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Parabidiminished Rhombicosidodecahedron
In geometry, the parabidiminished rhombicosidodecahedron is one of the Johnson solids (). It is also a canonical polyhedron. It can be constructed as a rhombicosidodecahedron with two opposing pentagonal cupolae removed. Related Johnson solids are the diminished rhombicosidodecahedron In geometry, the diminished rhombicosidodecahedron is one of the Johnson solids (). It can be constructed as a rhombicosidodecahedron with one pentagonal cupola In geometry, the pentagonal cupola is one of the Johnson solids (). It can be ob ... () where one cupola is removed, the metabidiminished rhombicosidodecahedron () where two non-opposing cupolae are removed, and the tridiminished rhombicosidodecahedron () where three cupolae are removed. Example External links * Johnson solids {{Polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Polyhedron
In geometry, the midsphere or intersphere of a polyhedron is a sphere which is tangent to every edge of the polyhedron. That is to say, it touches any given edge at exactly one point. Not every polyhedron has a midsphere, but for every convex polyhedron there is a combinatorially equivalent polyhedron, the canonical polyhedron, that does have a midsphere. The radius of the midsphere is called the midradius. Examples The uniform polyhedra, including the regular, quasiregular and semiregular polyhedra and their duals all have midspheres. In the regular polyhedra, the inscribed sphere, midsphere, and circumscribed sphere all exist and are concentric, and the midsphere touches each edge at its midpoint. Not every irregular tetrahedron has a midsphere. The tetrahedra that have a midsphere have been called "Crelle's tetrahedra"; they form a four-dimensional subfamily of the six-dimensional space of all tetrahedra (as parameterized by their six edge lengths). Tangent circles If is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johnson Solid
In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), vertex. An example of a Johnson solid is the square-based Pyramid (geometry), pyramid with equilateral sides (square pyramid, ); it has 1 square face and 4 triangular faces. Some authors require that the solid not be uniform polyhedron, uniform (i.e., not Platonic solid, Archimedean solid, prism (geometry), uniform prism, or uniform antiprism) before they refer to it as a “Johnson solid”. As in any strictly convex solid, at least three faces meet at every vertex, and the total of their angles is less than 360 degrees. Since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex. The pentagonal pyramid () is an example that has a degree-5 vertex. Although there is no obvious restriction tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bigyrate Diminished Rhombicosidodecahedron
In geometry, the bigyrate diminished rhombicosidodecahedron is one of the Johnson solids (). It can be constructed as a rhombicosidodecahedron with two pentagonal cupolae rotated through 36 degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...s, and a third pentagonal cupola removed. (None of the cupolae can be adjacent.) External links * Johnson solids {{Polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metabidiminished Rhombicosidodecahedron
In geometry, the metabidiminished rhombicosidodecahedron is one of the Johnson solids (). It can be constructed as a rhombicosidodecahedron with two non-opposing pentagonal cupolae () removed. Related Johnson solids are: * The diminished rhombicosidodecahedron () where one cupola is removed, * The parabidiminished rhombicosidodecahedron In geometry, the parabidiminished rhombicosidodecahedron is one of the Johnson solids (). It is also a canonical polyhedron. It can be constructed as a rhombicosidodecahedron with two opposing pentagonal cupolae removed. Related Johnson soli ... () where two opposing cupolae are removed, * The gyrate bidiminished rhombicosidodecahedron () where two non-opposing cupolae are removed and a third is rotated 36 degrees, * And the tridiminished rhombicosidodecahedron () where three cupolae are removed. External links * Johnson solids {{Polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-Collinearity, collinear, determine a unique triangle and simultaneously, a unique Plane (mathematics), plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Types of triangle The terminology for categorizing triangles is more than two thousand years old, having been defined on the very first page of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square (geometry)
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ''ABCD'' would be denoted . Characterizations A convex quadrilateral is a square if and only if it is any one of the following: * A rectangle with two adjacent equal sides * A rhombus with a right vertex angle * A rhombus with all angles equal * A parallelogram with one right vertex angle and two adjacent equal sides * A quadrilateral with four equal sides and four right angles * A quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other (i.e., a rhombus with equal diagonals) * A convex quadrilateral with successiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting. A self-intersecting ''regular pentagon'' (or ''star pentagon'') is called a pentagram. Regular pentagons A '' regular pentagon'' has Schläfli symbol and interior angles of 108°. A '' regular pentagon'' has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio to its sides. Given its side length t, its height H (distance from one side to the opposite vertex), width W (distance between two farthest separated points, which equals the diagonal length D) and circumradius R are given by: :\begin H &= \frac~t \approx 1.539~t, \\ W= D &= \frac~t\approx 1.618~t, \\ W &= \sqr ... [...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] |
Convex Polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others''Mathematical Programming'', by Melvyn W. Jeter (1986) p. 68/ref> (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In the influential textbooks of Grünbaum and Ziegler on the subject, as well as in many other texts in discrete geometry, convex polytopes are often simply called "polytopes". Grünbaum points out that this is solely to avoi ... [...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] |
Midsphere
In geometry, the midsphere or intersphere of a polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ... is a sphere which is tangent to every Edge (geometry), edge of the polyhedron. That is to say, it touches any given edge at exactly one point. Not every polyhedron has a midsphere, but for every convex polyhedron there is a combinatorially equivalent polyhedron, the canonical polyhedron, that does have a midsphere. The radius of the midsphere is called the midradius. Examples The uniform polyhedron, uniform polyhedra, including the regular polyhedron, regular, Quasiregular polyhedron, quasiregular and Semiregular polyhedron, semiregular polyhedra and their Dual polyhedron, duals all have midspheres. In the regular polyhedra, the inscribed sphere, midsphere, and circumscribe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhombicosidodecahedron
In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces. It has 20 regular triangular faces, 30 square (geometry), square faces, 12 regular pentagonal faces, 60 vertex (geometry), vertices, and 120 edge (geometry), edges. Names Johannes Kepler in Harmonices Mundi (1618) named this polyhedron a ''rhombicosidodecahedron'', being short for ''truncated icosidodecahedral rhombus'', with ''icosidodecahedral rhombus'' being his name for a rhombic triacontahedron. There are different truncations of a rhombic triacontahedron into a topology, topological rhombicosidodecahedron: Prominently its rectification (geometry), rectification (left), the one that creates the uniform solid (center), and the rectification of the dual icosidodecahedron (right), which is the core of the dual compound. It can also be called an ''Expansion (geometry), expanded'' or ''Cantellation (geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
