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Order-4 Square Tiling Honeycomb
In the geometry of hyperbolic 3-space, the order-4 square tiling honeycomb is one of 11 paracompact regular honeycombs. It is ''paracompact'' because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. Given by Schläfli symbol , it has four square tilings around each edge, and infinite square tilings around each vertex in a square tiling vertex figure.Coxeter ''The Beauty of Geometry'', 1999, Chapter 10, Table III Symmetry The order-4 square tiling honeycomb has many reflective symmetry constructions: as a regular honeycomb, ↔ with alternating types (colors) of square tilings, and with 3 types (colors) of square tilings in a ratio of 2:1:1. Two more half symmetry constructions with pyramidal domains have ,4,1+,4symmetry: ↔ , and ↔ . There are two high-index subgroups, both index 8: ,4,4*↔ 4,4,4,4,1+) with a pyramidal fundamental domain: (4,∞,4)),((4,∞,4))or ; and ,4*,4 with 4 orthogonal sets of ultra-parallel mirrors ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H3 444 FC Boundary
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Tiling 44-t1
A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, security guards, in some workplaces and schools, and by inmates in prisons. In some countries, some other officials also wear uniforms in their duties; such is the case of the Commissioned Corps of the United States Public Health Service or the French prefects. For some organizations, such as police, it may be illegal for non-members to wear the uniform. Etymology From the Latin ''unus'' (meaning one), and ''forma'' (meaning form). Variants Corporate and work uniforms Workers sometimes wear uniforms or corporate clothing of one nature or another. Workers required to wear a uniform may include retail workers, bank and post-office workers, public-security and health-care workers, blue-collar employees, personal trainers in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; for example, the symmetry group of each regular polyhedron is a finite Coxeter group. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups, and finite Coxeter groups were classified in 1935. Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H2 Tiling 24i-4
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-4 Apeirogonal Tiling
In geometry, the order-4 apeirogonal tiling is a List of regular polytopes#Hyperbolic tilings, regular Tessellation, tiling of the Hyperbolic geometry, hyperbolic plane. It has Schläfli symbol of . Symmetry This tiling represents the mirror lines of *2∞ symmetry. Its dual tiling represents the fundamental domains of orbifold notation *∞∞∞∞ symmetry, a square domain with four ideal vertices. : Uniform colorings Like the Euclidean Square tiling#Uniform colorings, square tiling there are 9 uniform colorings for this tiling, with 3 uniform colorings generated by triangle reflective domains. A fourth can be constructed from an infinite square symmetry (*∞∞∞∞) with 4 colors around a vertex. The checker board, r, coloring defines the fundamental domains of [(∞,4,4)], (*∞44) symmetry, usually shown as black and white domains of reflective orientations. Related polyhedra and tiling This tiling is also topologically related as a part of sequence of regular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypercycle (geometry)
In hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight line (its axis). Given a straight line and a point not on , one can construct a hypercycle by taking all points on the same side of as , with perpendicular distance to equal to that of . The line is called the ''axis'', ''center'', or ''base line'' of the hypercycle. The lines perpendicular to , which are also perpendicular to the hypercycle, are called the '' normals'' of the hypercycle. The segments of the normals between and the hypercycle are called the ''radii''. Their common length is called the ''distance'' or ''radius'' of the hypercycle. The hypercycles through a given point that share a tangent through that point converge towards a horocycle as their distances go towards infinity. Properties similar to those of Euclidean lines Hypercycles in hyperbolic geometry have some properties similar to those of lines i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H2 Tiling 2ii-4
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] |
Infinite-order Apeirogonal Tiling
The infinite-order apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of , which means it has countably infinitely many apeirogons around all its ideal vertices. Symmetry This tiling represents the fundamental domains of *∞ symmetry. Uniform colorings This tiling can also be alternately colored in the ∞,∞,∞)symmetry from 3 generator positions. Related polyhedra and tiling The union of this tiling and its dual can be seen as orthogonal red and blue lines here, and combined define the lines of a *2∞2∞ fundamental domain. : : a or = ∪ See also * Tilings of regular polygons * List of uniform planar tilings *List of regular polytopes References * John Horton Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many b ..., ... [...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. 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 any other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
