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Small Dodecicosahedron
In geometry, the small dodecicosahedron (or small dodekicosahedron) is a nonconvex uniform polyhedron, indexed as U50. It has 32 faces (20 hexagons and 12 decagons), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral. Related polyhedra It shares its vertex arrangement with the great stellated truncated dodecahedron. It additionally shares its edges with the small icosicosidodecahedron (having the hexagonal faces in common) and the small ditrigonal dodecicosidodecahedron In geometry, the small ditrigonal dodecicosidodecahedron (or small dodekified icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U43. It has 44 faces (20 triangles, 12 pentagrams and 12 decagons), 120 edges, and 60 vertices. Its ver ... (having the decagonal faces in common). References External links * Uniform polyhedra {{Polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Dodecicosahedron
In geometry, the small dodecicosahedron (or small dodekicosahedron) is a nonconvex uniform polyhedron, indexed as U50. It has 32 faces (20 hexagons and 12 decagons), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral. Related polyhedra It shares its vertex arrangement with the great stellated truncated dodecahedron. It additionally shares its edges with the small icosicosidodecahedron (having the hexagonal faces in common) and the small ditrigonal dodecicosidodecahedron In geometry, the small ditrigonal dodecicosidodecahedron (or small dodekified icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U43. It has 44 faces (20 triangles, 12 pentagrams and 12 decagons), 120 edges, and 60 vertices. Its ver ... (having the decagonal faces in common). References External links * Uniform polyhedra {{Polyhedron-stub ... [...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] |
Nonconvex Uniform Polyhedron
In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, or both. The complete set of 57 nonprismatic uniform star polyhedra includes the 4 regular ones, called the Kepler–Poinsot polyhedra, 5 quasiregular ones, and 48 semiregular ones. There are also two infinite sets of ''uniform star prisms'' and ''uniform star antiprisms''. Just as (nondegenerate) star polygons (which have polygon density greater than 1) correspond to circular polygons with overlapping tiles, star polyhedra that do not pass through the center have polytope density greater than 1, and correspond to spherical polyhedra with overlapping tiles; there are 47 nonprismatic such uniform star polyhedra. The remaining 10 nonprismatic uniform star polyhedra, those that pass through the center, are the hemipolyhedra as well as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexagon
In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regular hexagon A ''regular polygon, regular hexagon'' has Schläfli symbol and can also be constructed as a Truncation (geometry), truncated equilateral triangle, t, which alternates two types of edges. A regular hexagon is defined as a hexagon that is both equilateral polygon, equilateral and equiangular polygon, equiangular. It is bicentric polygon, bicentric, meaning that it is both cyclic polygon, cyclic (has a circumscribed circle) and tangential polygon, tangential (has an inscribed circle). The common length of the sides equals the radius of the circumscribed circle or circumcircle, which equals \tfrac times the apothem (radius of the inscribed figure, inscribed circle). All internal angles are 120 degree (angle), degrees. A regular hexago ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decagon
In geometry, a decagon (from the Greek δέκα ''déka'' and γωνία ''gonía,'' "ten angles") is a ten-sided polygon or 10-gon.. The total sum of the interior angles of a simple decagon is 1440°. A self-intersecting ''regular decagon'' is known as a decagram. Regular decagon A '' regular decagon'' has all sides of equal length and each internal angle will always be equal to 144°. Its Schläfli symbol is and can also be constructed as a truncated pentagon, t, a quasiregular decagon alternating two types of edges. Side length The picture shows a regular decagon with side length a and radius R of the circumscribed circle. * The triangle E_E_1M has to equally long legs with length R and a base with length a * The circle around E_1 with radius a intersects ]M\,E_ _in_a_point_P_(not_designated_in_the_picture)._ *_Now_the_triangle_\;_is_a_isosceles_triangle.html" ;"title="/math> in a point P (not designated in the picture). * Now the triangle \; is a isosceles triang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points around the face. When done, these lines form a complete circuit, i.e. a polygon, around the vertex. This polygon is the vertex figure. More precise formal definitions can vary quite widely, according to circumstance. For example Coxeter (e.g. 1948, 1954) varies his definition as convenient for the current area of discussion. Most of the following definitions of a vertex figure apply equally well to infinite tessellation, tilings or, by extension, to Honeycomb (geometry), space-filling tessellation with polytope Cell (geometry), cells and other higher-dimensional polytopes. As a flat slice Make a slice through the corner of the polyhedron, cutting through all the edges ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, derived from greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons (e.g. pentagon). Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices A, B, C and D is sometimes denoted as \square ABCD. Quadrilaterals are either simple (not self-intersecting), or complex (self-intersecting, or crossed). Simple quadrilaterals are either convex or concave. The interior angles of a simple (and planar) quadrilateral ''ABCD'' add up to 360 degrees of arc, that is :\angle A+\angle B+\angle C+\angle D=360^. This is a special case of the ''n''-gon interior angle sum formula: ''S'' = (''n'' − 2) × 180°. All non-self-crossing quadrilaterals tile the plane, b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Stellated Truncated Dodecahedron
