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List Of Convex Uniform 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 duals, each made from one type of irregular face. John Conway called these uniform duals ''Catalan tilings'', in parallel to the Catalan solid polyhedra. Uniform tilings are listed by their vertex configuration, the sequence of faces that exist on each vertex. For example ''4.8.8'' means one square and two octagons on a vertex. These 11 uniform tilings have 32 different ''uniform colorings''. A uniform coloring allows identical sided polygons at a vertex to be colored differently, while still maintaining vertex-uniformity and transformational congruence between vertices. (Note: Some of the tiling images shown below are ''not'' color-uniform.) In addition to the 11 convex uniform tilings, there are also 14 known nonconvex tilings, using s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fritz Laves
Fritz Henning Emil Paul Berndt Laves (27 February 1906 – 12 August 1978) was a German crystallographer who served as the president of the German Mineralogical Society from 1956 to 1958. He is the namesake of Laves phases and the Laves tilings; the Laves graph, a highly-symmetrical three-dimensional crystal structure that he studied, was named after him by H. S. M. Coxeter. Education and career Laves was born in Hanover, the son of a judge and the great-grandson of architect Georg Ludwig Friedrich Laves. He grew up in Göttingen, where his interests included piano music as well as collecting rocks and minerals. He began his university studies in geology in 1924 at the University of Innsbruck, and continued at the University of Göttingen before moving to ETH Zurich for doctoral studies under Paul Niggli. In 1929 he took a faculty position under Victor Goldschmidt at Göttingen. He tried unsuccessfully to prevent Goldschmidt from being dismissed in 1933, and later had diffi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tiling 4a Dual Face
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) Packing may refer to: Law and politics * Jury packing, selecting biased jurors for a court case * Packing and cracking, a method of creating voting districts to give a political party an advantage Other uses * Packing (firestopping), the proces ... * Tiling puzzle {{disambiguation, surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dihedral Group
In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. The notation for the dihedral group differs in geometry and abstract algebra. In geometry, or refers to the symmetries of the n-gon, -gon, a group of order . In abstract algebra, refers to this same dihedral group. This article uses the geometric convention, . Definition The word "dihedral" comes from "di-" and "-hedron". The latter comes from the Greek word hédra, which means "face of a geometrical solid". Overall it thus refers to the two faces of a polygon. Elements A regular polygon with n sides has 2n different symmetries: n rotational symmetry, rotational symmetries and n reflection symmetry, reflection symmetries. Usually, we take n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Face Configuration
In geometry, a vertex configuration is a shorthand notation for representing a polyhedron or Tessellation, tiling as the sequence of Face (geometry), faces around a Vertex (geometry), vertex. It has variously been called a vertex description, vertex type, vertex symbol, vertex arrangement, vertex pattern, face-vector, vertex sequence. It is also called a Cundy and Rollett symbol for its usage for the Archimedean solids in their 1952 book ''Mathematical Models (Cundy and Rollett), Mathematical Models''.Laughlin (2014), p. 16 For uniform polyhedron, uniform polyhedra, there is only one vertex type and therefore the vertex configuration fully defines the polyhedron. (Chirality (mathematics), Chiral polyhedra exist in mirror-image pairs with the same vertex configuration.) For example, "" indicates a vertex belonging to 4 faces, alternating triangles and pentagons. This vertex configuration defines the vertex-transitive icosidodecahedron. The notation is cyclic and therefore is equival ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stereohedron
In geometry and crystallography, a stereohedron is a convex polyhedron that isohedral tiling, fills space isohedrally, meaning that the symmetries of the tiling take any copy of the stereohedron to any other copy. Two-dimensional analogues to the stereohedra are called planigons. Higher dimensional polytopes can also be stereohedra, while they would more accurately be called stereotopes. Plesiohedra A subset of stereohedra are called plesiohedrons, defined as the Voronoi diagram, Voronoi cells of a symmetric Delone set. Parallelohedrons are plesiohedra which are space-filling by translation only. Edges here are colored as parallel vectors. Other periodic stereohedra The catoptric tessellation contain stereohedra cells. Dihedral angles are integer divisors of 180°, and are colored by their order. The first three are the fundamental domains of _3, _3, and _3 symmetry, represented by Coxeter-Dynkin diagrams: , and . _3 is a half symmetry of _3, and _3 is a quarter symmetry. An ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planigon
In geometry, a planigon is a convex polygon that can fill the plane with only copies of itself (Homotopy#Isotopy, isotopic to the Prototile, fundamental units of Monohedral tiling, monohedral tessellations). In the Euclidean plane there are 3 regular planigons; equilateral triangle, squares, and regular hexagons; and 8 List of Euclidean uniform tilings, semiregular planigons; and 4 Euclidean tilings by convex regular polygons, demiregular planigons which can tile the plane only with other planigons. All angles of a planigon are whole divisors of 360°. Tilings are made by edge-to-edge connections by perpendicular bisectors of the edges of the original uniform lattice, or centroids along common edges (they coincide). Tilings made from planigons can be seen as Dual polyhedron, dual tilings to the List of Euclidean uniform tilings, regular, semiregular, and Euclidean tilings by convex regular polygons, demiregular tilings of the plane by regular polygons. History In the 1987 book, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prototile
In mathematics, a prototile is one of the shapes of a tile in a tessellation. Definition A tessellation of the plane or of any other space is a cover of the space by closed shapes, called tiles, that have disjoint interiors. Some of the tiles may be congruent to one or more others. If is the set of tiles in a tessellation, a set of shapes is called a set of prototiles if no two shapes in are congruent to each other, and every tile in is congruent to one of the shapes in . It is possible to choose many different sets of prototiles for a tiling: translating or rotating any one of the prototiles produces another valid set of prototiles. However, every set of prototiles has the same cardinality, so the number of prototiles is well defined. A tessellation is said to be ''monohedral'' if it has exactly one prototile. Aperiodicity A set of prototiles is said to be aperiodic if every tiling with those prototiles is an aperiodic tiling. In March 2023, four researchers, Chaim Goodm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
