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Order-5 Cubic Honeycomb
In hyperbolic geometry, the order-5 cubic honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol it has five cubes around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral honeycomb. Description Symmetry It has a radial subgroup symmetry construction with dodecahedral fundamental domains: Coxeter notation: ,(3,5)* index 120. Related polytopes and honeycombs The order-5 cubic honeycomb has a related alternated honeycomb, ↔ , with icosahedron and tetrahedron cells. The honeycomb is also one of four regular compact honeycombs in 3D hyperbolic space: There are fifteen uniform honeycombs in the ,3,4Coxeter group family, including the order-5 cubic honeycomb as the regular form: The order-5 cubic honeycomb is in a sequence of regular polychora and honeycombs with icosahedral vertex figures. It is also in a sequence of regular polychora and honeycombs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H3 435 CC Center
H3, H03 or H-3 may refer to: Entertainment * ''Happy Hustle High'', a manga series by Rie Takada, originally titled "H3 School!" * H3 (film), ''H3'' (film), a 2001 film about the 1981 Irish hunger strike * h3h3Productions, styled "[h3]", a satirical YouTube channel Science * Triatomic hydrogen (H3), an unstable molecule * Trihydrogen cation (H3+), one of the most abundant ions in the universe * Tritium (Hydrogen-3, or H-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, Histamine H3 receptor, a human gene * Histone H3, a component of DNA higher structure in eukaryotic cells * *h3, , one of the three laryngeals in the reconstructed Proto-Indo-European language * Hekla 3 eruption, a huge volcanic eruption around 1000 BC Computing * , the level-3 HTML element#heading ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-4 Dodecahedral Honeycomb
In hyperbolic geometry, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs) of hyperbolic 3-space. With Schläfli symbol it has four dodecahedra around each edge, and 8 dodecahedra around each vertex in an octahedral arrangement. Its vertices are constructed from 3 orthogonal axes. Its dual is the order-5 cubic honeycomb. Description The dihedral angle of a regular dodecahedron is ~116.6°, so it is impossible to fit 4 of them on an edge in Euclidean 3-space. However in hyperbolic space a properly-scaled regular dodecahedron can be scaled so that its dihedral angles are reduced to 90 degrees, and then four fit exactly on every edge. Symmetry It has a half symmetry construction, , with two types (colors) of dodecahedra in the Wythoff construction. ↔ . Images A view of the order-4 dodecahedral honeycomb under the Beltrami-Klein model Related polytopes and honeycombs There are four regular compact honeycom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H2-5-4-primal
H, or h, is the eighth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''aitch'' (pronounced , plural ''aitches''), or regionally ''haitch'' ."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. History The original Semitic letter Heth most likely represented the voiceless pharyngeal fricative (). The form of the letter probably stood for a fence or posts. The Greek Eta 'Η' in archaic Greek alphabets, before coming to represent a long vowel, , still represented a similar sound, the voiceless glottal fricative . In this context, the letter eta is also known as Heta to underline this fact. Thus, in the Old Italic alphabets, the letter Heta of the Euboean alphabet was adopted with its original sound value . While Etruscan and La ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-4 Dodecahedral Honeycomb
In hyperbolic geometry, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs) of hyperbolic 3-space. With Schläfli symbol it has four dodecahedra around each edge, and 8 dodecahedra around each vertex in an octahedral arrangement. Its vertices are constructed from 3 orthogonal axes. Its dual is the order-5 cubic honeycomb. Description The dihedral angle of a regular dodecahedron is ~116.6°, so it is impossible to fit 4 of them on an edge in Euclidean 3-space. However in hyperbolic space a properly-scaled regular dodecahedron can be scaled so that its dihedral angles are reduced to 90 degrees, and then four fit exactly on every edge. Symmetry It has a half symmetry construction, , with two types (colors) of dodecahedra in the Wythoff construction. ↔ . Images A view of the order-4 dodecahedral honeycomb under the Beltrami-Klein model Related polytopes and honeycombs There are four regular compact honeycom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Polytope
In geometry, every polyhedron is associated with a second dual structure, where the Vertex (geometry), vertices of one correspond to the Face (geometry), faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or Abstract polytope, abstract polyhedra, but not all can also be constructed as geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the Symmetry, symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a corresponding symmetry class. For example, the regular polyhedrathe (convex) Platonic solids and (star) Kepler–Poinsot polyhedraform dual pairs, where the regular tetrahedron is #Self-dual polyhedra, self-dual. The dual of an Isogonal figure, isogonal polyhedron (one in which any two vertices are equivalent under symmetries of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex (geometry)
In geometry, a vertex (in plural form: vertices or vertexes) is a point (geometry), point where two or more curves, line (geometry), lines, or edge (geometry), edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedron, polyhedra are vertices. Definition Of an angle The ''vertex'' of an angle is the point where two Line (mathematics)#Ray, rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments, and lines that result in two straight "sides" meeting at one place. :(3 vols.): (vol. 1), (vol. 2), (vol. 3). Of a polytope A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection (Euclidean geometry), intersection of Edge (geometry), edges, face (geometry), faces or facets of the object. In a polygon, a vertex is called "convex set, convex" if the internal an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge (geometry)
In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a line segment on the boundary, and is often called a polygon side. In a polyhedron or more generally a polytope, an edge is a line segment where two faces (or polyhedron sides) meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal. Relation to edges in graphs In graph theory, an edge is an abstract object connecting two graph vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment. However, any polyhedron can be represented by its skeleton or edge-skeleton, a graph whose vertices are the geometric vertices of the polyhedron and whose edges correspond to the geometric edges. Conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized by Steinitz's theore ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. There are many ways to construct it as an open subset of \mathbb R^n with an explicitly written Riemannian metric; such constructions are referred to as models. Hyperbolic 2-space, H2, which was the first instance studied, is also called the hyperbolic plane. It is also sometimes referred to as Lobachevsky space or Bolyai–Lobachevsky space after the names of the author who first published on the topic of hyperbolic geometry. Sometimes the qualificative "real" is added to differentiate it from complex hyperbolic spaces, quaternionic hyperbolic spaces and the octononic hyperbolic plane which are the other symmetric spaces of negative curvature. Hyperbolic space serves as the prototype of a Gromov hyperbolic space which is a far-reachin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Honeycomb (geometry)
In geometry, a honeycomb is a ''space filling'' or ''close packing'' of polyhedral or higher-dimensional ''cells'', so that there are no gaps. It is an example of the more general mathematical ''tiling'' or ''tessellation'' in any number of dimensions. Its dimension can be clarified as ''n''-honeycomb for a honeycomb of ''n''-dimensional space. Honeycombs are usually constructed in ordinary Euclidean ("flat") space. They may also be constructed in non-Euclidean spaces, such as hyperbolic honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. Classification There are infinitely many honeycombs, which have only been partially classified. The more regular ones have attracted the most interest, while a rich and varied assortment of others continue to be discovered. The simplest honeycombs to build are formed from stacked layers or ''slabs'' of prisms based on some tessellations of the plane. In particula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