In geometry, the great stellated truncated dodecahedron (or quasitruncated great stellated dodecahedron or great stellatruncated dodecahedron) is a nonconvex uniform polyhedron, indexed as U66. It has 32 faces (20 triangles and 12 decagrams), 90 edges, and 60 vertices. It is given a Schläfli symbol Related polyhedra It shares its vertex arrangement with three other uniform polyhedra: the small icosicosidodecahedron, the small ditrigonal dodecicosidodecahedron, and the small dodecicosahedron: Cartesian coordinates Cartesian coordinates for the vertices of a great stellated truncated dodecahedron are all the even permutations of \begin \Bigl(& 0,& \pm\,\varphi,& \pm \bigl -\frac\bigr&\Bigr) \\ \Bigl(& \pm\,\varphi,& \pm\,\frac,& \pm\,\frac &\Bigr) \\ \Bigl(& \pm\,\frac,& \pm\,\frac,& \pm\,2 &\Bigr) \end where \varphi = \tfrac is the golden ratio. See also * List of uniform polyhedra In geometry, a uniform polyhedron is a polyhedron which has regular pol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Icosicosidodecahedron
In geometry, the small icosicosidodecahedron (or small icosified icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U31. It has 52 faces (20 triangles, 12 pentagrams, and 20 hexagons), 120 edges, and 60 vertices. Related polyhedra It shares its vertex arrangement with the great stellated truncated dodecahedron. It additionally shares its edges with the small ditrigonal dodecicosidodecahedron (having the triangular and pentagrammic faces in common) and the small dodecicosahedron (having the hexagonal faces in common). See also * List of uniform polyhedra In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive ( transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are c ... References External links * {{Polyhedron-stub Uniform polyhedra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Ditrigonal Dodecicosidodecahedron
In geometry, the small ditrigonal dodecicosidodecahedron (or small dodekified icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U43. It has 44 faces (20 triangles, 12 pentagrams and 12 decagons), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral. Related polyhedra It shares its vertex arrangement with the great stellated truncated dodecahedron. It additionally shares its edges with the small icosicosidodecahedron (having the triangular and pentagrammic faces in common) and the small dodecicosahedron (having the decagonal faces in common). See also * List of uniform polyhedra In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive ( transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are c ... References External links * Uniform polyhedra {{Polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Stellated Truncated Dodecahedron
In geometry, the great stellated truncated dodecahedron (or quasitruncated great stellated dodecahedron or great stellatruncated dodecahedron) is a nonconvex uniform polyhedron, indexed as U66. It has 32 faces (20 triangles and 12 decagrams), 90 edges, and 60 vertices. It is given a Schläfli symbol Related polyhedra It shares its vertex arrangement with three other uniform polyhedra: the small icosicosidodecahedron, the small ditrigonal dodecicosidodecahedron, and the small dodecicosahedron: Cartesian coordinates Cartesian coordinates for the vertices of a great stellated truncated dodecahedron are all the even permutations of \begin \Bigl(& 0,& \pm\,\varphi,& \pm \bigl -\frac\bigr&\Bigr) \\ \Bigl(& \pm\,\varphi,& \pm\,\frac,& \pm\,\frac &\Bigr) \\ \Bigl(& \pm\,\frac,& \pm\,\frac,& \pm\,2 &\Bigr) \end where \varphi = \tfrac is the golden ratio. See also * List of uniform polyhedra In geometry, a uniform polyhedron is a polyhedron which has regular pol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Icosicosidodecahedron
In geometry, the small icosicosidodecahedron (or small icosified icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U31. It has 52 faces (20 triangles, 12 pentagrams, and 20 hexagons), 120 edges, and 60 vertices. Related polyhedra It shares its vertex arrangement with the great stellated truncated dodecahedron. It additionally shares its edges with the small ditrigonal dodecicosidodecahedron (having the triangular and pentagrammic faces in common) and the small dodecicosahedron (having the hexagonal faces in common). See also * List of uniform polyhedra In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive ( transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are c ... References External links * {{Polyhedron-stub Uniform polyhedra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
