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Snub Hexagonal Tiling
In geometry, the snub hexagonal tiling (or ''snub trihexagonal tiling'') is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex. It has Schläfli symbol ''sr''. The snub tetrahexagonal tiling is a related hyperbolic tiling with Schläfli symbol ''sr''. Conway calls it a snub hextille, constructed as a snub operation applied to a hexagonal tiling (hextille). There are three regular and eight semiregular tilings in the plane. This is the only one which does not have a reflection as a symmetry. There is only one uniform coloring of a snub trihexagonal tiling. (Labeling the colors by numbers, "3.3.3.3.6" gives "11213".) Circle packing The snub trihexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 5 other circles in the packing (kissing number).Order in Space: A design source book, Keith Critchlow, p.74-75, pattern E The lattice domain (red ... [...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] |
Orbifold Notation
In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it follows William Thurston in describing the orbifold obtained by taking the quotient of Euclidean space by the group under consideration. Groups representable in this notation include the point groups on the sphere (S^2), the frieze groups and wallpaper groups of the Euclidean plane (E^2), and their analogues on the hyperbolic plane (H^2). Definition of the notation The following types of Euclidean transformation can occur in a group described by orbifold notation: * reflection through a line (or plane) * translation by a vector * rotation of finite order around a point * infinite rotation around a line in 3- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagonal Tiling
In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon. A regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a regular pentagon, 108°, is not a divisor of 360°, the angle measure of a whole turn. However, regular pentagons can tile the hyperbolic plane with four pentagons around each vertex ( or more) and sphere with three pentagons; the latter produces a tiling topologically equivalent to the dodecahedron. Monohedral convex pentagonal tilings Fifteen types of convex pentagons are known to tile the plane monohedrally (i.e. with one type of tile). The most recent one was discovered in 2015. This list has been shown to be complete by (result subject to peer-review). showed that there are only eight edge-to-edge convex types, a result obtained independently by . Michaël Rao of the École normale supérieure de Lyon claimed in May 2017 to have found the proof that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P7 Dual
P7, P.7, or P-7 may refer to: Aircraft * Boeing XP-7, a prototype United States biplane fighter of the 1920s * HAT-P-7b, an extrasolar planet discovered in 2008 * Lockheed P-7, a proposed patrol aircraft ordered by the U.S. Navy * Piaggio P.7, an Italian hydrofoil racing seaplane of 1929 * PZL P.7, a 1930 Polish fighter aircraft Science * p7 protein, a protein providing nucleocapsid of HIV * Period 7 of the periodic table * Seoul Semiconductor, high-energy quad-chip LED light (SSC P7) Computer science * Intel Itanium processor code-name * Intel NetBurst microarchitecture, a seventh generation x86 microarchitecture * X.400 protocol for access of a message store from a user agent Other * Cinturato P7 tire developed by Pirelli * DR P7 Mix, a former radio station operated by the Danish Broadcasting Corporation * Heckler & Koch P7, a semi-automatic pistol manufactured in Germany * ProSieben, a German television channel * Russian Sky Airlines IATA airline code * South Scanian Reg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Planar Symmetry Groups
This article summarizes the classes of discrete symmetry groups of the Euclidean plane. The symmetry groups are named here by three naming schemes: International notation, orbifold notation, and Coxeter notation. There are three kinds of symmetry groups of the plane: *2 families of rosette groups – 2D point groups *7 frieze groups – 2D line groups *17 wallpaper groups – 2D space groups. Rosette groups There are two families of discrete two-dimensional point groups, and they are specified with parameter ''n'', which is the order of the group of the rotations in the group. Frieze groups The 7 frieze groups, the two-dimensional line groups, with a direction of periodicity are given with five notational names. The Schönflies notation is given as infinite limits of 7 dihedral groups. The yellow regions represent the infinite fundamental domain in each. Wallpaper groups The 17 wallpaper groups, with finite fundamental domains, are given by International notation, or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Face (geometry)
In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a ''polyhedron''. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions).. Polygonal face In elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include polyhedron side and Euclidean plane ''tile''. For example, any of the six squares that bound a cube is a face of the cube. Sometimes "face" is also used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells. Number of polygonal faces of a polyhedron Any convex polyhedron's surface has Euler characteristic :V - E + F = 2, where ''V'' is the number of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flower
