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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. : 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. There are 15 small index subgroups (7 unique) constructed from [(∞,∞,â ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H2-I-3-dual
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iii Symmetry 00a
III or iii may refer to: Companies * Information International, Inc., a computer technology company * Innovative Interfaces, Inc., a library-software company * 3i, formerly Investors in Industry, a British investment company Other uses * Institute for Information Industry, research institute in Taiwan * Insurance Information Institute, a US industry organization * Insurance Institute of India, an Indian organisation for training * Intelligence and Information Institute, a fictional US government organization in the comic version of ''Transformers'' * Interactive Investor International * Interstate Identification Index, an index of criminal records maintained by the FBI See also * 3 (other), including all uses of the Roman numeral "III" as a number *1/3 (other) *Number Three (other) *The Third (other) *Third (other) *Third party (other) *Third person (other) Third person, or third-person, may refer to: * Third pe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iii Symmetry 0a0
III or iii may refer to: Companies * Information International, Inc., a computer technology company * Innovative Interfaces, Inc., a library-software company * 3i, formerly Investors in Industry, a British investment company Other uses * Institute for Information Industry, research institute in Taiwan * Insurance Information Institute, a US industry organization * Insurance Institute of India, an Indian organisation for training * Intelligence and Information Institute, a fictional US government organization in the comic version of ''Transformers'' * Interactive Investor International * Interstate Identification Index, an index of criminal records maintained by the FBI See also * 3 (other), including all uses of the Roman numeral "III" as a number *1/3 (other) *Number Three (other) *The Third (other) *Third (other) *Third party (other) *Third person (other) Third person, or third-person, may refer to: * Third pe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iii Symmetry A00
III or iii may refer to: Companies * Information International, Inc., a computer technology company * Innovative Interfaces, Inc., a library-software company * 3i, formerly Investors in Industry, a British investment company Other uses * Institute for Information Industry, research institute in Taiwan * Insurance Information Institute, a US industry organization * Insurance Institute of India, an Indian organisation for training * Intelligence and Information Institute, a fictional US government organization in the comic version of ''Transformers'' * Interactive Investor International * Interstate Identification Index, an index of criminal records maintained by the FBI See also * 3 (other), including all uses of the Roman numeral "III" as a number *1/3 (other) *Number Three (other) *The Third (other) *Third (other) *Third party (other) *Third person (other) Third person, or third-person, may refer to: * Third pe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iii Symmetry 000
III or iii may refer to: Companies * Information International, Inc., a computer technology company * Innovative Interfaces, Inc., a library-software company * 3i, formerly Investors in Industry, a British investment company Other uses * Institute for Information Industry, research institute in Taiwan * Insurance Information Institute, a US industry organization * Insurance Institute of India, an Indian organisation for training * Intelligence and Information Institute, a fictional US government organization in the comic version of ''Transformers'' * Interactive Investor International * Interstate Identification Index, an index of criminal records maintained by the FBI See also * 3 (other), including all uses of the Roman numeral "III" as a number *1/3 (other) *Number Three (other) *The Third (other) *Third (other) *Third party (other) *Third person (other) Third person, or third-person, may refer to: * Third pe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgroup Index
In mathematics, specifically group theory, the index of a subgroup ''H'' in a group ''G'' is the number of left cosets of ''H'' in ''G'', or equivalently, the number of right cosets of ''H'' in ''G''. The index is denoted , G:H, or :H/math> or (G:H). Because ''G'' is the disjoint union of the left cosets and because each left coset has the same size as ''H'', the index is related to the orders of the two groups by the formula :, G, = , G:H, , H, (interpret the quantities as cardinal numbers if some of them are infinite). Thus the index , G:H, measures the "relative sizes" of ''G'' and ''H''. For example, let G = \Z be the group of integers under addition, and let H = 2\Z be the subgroup consisting of the even integers. Then 2\Z has two cosets in \Z, namely the set of even integers and the set of odd integers, so the index , \Z:2\Z, is 2. More generally, , \Z:n\Z, = n for any positive integer ''n''. When ''G'' is finite, the formula may be written as , G:H, = , G, /, H, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
I32 Symmetry
In geometry, the truncated triapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of tr. Symmetry The dual of this tiling represents the fundamental domains of ˆž,3 *∞32 symmetry. There are 3 small index subgroup constructed from ˆž,3by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. A special index 4 reflective subgroup, is ∞,∞,3) (*∞∞3), and its direct subgroup ∞,∞,3)sup>+, (∞∞3), and semidirect subgroup ∞,∞,3+) (3*∞).Norman W. Johnson and Asia Ivic Weiss, ''Quadratic Integers and Coxeter Groups'', Can. J. Math. Vol. 51 (6), 1999 pp. 1307–133/ref> Given ˆž,3with generating mirrors , then its index 4 subgroup has generators . An index 6 subgroup constructed as ˆž,3* becomes ∞,∞,∞) (*∞∞∞). Related polyhedra and tiling This tiling can be considered a member of a sequence of uniform p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ii2 Symmetry
In geometry, the truncated order-4 apeirogonal tiling is a uniform tiling of the Hyperbolic geometry, hyperbolic plane. It has Schläfli symbol of t. Uniform colorings A half symmetry coloring is tr, has two types of apeirogons, shown red and yellow here. If the apeirogonal curvature is too large, it doesn't converge to a single ideal point, like the right image, red apeirogons below. Coxeter diagram are shown with dotted lines for these divergent, ultraparallel mirrors. Symmetry From [∞,∞] symmetry, there are 15 small index subgroup by mirror removal and alternation. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The symmetry can be doubled as i42 symmetry, ∞42 symmetry by adding a mirror bisecting the f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite-order Triangular Tiling
In geometry, the infinite-order triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of . All vertices are ''ideal'', located at "infinity" and seen on the boundary of the Poincaré hyperbolic disk projection. Symmetry A lower symmetry form has alternating colors, and represented by cyclic symbol , . The tiling also represents the fundamental domains of the *∞∞∞ symmetry, which can be seen with 3 colors of lines representing 3 mirrors of the construction. Related polyhedra and tiling This tiling is topologically related as part of a sequence of regular polyhedra with Schläfli symbol . Other infinite-order triangular tilings A nonregular infinite-order triangular tiling can be generated by a recursive process from a central triangle as shown here: : See also * Infinite-order tetrahedral honeycomb *List of regular polytopes *List of uniform planar tilings *Tilings of regular polygons *Triangular tiling *Uniform tilings in hyperbol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H2checkers 2ii
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] |
H2checkers 23i
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] |
