolic honeycombs with order-7 triangular tiling vertex figures, . Order-8 cubic honeycomb In the geometry of Hyperbolic space, hyperbolic 3-space, the order-8 cubic honeycomb a regular space-filling tessellation (or honeycomb (geometry), honeycomb). With Schläfli symbol . It has eight cubes around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many cubes existing around each vertex i ... [...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 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 tilings or, by extension, to space-filling tessellation with polytope cells and other higher-dimensional polytopes. As a flat slice Make a slice through the corner of the polyhedron, cutting through all the edges connected to the vertex. The cut surface is the vertex figure. This i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cell (geometry)
In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a ''polyhedron''. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions).. Polygonal face In elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include polyhedron side and Euclidean plane ''tile''. For example, any of the six squares that bound a cube is a face of the cube. Sometimes "face" is also used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells. Number of polygonal faces of a polyhedron Any convex polyhedron's surface has Euler characteristic :V - E + F = 2, where ''V'' is the number of ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Polychora
In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions. There are six convex and ten star regular 4-polytopes, giving a total of sixteen. History The convex regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. He discovered that there are precisely six such figures. Schläfli also found four of the regular star 4-polytopes: the grand 120-cell, great stellated 120-cell, grand 600-cell, and great grand stellated 120-cell. He skipped the remaining six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zero-hole tori: ''F'' − ''E'' + ''V'' 2). That excludes cells and vertex figures such as the great dodecahedron and small stellated dodecahedron . Edmund Hess (1843–1903) published ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Half-space Model
Poincaré is a French surname. Notable people with the surname include: * Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science * Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré * Lucien Poincaré (1862–1920), physicist, brother of Raymond and cousin of Henri * Raymond Poincaré (1860–1934), French Prime Minister or President ''inter alia'' from 1913 to 1920, cousin of Henri See also *List of things named after Henri Poincaré In physics and mathematics, a number of ideas are named after Henri Poincaré: * Euler–Poincaré characteristic * Hilbert–Poincaré series * Poincaré–Bendixson theorem * Poincaré–Birkhoff theorem * Poincaré–Birkhoff–Witt theorem, u .... * * {{DEFAULTSORT:Poincare French-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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