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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] |
List Of Regular Polytopes
This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces. Overview This table shows a summary of regular polytope counts by rank. There are no Euclidean regular star tessellations in any number of dimensions. 1-polytopes There is only one polytope of rank 1 (1-polytope), the closed line segment bounded by its two endpoints. Every realization of this 1-polytope is regular. It has the Schläfli symbol , or a Coxeter diagram with a single ringed node, . Norman Johnson calls it a ''dion'' and gives it the Schläfli symbol . Although trivial as a polytope, it appears as the edges of polygons and other higher dimensional polytopes. It is used in the definition of uniform prisms like Schläfli symbol ×, or Coxeter diagram as a Cartesian product of a line segment and a regular polygon. 2-polytopes (polygons) The polytopes of rank 2 (2-polytopes) are called polygons. Regular polygons are equilateral and cyclic. A -gonal regular polygon is repre ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-3 Apeirogonal Tiling
In geometry, the order-3 apeirogonal tiling is a regular hyperbolic tiling, regular tiling of the Hyperbolic geometry, hyperbolic plane. It is represented by the Schläfli symbol , having three regular Apeirogon#Hyperbolic geometry, apeirogons around each vertex. Each apeirogon is Circumscribed circle, inscribed in a horocycle. The order-2 apeirogonal tiling represents an infinite dihedron in the Euclidean plane as . Images Each apeirogon face is circumscribed by a horocycle, which looks like a circle in a Poincaré disk model, internally tangent to the projective circle boundary. : The edges of the tiling, shown in blue, form an order-3 Cayley tree. Uniform colorings Like the Euclidean Hexagonal tiling#Uniform colorings, hexagonal tiling, there are 3 uniform colorings of the ''order-3 apeirogonal tiling'', each from different reflective triangle group domains: Symmetry The dual to this tiling represents the fundamental domains of [(∞,∞,∞)] (*∞∞∞) symmetry. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apeirogon
In geometry, an apeirogon () or infinite polygon is a polygon with an infinite number of sides. Apeirogons are the rank 2 case of infinite polytopes. In some literature, the term "apeirogon" may refer only to the regular apeirogon, with an infinite dihedral group of symmetries. Definitions Geometric apeirogon Given a point ''A''0 in a Euclidean space and a translation ''S'', define the point ''Ai'' to be the point obtained from ''i'' applications of the translation ''S'' to ''A''0, so ''Ai'' = ''Si''(''A''0). The set of vertices ''Ai'' with ''i'' any integer, together with edges connecting adjacent vertices, is a sequence of equal-length segments of a line, and is called the regular apeirogon as defined by H. S. M. Coxeter. A regular apeirogon can be defined as a partition of the Euclidean line ''E''1 into infinitely many equal-length segments. It generalizes the regular ''n''-gon, which may be defined as a partition of the circle ''S''1 into ''finitely'' many equal-length ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H2-I-3-dual
H, or h, is the eighth letter of the Latin alphabet, used in the modern English alphabet, including the alphabets of other western European languages and others worldwide. Its name in English is ''aitch'' (pronounced , plural ''aitches''), or regionally ''haitch'' (pronounced , plural ''haitches'')''.''"H" ''Oxford English Dictionary,'' 2nd edition (1989); ''Merriam-Webster's Third New International Dictionary of the English Language, Unabridged'' (1993); "aitch" or "haitch", op. cit. Name English For most English speakers, the name for the letter is pronounced as and spelled "aitch" or occasionally "eitch". The pronunciation and the associated spelling "haitch" are often considered to be h-adding and are considered non-standard in England. It is, however, a feature of Hiberno-English, and occurs sporadically in various other dialects. The perceived name of the letter affects the choice of indefinite article before initialisms beginning with H: for example "an H-bomb" ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-3 Apeirogonal Tiling
In geometry, the order-3 apeirogonal tiling is a regular hyperbolic tiling, regular tiling of the Hyperbolic geometry, hyperbolic plane. It is represented by the Schläfli symbol , having three regular Apeirogon#Hyperbolic geometry, apeirogons around each vertex. Each apeirogon is Circumscribed circle, inscribed in a horocycle. The order-2 apeirogonal tiling represents an infinite dihedron in the Euclidean plane as . Images Each apeirogon face is circumscribed by a horocycle, which looks like a circle in a Poincaré disk model, internally tangent to the projective circle boundary. : The edges of the tiling, shown in blue, form an order-3 Cayley tree. Uniform colorings Like the Euclidean Hexagonal tiling#Uniform colorings, hexagonal tiling, there are 3 uniform colorings of the ''order-3 apeirogonal tiling'', each from different reflective triangle group domains: Symmetry The dual to this tiling represents the fundamental domains of [(∞,∞,∞)] (*∞∞∞) symmetry. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H3 835 UHS Plane At Infinity
H3, H03 or H-3 may refer to: Entertainment * ''H3'' (film), a 2001 film about the 1981 Irish hunger strike * ''H3 – Halloween Horror Hostel'', a 2008 German horror-parody television film * ''Happy Hustle High'', a manga series by Rie Takada, originally titled "H3 School!" * h3h3Productions, styled " 3, a satirical YouTube channel Science * Triatomic hydrogen (H3), an unstable molecule * Trihydrogen cation (), one of the most abundant ions in the universe * Tritium (3H), or hydrogen-3, an isotope of hydrogen * ATC code H03 ''Thyroid therapy'', a subgroup of the Anatomical Therapeutic Chemical Classification System * British NVC community H3, a heath community of the British National Vegetation Classification system * Histamine H3 receptor, a human gene * Histone H3, a component of DNA higher structure in eukaryotic cells * Hekla 3 eruption, a huge volcanic eruption around 1000 BC Computing * HTTP/3, the third major version of the Hypertext Transfer Protocol * Socket H3 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Honeycomb 8-3-5 Poincare Vc
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Hyperbolic may refer to: * of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics ** Hyperbolic geometry, a non-Euclidean geometry ** Hyperbolic functions, analogues of ordinary trigonometric functions, defined using the hyperbola * of or pertaining to hyperbole, the use of exaggeration as a rhetorical device or figure of speech * ''Hyperbolic'' (album), by Pnau, 2024 See also * Exaggeration * Hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-8 Dodecahedral Honeycomb
In the geometry of Hyperbolic space, hyperbolic 3-space, the order-7 dodecahedral honeycomb is 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 boun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