A flower, sometimes known as a bloom or blossom, is the reproductive structure found in flowering plants (plants of the division Angiospermae). The biological function of a flower is to facilitate reproduction, usually by providing a mechanism for the union of sperm with eggs. Flowers may facilitate outcrossing (fusion of sperm and eggs from different individuals in a population) resulting from cross-pollination or allow selfing (fusion of sperm and egg from the same flower) when self-pollination occurs. There are two types of pollination: self-pollination and cross-pollination. Self-pollination occurs when the pollen from the anther is deposited on the stigma of the same flower, or another flower on the same plant. Cross-pollination is when pollen is transferred from the anther of one flower to the stigma of another flower on a different individual of the same species. Self-pollination happens in flowers where the stamen and carpel mature at the same time, and are positi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagon Tiling
In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon. A regular tiling, regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a Pentagon#Regular pentagons, regular pentagon, 108°, is not a divisor of 360°, the angle measure of a whole turn (geometry), turn. However, regular pentagons can tile the hyperbolic plane with Order-4 pentagonal tiling, four pentagons around each vertex (Order-5 pentagonal tiling, or more) and sphere with Spherical dodecahedron, three pentagons; the latter produces a tiling topologically equivalent to the regular dodecahedron, dodecahedron. Monohedral convex pentagonal tilings Fifteen types of convex pentagons are known to tile the plane monohedral tiling, monohedrally (i.e. with one type of tile). The most recent one was discovered in 2015. This list has been shown to be complete by (result subject to peer-review). showed that there are only eigh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isohedral Figure
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent but must be ''transitive'', i.e. must lie within the same '' symmetry orbit''. In other words, for any two faces and , there must be a symmetry of the ''entire'' figure by translations, rotations, and/or reflections that maps onto . For this reason, convex isohedral polyhedra are the shapes that will make fair dice. Isohedral polyhedra are called isohedra. They can be described by their face configuration. An isohedron has an even number of faces. The dual of an isohedral polyhedron is vertex-transitive, i.e. isogonal. The Catalan solids, the bipyramids, and the trapezohedra are all isohedral. They are the duals of the (isogonal) Archimedean solids, prisms, and antiprisms, respectively. The Platonic solids, which are either self-du ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chirality (mathematics)
In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that is not chiral is said to be ''achiral''. A chiral object and its mirror image are said to be enantiomorphs. The word ''chirality'' is derived from the Greek (cheir), the hand, the most familiar chiral object; the word ''enantiomorph'' stems from the Greek (enantios) 'opposite' + (morphe) 'form'. Examples Some chiral three-dimensional objects, such as the helix, can be assigned a right or left handedness, according to the right-hand rule. Many other familiar objects exhibit the same chiral symmetry of the human body, such as gloves and shoes. Right shoes differ from left shoes only by being mirror images of each other. In contrast thin gloves may not be considered chiral if you can wear them inside-out. The J, L, S and Z-shaped ''tetrominoes'' of the popul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Face-transitive
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent but must be ''transitive'', i.e. must lie within the same '' symmetry orbit''. In other words, for any two faces and , there must be a symmetry of the ''entire'' figure by translations, rotations, and/or reflections that maps onto . For this reason, convex isohedral polyhedra are the shapes that will make fair dice. Isohedral polyhedra are called isohedra. They can be described by their face configuration. An isohedron has an even number of faces. The dual of an isohedral polyhedron is vertex-transitive, i.e. isogonal. The Catalan solids, the bipyramids, and the trapezohedra are all isohedral. They are the duals of the (isogonal) Archimedean solids, prisms, and antiprisms, respectively. The Platonic solids, which are either self-du ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tiling Snub 3-6 Left Dual Face
{{disambiguation, surname ...
Tiling may refer to: *The physical act of laying tiles *Tessellations Computing *The compiler optimization of loop tiling *Tiled rendering, the process of subdividing an image by regular grid *Tiling window manager People *Heinrich Sylvester Theodor Tiling (1818–1871), physician and botanist *Reinhold Tiling (1893–1933), German rocket pioneer Other uses *Neuronal tiling *Tile drainage, an agriculture practice that removes excess water from soil *Tiling (crater), a small, undistinguished crater on the far side of the Moon See also *Brickwork *Packing (other) *Tiling puzzle Tiling puzzles are puzzles involving two-dimensional packing problems in which a number of flat shapes have to be assembled into a larger given shape without overlaps (and often without gaps). Some tiling puzzles ask you to dissect a given ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
