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842 Symmetry
In geometry, the truncated tetraoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr. Dual tiling Symmetry There are 15 subgroups constructed from ,4by 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 subgroup index-8 group, +,8,1+,4,1+(4242) is the commutator subgroup of ,4 A larger subgroup is constructed as ,4* index 8, as ,4+ (4*4) with gyration points removed, becomes (*4444) or (*44), and another *,4 index 16 as +,4 (8*2) with gyration points removed as (*22222222) or (*28). And their direct subgroups ,4*sup>+, *,4sup>+, subgroup indices 16 and 3 ... [...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] |
842 Symmetry A0b
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84 may refer to: * 84 (number) * one of the years 84 BC, AD 84, 1984, AD 2084 * Eighty Four, Pennsylvania, an unincorporated census-designated place in Washington County, Pennsylvania, United States * Seksendört, a Turkish pop group whose name means 84 See also * * List of highways numbered A ''list'' is any set of items in a row. List or lists may also refer to: People * List (surname) Organizations * List College, an undergraduate division of the Jewish Theological Seminary of America * SC Germania List, German rugby union ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4242 Symmetry
In geometry, the tetraoctagonal tiling is a uniform tiling of the hyperbolic plane. Constructions There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the ,4or (*842) orbifold symmetry. Removing the mirror between the order 2 and 4 points, ,4,1+ gives ,8 (*882). Removing the mirror between the order 2 and 8 points, +,8,4 gives 4,4,4) (*444). Removing both mirrors, +,8,4,1+ leaves a rectangular fundamental domain, ∞,4,∞,4) (*4242). Symmetry The dual tiling has face configuration V4.8.4.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4242), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*42) orbifold. Related polyhedra and tiling See also * Square tiling *Tilings of regular polygons *List of uniform planar tilings *List of regular polytopes References * John H. Conway John Horton Conway (26 December 1937 – 11 April 2020) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4222 Symmetry
In geometry, the rhombitetraoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr. It can be seen as constructed as a rectified tetraoctagonal tiling, r, as well as an expanded order-4 octagonal tiling or expanded order-8 square tiling. Constructions There are two uniform constructions of this tiling, one from ,4or (*842) symmetry, and secondly removing the mirror middle, ,1+,4 gives a rectangular fundamental domain ��,4,∞ (*4222). Symmetry A lower symmetry construction exists, with (*4222) orbifold symmetry. This symmetry can be seen in the dual tiling, called a ''deltoidal tetraoctagonal tiling'', alternately colored here. Its fundamental domain is a Lambert quadrilateral, with 3 right angles. With edge-colorings there is a half symmetry form (4*4) orbifold notation. The octagons can be considered as truncated squares, t with two types of edges. It has Coxeter diagram , Schläfli symbol s2. The squares can be distorted into ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
882 Symmetry
In geometry, the truncated order-4 octagonal tiling is a uniform tiling of the Hyperbolic geometry, hyperbolic plane. It has Schläfli symbol of t0,1. A secondary construction t0,1,2 is called a truncated octaoctagonal tiling with two colors of hexakaidecagons. Constructions There are two uniform constructions of this tiling, first by the [8,4] kaleidoscope, and second by removing the last mirror, [8,4,1+], gives [8,8], (*882). Dual tiling Symmetry The dual of the tiling represents the fundamental domains of (*882) Orbifold notation, orbifold symmetry. From [8,8] symmetry, there are 15 small index subgroup 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 unique mirrors are colored red, green, and blue, and alternatively colored triangles show the locat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
444 Symmetry
In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of ', having eight regular triangles around each vertex. Uniform colorings The half symmetry +,8,3= 4,3,3)can be shown with alternating two colors of triangles: : Symmetry From 4,4,4)symmetry, there are 15 small index subgroups (7 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 images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. Adding 3 bisecting mirrors across each fundamental domains creates 832 symmetry. The subgroup index-8 group, 1+,4,1+,4,1+,4)(222222) is the commutator subgroup of 4,4,4) A larger subgroup is constructed 4,4,4*) index 8, as (2*2222) with gyration points remo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
