|
Heptagonal Tiling Honeycomb
In the geometry of hyperbolic 3-space, the heptagonal tiling honeycomb or 7,3,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a heptagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere. Geometry The Schläfli symbol of the heptagonal tiling honeycomb is , with three heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a tetrahedron, . Related polytopes and honeycombs It is a part of a series of regular polytopes and honeycombs with Schläfli symbol, and tetrahedral vertex figures: It is a part of a series of regular honeycombs, . It is a part of a series of regular honeycombs, with . Octagonal tiling honeycomb In the geometry of hyperbolic 3-space, the octagonal tiling honeycomb or 8,3,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an octagonal tiling whose vertices lie on a 2-hype ... [...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] |
Poincaré Disk Model
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or diameters of the unit circle. The group of orientation preserving isometries of the disk model is given by the projective special unitary group , the quotient of the special unitary group SU(1,1) by its center . Along with the Klein model and the Poincaré half-space model, it was proposed by Eugenio Beltrami who used these models to show that hyperbolic geometry was equiconsistent with Euclidean geometry. It is named after Henri Poincaré, because his rediscovery of this representation fourteen years later became better known than the original work of Beltrami. The Poincaré ball model is the similar model for ''3'' or ''n''-dimensional hyperbolic geometry in which the points of the geometry ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Honeycomb 7-3-7 Poincare
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 fin ... [...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-3-6 Heptagonal Honeycomb
In the geometry of Hyperbolic space, hyperbolic 3-space, the order-3-6 heptagonal honeycomb a regular space-filling tessellation (or honeycomb (geometry), honeycomb). Each infinite cell consists of a heptagonal tiling whose vertices lie on a Hypercycle (geometry), 2-hypercycle, each of which has a limiting circle on the ideal sphere. Geometry The Schläfli symbol of the ''order-3-6 heptagonal honeycomb'' is , with six heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is an triangular tiling, . It has a quasiregular polytope, quasiregular construction, , which can be seen as alternately colored cells. Related polytopes and honeycombs It is a part of a series of regular polytopes and honeycombs with Schläfli symbol, and triangular tiling vertex figures. Order-3-6 octagonal honeycomb In the geometry of Hyperbolic space, hyperbolic 3-space, the order-3-6 octagonal honeycomb a regular space-filling tessellation (or honeycomb (geometry), honeycomb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-3-5 Heptagonal Honeycomb
In the geometry of Hyperbolic space, hyperbolic 3-space, the order-3-5 heptagonal honeycomb a regular space-filling tessellation (or honeycomb (geometry), honeycomb). Each infinite cell consists of a heptagonal tiling whose vertices lie on a Hypercycle (geometry), 2-hypercycle, each of which has a limiting circle on the ideal sphere. Geometry The Schläfli symbol of the order-3-5 heptagonal honeycomb is , with five heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is an icosahedron, . Related polytopes and honeycombs It is a part of a series of regular polytopes and honeycombs with Schläfli symbol, and icosahedral vertex figures. Order-3-5 octagonal honeycomb In the geometry of Hyperbolic space, hyperbolic 3-space, the order-3-5 octagonal honeycomb a regular space-filling tessellation (or honeycomb (geometry), honeycomb). Each infinite cell consists of an octagonal tiling whose vertices lie on a Hypercycle (geometry), 2-hypercycle, each of w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-3-4 Heptagonal Honeycomb
In the geometry of Hyperbolic space, hyperbolic 3-space, the order-3-4 heptagonal honeycomb or 7,3,4 honeycomb a regular space-filling tessellation (or honeycomb (geometry), honeycomb). Each infinite cell consists of a heptagonal tiling whose vertices lie on a Hypercycle (geometry), 2-hypercycle, each of which has a limiting circle on the ideal sphere. Geometry The Schläfli symbol of the order-3-4 heptagonal honeycomb is , with four heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octahedron, . Related polytopes and honeycombs It is a part of a series of regular polytopes and honeycombs with Schläfli symbol, and octahedral vertex figures: Order-3-4 octagonal honeycomb In the geometry of Hyperbolic space, hyperbolic 3-space, the order-3-4 octagonal honeycomb or 8,3,4 honeycomb a regular space-filling tessellation (or honeycomb (geometry), honeycomb). Each infinite cell consists of an octagonal tiling whose vertices lie on a Hypercycle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
