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862 Symmetry
In geometry, the truncated hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one dodecagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr. Dual tiling Symmetry There are six reflective subgroup kaleidoscopic constructed from ,6by removing one or two of three mirrors. 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 subgroup index-8 group, +,8,1+,6,1+(4343) is the commutator subgroup of ,6 A radical subgroup is constructed as ,6* index 12, as ,6+ (6*4) with gyration points removed, becomes (*444444), and another *,6 index 16 as +,6 (8*3) with gyration points removed as (*33333333). Related polyhedra and tilings From a Wythoff con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4232 Symmetry
In geometry, the rhombihexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. It has Schläfli symbol of rr. Symmetry The dual tiling, called a deltoidal hexaoctagonal tiling represent the fundamental domains of *4232 symmetry, a half symmetry of ,6 (*862) as ,1+,6 : Related polyhedra and tilings From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-6 octagonal 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 ,6symmetry, and 7 with subsymmetry. See also * Tilings of regular polygons * List of uniform planar tilings This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings. There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their dua ... References * ... [...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] |
862 Symmetry Z0z
86 may refer to: * 86 (number), a natural number * 86 (term), a slang term for getting rid of something Dates * 86 BC, a year of the pre-Julian Roman calendar * AD 86, a common year of the Julian calendar * 1986, a common year of the Gregorian calendar * 2086, a common year of the Gregorian calendar Art and entertainment * ''86'' (novel series), a Japanese light novel series and anime series * "86", a song by Green Day from ''Insomniac'' * Agent 86 or Maxwell Smart, a character on ''Get Smart'' * ''Eighty-Sixed'', a 2017 web series created by Cazzie David and Elisa Kalani * ''Eighty-Sixed'', a 1989 novel by David B. Feinberg * ''86'd'', a 2009 novel by Dan Fante * "86" (Dawn Richard song) Transportation * Toyota 86, sports car * List of highways numbered 86 * 86 (MBTA bus) * 86 (New Jersey bus) 86 may refer to: * 86 (number), a natural number * 86 (term), a slang term for getting rid of something Dates * 86 BC, a year of the pre-Julian Roman calendar * AD 86, a com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4343 Symmetry
In geometry, the hexaoctagonal tiling is a uniform tiling of the hyperbolic plane. Constructions There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the ,6 kaleidoscope. Removing the mirror between the order 2 and 4 points, ,6,1+ gives 8,8,3) (*883). Removing the mirror between the order 2 and 8 points, +,8,6 gives 4,6,6) (*664). Removing two mirrors as ,1+,6,1+ leaves remaining mirrors (*4343). Symmetry The dual tiling has face configuration V6.8.6.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4343), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*43) orbifold. These are subsymmetries of ,6 Related polyhedra and tiling See also * Square tiling *Tilings of regular polygons *List of uniform planar tilings This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
883 Symmetry
In geometry, the truncated order-6 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t. Uniform colorings A secondary construction t is called a truncated trioctaoctagonal tiling: : Symmetry The dual to this tiling represent the fundamental domains of 8,8,3)(*883) symmetry. There are 3 small index subgroup symmetries constructed from 8,8,3)by mirror removal and alternation. 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 862 symmetry by adding a mirror bisecting the fundamental domain. Related polyhedra and tiling References * John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations) * See also *Tilings of regular polygons *List of uniform planar tilings This table shows the 11 convex uniform tilings (regular and semiregula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
664 Symmetry
In geometry, the truncated order-8 hexagonal tiling is a semiregular tiling of the hyperbolic plane. It has Schläfli symbol of t. Uniform colorings This tiling can also be constructed from *664 symmetry, as t. : Related polyhedra and tilings From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-6 octagonal 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 ,6symmetry, and 7 with subsymmetry. Symmetry The dual of the tiling represents the fundamental domains of (*664) orbifold symmetry. From 6,6,4)(*664) symmetry, there are 15 small index subgroup (11 unique) 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 im ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
