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Order-6 Dodecahedral Honeycomb
The order-6 dodecahedral honeycomb is one of 11 paracompact regular honeycomb (geometry), honeycombs in Hyperbolic space, hyperbolic 3-space. It is ''paracompact'' because it has vertex figures composed of an infinite number of faces, with all vertices as ideal points at infinity. It has Schläfli symbol , with six ideal polyhedron, ideal regular dodecahedron, dodecahedral cells surrounding each edge of the honeycomb. Each vertex is ideal, and surrounded by infinitely many dodecahedra. The honeycomb has a triangular tiling vertex figure. Symmetry A half symmetry construction exists as with alternately colored dodecahedral cells. Images The order-6 dodecahedral honeycomb is similar to the 2D hyperbolic infinite-order pentagonal tiling, , with pentagonal faces, and with vertices on the ideal surface. : Related polytopes and honeycombs The order-6 dodecahedral honeycomb is a List of regular polytopes#Tessellations of hyperbolic 3-space, regular hyperbolic honeycomb in 3-sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H3 536 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), a 2001 film about the 1981 Irish hunger strike * h3h3Productions, styled " 3, 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, a human gene * Histone H3, a component of DNA higher structure in eukaryotic cells * , 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 heading markup element * HTTP/3, the third revision ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-5 Hexagonal Tiling Honeycomb
In the field of hyperbolic geometry, the order-5 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is ''paracompact'' because it has cells composed of an infinite number of faces. Each cell consists of a hexagonal tiling whose vertices lie on a horosphere, a flat plane in hyperbolic space that approaches a single ideal point at infinity. The Schläfli symbol of the order-5 hexagonal tiling honeycomb is . Since that of the hexagonal tiling is , this honeycomb has five such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the icosahedron is , the vertex figure of this honeycomb is an icosahedron. Thus, 20 hexagonal tilings meet at each vertex of this honeycomb.Coxeter ''The Beauty of Geometry'', 1999, Chapter 10, Table III Symmetry A lower-symmetry construction of index 120, ,(3,5)* exists with regular dodecahedral fundamental domains, and an icosahedral Coxeter-Dynkin diagram with 6 axia ... [...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 th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-5 Hexagonal Tiling Honeycomb
In the field of hyperbolic geometry, the order-5 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is ''paracompact'' because it has cells composed of an infinite number of faces. Each cell consists of a hexagonal tiling whose vertices lie on a horosphere, a flat plane in hyperbolic space that approaches a single ideal point at infinity. The Schläfli symbol of the order-5 hexagonal tiling honeycomb is . Since that of the hexagonal tiling is , this honeycomb has five such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the icosahedron is , the vertex figure of this honeycomb is an icosahedron. Thus, 20 hexagonal tilings meet at each vertex of this honeycomb.Coxeter ''The Beauty of Geometry'', 1999, Chapter 10, Table III Symmetry A lower-symmetry construction of index 120, ,(3,5)* exists with regular dodecahedral fundamental domains, and an icosahedral Coxeter-Dynkin diagram with 6 axia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Uniform Honeycombs In Hyperbolic Space
In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by permutations of rings of the Coxeter diagrams for each family. Hyperbolic uniform honeycomb families Honeycombs are divided between compact and paracompact forms defined by Coxeter groups, the first category only including finite cells and vertex figures (finite subgroups), and the second includes affine subgroups. Compact uniform honeycomb families The nine compact Coxeter groups are listed here with their Coxeter diagrams, in order of the relative volumes of their fundamental simplex domains. These 9 families generate a total of 76 unique uniform honeycombs. The full list of hyperbolic uniform honeycombs has not been proven and an unknown number of non-Wythoffian forms exist. Two known ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H2 Tiling 25i-4
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] |
Infinite-order Pentagonal Tiling
In 2-dimensional hyperbolic geometry, the infinite-order pentagonal tiling is a regular tiling. It has Schläfli symbol of . All vertices are ''ideal'', located at "infinity", seen on the boundary of the Poincaré hyperbolic disk projection. Symmetry There is a half symmetry form, , seen with alternating colors: : Related polyhedra and tiling This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (5n). See also *Pentagonal tiling *Uniform tilings in hyperbolic plane *List of regular polytopes This article lists the regular polytopes and regular polytope compounds in Euclidean geometry, Euclidean, spherical geometry, spherical and hyperbolic geometry, hyperbolic spaces. The Schläfli symbol describes every regular tessellation of an ' ... References * * External links * * Hyperbolic and Spherical Tiling Gallery {{Tessellation Hyperbolic tilings Infinite-order tilings Isogonal tilings Isohedral t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-6 Dodecahedral Honeycomb
The order-6 dodecahedral honeycomb is one of 11 paracompact regular honeycomb (geometry), honeycombs in Hyperbolic space, hyperbolic 3-space. It is ''paracompact'' because it has vertex figures composed of an infinite number of faces, with all vertices as ideal points at infinity. It has Schläfli symbol , with six ideal polyhedron, ideal regular dodecahedron, dodecahedral cells surrounding each edge of the honeycomb. Each vertex is ideal, and surrounded by infinitely many dodecahedra. The honeycomb has a triangular tiling vertex figure. Symmetry A half symmetry construction exists as with alternately colored dodecahedral cells. Images The order-6 dodecahedral honeycomb is similar to the 2D hyperbolic infinite-order pentagonal tiling, , with pentagonal faces, and with vertices on the ideal surface. : Related polytopes and honeycombs The order-6 dodecahedral honeycomb is a List of regular polytopes#Tessellations of hyperbolic 3-space, regular hyperbolic honeycomb in 3-sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal Polyhedron
In three-dimensional hyperbolic geometry, an ideal polyhedron is a convex polyhedron all of whose vertices are ideal points, points "at infinity" rather than interior to three-dimensional hyperbolic space. It can be defined as the convex hull of a finite set of ideal points. An ideal polyhedron has ideal polygons as its faces, meeting along lines of the hyperbolic space. The Platonic solids and Archimedean solids have ideal versions, with the same combinatorial structure as their more familiar Euclidean versions. Several uniform hyperbolic honeycombs divide hyperbolic space into cells of these shapes, much like the familiar division of Euclidean space into cubes. However, not all polyhedra can be represented as ideal polyhedra – a polyhedron can be ideal only when it can be represented in Euclidean geometry with all its vertices on a circumscribed sphere. Using linear programming, it is possible to test whether a given polyhedron has an ideal version, in polynomial time. Eve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal Point
In hyperbolic geometry, an ideal point, omega point or point at infinity is a well-defined point outside the hyperbolic plane or space. Given a line ''l'' and a point ''P'' not on ''l'', right- and left-limiting parallels to ''l'' through ''P'' converge to ''l'' at ''ideal points''. Unlike the projective case, ideal points form a boundary, not a submanifold. So, these lines do not intersect at an ideal point and such points, although well-defined, do not belong to the hyperbolic space itself. The ideal points together form the Cayley absolute or boundary of a hyperbolic geometry. For instance, the unit circle forms the Cayley absolute of the Poincaré disk model and the Klein disk model. While the real line forms the Cayley absolute of the Poincaré half-plane model . Pasch's axiom and the exterior angle theorem still hold for an omega triangle, defined by two points in hyperbolic space and an omega point. Properties * The hyperbolic distance between an ideal point and an ... [...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 (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 ... [...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] |
