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Octadecagon
In geometry, an octadecagon (or octakaidecagon) or 18-gon is an eighteen-sided polygon. Regular octadecagon A '' regular octadecagon'' has a Schläfli symbol and can be constructed as a quasiregular truncated enneagon, t, which alternates two types of edges. Construction As 18 = 2 × 32, a regular octadecagon cannot be constructed using a compass and straightedge. However, it is constructible using neusis, or an angle trisection with a tomahawk. The following approximate construction is very similar to that of the enneagon, as an octadecagon can be constructed as a truncated enneagon. It is also feasible with exclusive use of compass and straightedge. Symmetry The ''regular octadecagon'' has Dih18 symmetry, order 36. There are 5 subgroup dihedral symmetries: Dih9, (Dih6, Dih3), and (Dih2 Dih1), and 6 cyclic group symmetries: (Z18, Z9), (Z6, Z3), and (Z2, Z1). These 15 symmetries can be seen in 12 distinct symmetries on the octadecagon. John Conway labels these by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
18-gon Rhombic Dissection-size2
In geometry, an octadecagon (or octakaidecagon) or 18-gon is an eighteen-sided polygon. Regular octadecagon A '' regular octadecagon'' has a Schläfli symbol and can be constructed as a quasiregular truncated enneagon, t, which alternates two types of edges. Construction As 18 = 2 × 32, a regular octadecagon cannot be constructed using a compass and straightedge. However, it is constructible using neusis, or an angle trisection with a tomahawk. The following approximate construction is very similar to that of the enneagon, as an octadecagon can be constructed as a truncated enneagon. It is also feasible with exclusive use of compass and straightedge. Symmetry The ''regular octadecagon'' has Dih18 symmetry, order 36. There are 5 subgroup dihedral symmetries: Dih9, (Dih6, Dih3), and (Dih2 Dih1), and 6 cyclic group symmetries: (Z18, Z9), (Z6, Z3), and (Z2, Z1). These 15 symmetries can be seen in 12 distinct symmetries on the octadecagon. John Conway labels these ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
18-gon Rhombic Dissectionx
In geometry, an octadecagon (or octakaidecagon) or 18-gon is an eighteen-sided polygon. Regular octadecagon A '' regular octadecagon'' has a Schläfli symbol and can be constructed as a quasiregular truncated enneagon, t, which alternates two types of edges. Construction As 18 = 2 × 32, a regular octadecagon cannot be constructed using a compass and straightedge. However, it is constructible using neusis, or an angle trisection with a tomahawk. The following approximate construction is very similar to that of the enneagon, as an octadecagon can be constructed as a truncated enneagon. It is also feasible with exclusive use of compass and straightedge. Symmetry The ''regular octadecagon'' has Dih18 symmetry, order 36. There are 5 subgroup dihedral symmetries: Dih9, (Dih6, Dih3), and (Dih2 Dih1), and 6 cyclic group symmetries: (Z18, Z9), (Z6, Z3), and (Z2, Z1). These 15 symmetries can be seen in 12 distinct symmetries on the octadecagon. John Conway labels these ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Polygon
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon (effectively a straight line), if the edge length is fixed. General properties ''These properties apply to all regular polygons, whether convex or star.'' A regular ''n''-sided polygon has rotational symmetry of order ''n''. All vertices of a regular polygon lie on a common circle (the circumscribed circle); i.e., they are concyclic points. That is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a tangential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructible Polygon
In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinitely many constructible polygons, but only 31 with an odd number of sides are known. Conditions for constructibility Some regular polygons are easy to construct with compass and straightedge; others are not. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides, and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.Bold, Benjamin. ''Famous Problems of Geometry and How to Solve Them'', Dover Publications, 1982 (orig. 1969). This led to the question being posed: is it possible to construct ''all'' regular polygons with compass and straightedge? If not, which ''n''-gons (that is, polygons with ''n'' edges) are constructible and which are not? Carl ... [...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 geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chaim Goodman-Strauss
Chaim Goodman-Strauss (born June 22, 1967 in Austin TX) is an American mathematician who works in convex geometry, especially aperiodic tiling. He is on the faculty of the University of Arkansas and is a co-author with John H. Conway of ''The Symmetries of Things'', a comprehensive book surveying the mathematical theory of patterns. Education and career Goodman-Strauss received both his B.S. (1988) and Ph.D. (1994) in mathematics from the University of Texas at Austin.Chaim Goodman-Strauss The College Board His doctoral advisor was John Edwin Luecke. He joined the faculty at the |
Tiling By Regular Polygons
Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his ''Harmonices Mundi'' (Latin: ''The Harmony of the World'', 1619). Notation of Euclidean tilings Euclidean tilings are usually named after Cundy & Rollett’s notation. This notation represents (i) the number of vertices, (ii) the number of polygons around each vertex (arranged clockwise) and (iii) the number of sides to each of those polygons. For example: 36; 36; 34.6, tells us there are 3 vertices with 2 different vertex types, so this tiling would be classed as a ‘3-uniform (2-vertex types)’ tiling. Broken down, 36; 36 (both of different transitivity class), or (36)2, tells us that there are 2 vertices (denoted by the superscript 2), each with 6 equilateral 3-sided polygons (triangles). With a final vertex 34.6, 4 more contiguous equilateral triangles and a single regular hexagon. However, this notation has two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
9-cube
In geometry, a 9-cube is a nine-dimensional hypercube with 512 vertices, 2304 edges, 4608 square faces, 5376 cubic cells, 4032 tesseract 4-faces, 2016 5-cube 5-faces, 672 6-cube 6-faces, 144 7-cube 7-faces, and 18 8-cube 8-faces. It can be named by its Schläfli symbol , being composed of three 8-cubes around each 7-face. It is also called an enneract, a portmanteau of tesseract (the ''4-cube'') and ''enne'' for nine (dimensions) in Greek. It can also be called a regular octadeca-9-tope or octadecayotton, as a nine-dimensional polytope constructed with 18 regular facets. It is a part of an infinite family of polytopes, called hypercubes. The dual of a 9-cube can be called a 9-orthoplex, and is a part of the infinite family of cross-polytopes. Cartesian coordinates Cartesian coordinates for the vertices of a 9-cube centered at the origin and edge length 2 are : (±1,±1,±1,±1,±1,±1,±1,±1,±1) while the interior of the same consists of all points (''x''0, ''x''1, '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Petrie Polygon
In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a regular polyhedron is a skew polygon such that every two consecutive sides (but no three) belongs to one of the faces. Petrie polygons are named for mathematician John Flinders Petrie. For every regular polytope there exists an orthogonal projection onto a plane such that one Petrie polygon becomes a regular polygon with the remainder of the projection interior to it. The plane in question is the Coxeter plane of the symmetry group of the polygon, and the number of sides, , is the Coxeter number of the Coxeter group. These polygons and projected graphs are useful in visualizing symmetric structure of the higher-dimensional regular polytopes. Petrie polygons can be defined more generally for any embedded graph. They form the faces of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
