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642 Symmetry
In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one dodecagon on each vertex. It has Schläfli symbol of tr. Dual tiling Related polyhedra and tilings From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-4 hexagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full ,4symmetry, and 7 with subsymmetry. Symmetry The dual of the tiling represents the fundamental domains of (*642) orbifold symmetry. From ,4symmetry, there are 15 small index subgroup by mirror removal and alternation operators. 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 unique mirrors ... [...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 geom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3333 Symmetry
In geometry, the order-6 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of . Symmetry This tiling represents a hyperbolic kaleidoscope of 4 mirrors meeting as edges of a square, with six squares around every vertex. This symmetry by orbifold notation is called (*3333) with 4 order-3 mirror intersections. In Coxeter notation can be represented as ,4* removing two of three mirrors (passing through the square center) in the ,4symmetry. The *3333 symmetry can be doubled to 663 symmetry by adding a mirror bisecting the fundamental domain. This bicolored square tiling shows the even/odd reflective fundamental square domains of this symmetry. This bicolored tiling has a wythoff construction t1. A second 6-color symmetry can be constructed from a hexagonal symmetry domain. Example artwork Around 1956, M.C. Escher explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician H.S.M. Coxeter i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3232 Symmetry
In geometry, the tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol r. Constructions There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the ,4kaleidoscope. Removing the last mirror, ,4,1+ gives ,6 (*662). Removing the first mirror +,6,4 gives 4,4,3) (*443). Removing both mirror as +,6,4,1+ leaving 3,∞,3,∞)(*3232). Symmetry The dual tiling, called a rhombic tetrahexagonal tiling, with face configuration V4.6.4.6, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*3232), shown here in two different centered views. Adding a 2-fold rotation point in the center of each rhombi represents a (2*32) orbifold. : Related polyhedra and tiling See also *Square tiling *Tilings of regular polygons *List of uniform planar tilings *List of regular polytopes This article lists the regular polytopes and regular polytope compounds in Euclidean, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3222 Symmetry
In geometry, the rhombitetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr. It can be seen as constructed as a rectified tetrahexagonal tiling, r, as well as an expanded order-4 hexagonal tiling or expanded order-6 square tiling. Constructions There are two uniform constructions of this tiling, one from ,4or (*642) symmetry, and secondly removing the mirror middle, ,1+,4 gives a rectangular fundamental domain ��,3,∞ (*3222). There are 3 lower symmetry forms seen by including edge-colorings: sees the hexagons as truncated triangles, with two color edges, with ,4+(4*3) symmetry. sees the yellow squares as rectangles, with two color edges, with +,4(6*2) symmetry. A final quarter symmetry combines these colorings, with +,4+(32×) symmetry, with 2 and 3 fold gyration points and glide reflections. This four color tiling is related to a semiregular infinite skew polyhedron with the same vertex figure in Euclidean 3-space with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
662 Symmetry
In geometry, the truncated order-4 hexagonal tiling is a uniform tiling of the Hyperbolic geometry, hyperbolic plane. It has Schläfli symbol of t. A secondary construction tr is called a truncated hexahexagonal tiling with two colors of dodecagons. Constructions There are two uniform constructions of this tiling, first from [6,4] kaleidoscope, and a lower symmetry by removing the last mirror, [6,4,1+], gives [6,6], (*662). Dual tiling Related polyhedra and tiling Symmetry The dual of the tiling represents the fundamental domains of (*662) Orbifold notation, orbifold symmetry. From [6,6] (*662) symmetry, there are 15 small index subgroup (12 unique) by mirror removal and alternation (geometry), alternation operators. 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 bla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
443 Symmetry
In geometry, the truncated order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t. Uniform colorings Symmetry The dual tiling represents the fundamental domains of the *443 orbifold symmetry. There are two reflective subgroup kaleidoscopic constructed from 4,4,3)by removing one or two of three mirrors. In these images fundamental domains are alternately colored black and cyan, and mirrors exist on the boundaries between colors. A larger subgroup is constructed 4,4,3*) index 6, as (3*22) with gyration points removed, becomes (*222222). The symmetry can be doubled as 642 symmetry by adding a mirror bisecting the fundamental domain. Related polyhedra and tilings From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular order-4 hexagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Notation
In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson. Reflectional groups For Coxeter groups, defined by pure reflections, there is a direct correspondence between the bracket notation and Coxeter-Dynkin diagram. The numbers in the bracket notation represent the mirror reflection orders in the branches of the Coxeter diagram. It uses the same simplification, suppressing 2s between orthogonal mirrors. The Coxeter notation is simplified with exponents to represent the number of branches in a row for linear diagram. So the ''A''''n'' group is represented by ''n''−1 to imply ''n'' nodes connected by ''n−1'' order-3 branches. Exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
642 Symmetry Xxx
64 or sixty-four or ''variation'', may refer to: * 64 (number) Dates * one of the years 64 BC, AD 64, 1864, 1964, 2064, etc. * June 4th (6/4) ** the date of the 1989 Tiananmen Square protests and massacre * April 6th (6/4) * April 6 AD (6/4) * June 4 AD (6/4) Places * Highway 64, see list of highways numbered 64 ** Interstate 64, a national route in the United States * +64, country code dialing code of New Zealand; see Telephone numbers in New Zealand * 64 Angelina (asteroid 64), a main-belt asteroid Other uses * Nintendo 64, the third home console by Nintendo, released in 1996 * Commodore 64 * 64-bit computing * "64" (song), a 2011 song by hip hop band Odd Future * ''Sixty Four'' (album), a 2004 album recorded in 1964 by Donovan * Sixty-four (ship), a type of sailing warship * A /64 Classless Inter-Domain Routing Classless Inter-Domain Routing (CIDR ) is a method for allocating IP addresses and for IP routing. The Internet Engineering Task Force introduced CIDR in 199 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
642 Symmetry A0b
64 or sixty-four or ''variation'', may refer to: * 64 (number) Dates * one of the years 64 BC, AD 64, 1864, 1964, 2064, etc. * June 4th (6/4) ** the date of the 1989 Tiananmen Square protests and massacre * April 6th (6/4) * April 6 AD (6/4) * June 4 AD (6/4) Places * Highway 64, see list of highways numbered 64 ** Interstate 64, a national route in the United States * +64, country code dialing code of New Zealand; see Telephone numbers in New Zealand * 64 Angelina (asteroid 64), a main-belt asteroid Other uses * Nintendo 64, the third home console by Nintendo, released in 1996 * Commodore 64 * 64-bit computing * "64" (song), a 2011 song by hip hop band Odd Future * ''Sixty Four'' (album), a 2004 album recorded in 1964 by Donovan * Sixty-four (ship) In the rating system of the Royal Navy, a third rate was a ship of the line which from the 1720s mounted between 64 and 80 guns, typically built with two gun decks (thus the related term two-decker). Years of experie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
642 Symmetry 0a0
64 or sixty-four or ''variation'', may refer to: * 64 (number) Dates * one of the years 64 BC, AD 64, 1864, 1964, 2064, etc. * June 4th (6/4) ** the date of the 1989 Tiananmen Square protests and massacre * April 6th (6/4) * April 6 AD (6/4) * June 4 AD (6/4) Places * Highway 64, see list of highways numbered 64 ** Interstate 64, a national route in the United States * +64, country code dialing code of New Zealand; see Telephone numbers in New Zealand * 64 Angelina (asteroid 64), a main-belt asteroid Other uses * Nintendo 64, the third home console by Nintendo, released in 1996 * Commodore 64 * 64-bit computing * "64" (song), a 2011 song by hip hop band Odd Future * ''Sixty Four'' (album), a 2004 album recorded in 1964 by Donovan * Sixty-four (ship), a type of sailing warship * A /64 Classless Inter-Domain Routing Classless Inter-Domain Routing (CIDR ) is a method for allocating IP addresses and for IP routing. The Internet Engineering Task Force introduced CIDR in 199 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
