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Parabigyrate Rhombicosidodecahedron
In geometry, the parabigyrate rhombicosidodecahedron is one of the Johnson solids (). It can be constructed as a rhombicosidodecahedron with two opposing pentagonal cupolae rotated through 36 degrees. It is also a canonical polyhedron. Alternative Johnson solids, constructed by rotating different cupolae of a rhombicosidodecahedron, are: * The gyrate rhombicosidodecahedron () where only one cupola is rotated; * The metabigyrate rhombicosidodecahedron () where two non-opposing cupolae are rotated; * And the trigyrate rhombicosidodecahedron In geometry, the trigyrate rhombicosidodecahedron is one of the Johnson solids (). It contains 20 triangles, 30 squares and 12 pentagons. It is also a canonical polyhedron. It can be constructed as a rhombicosidodecahedron with three pentag ... () where three cupolae are rotated. External links * Johnson solids {{Polyhedron-stub ... [...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 each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides ( ); it has 1 square face and 4 triangular faces. Some authors require that the solid not be uniform (i.e., not Platonic solid, Archimedean solid, 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 that any given regular polygon cannot be a face of a Johnson solid, it turns out that the face ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gyrate Rhombicosidodecahedron
In geometry, the gyrate rhombicosidodecahedron is one of the Johnson solids (). It is also a canonical polyhedron. Related polyhedron It can be constructed as a rhombicosidodecahedron with one pentagonal cupola rotated through 36 degrees. They have the same faces around each vertex, but vertex configurations along the rotation become a different order, . Alternative Johnson solids, constructed by rotating different cupolae of a rhombicosidodecahedron, are: * The parabigyrate rhombicosidodecahedron () where two opposing cupolae are rotated; * The metabigyrate rhombicosidodecahedron () where two non-opposing cupolae are rotated; * And the trigyrate rhombicosidodecahedron In geometry, the trigyrate rhombicosidodecahedron is one of the Johnson solids (). It contains 20 triangles, 30 squares and 12 pentagons. It is also a canonical polyhedron. It can be constructed as a rhombicosidodecahedron with three pentag ... () where three cupolae are rotated. External links * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metabigyrate Rhombicosidodecahedron
In geometry, the metabigyrate rhombicosidodecahedron is one of the Johnson solids (). It can be constructed as a rhombicosidodecahedron with two non-opposing pentagonal cupolae rotated through 36 degrees. It is also a canonical polyhedron. Alternative Johnson solids, constructed by rotating different cupolae of a rhombicosidodecahedron, are: * The gyrate rhombicosidodecahedron () where only one cupola is rotated; * The parabigyrate rhombicosidodecahedron () where two opposing cupolae are rotated; * And the trigyrate rhombicosidodecahedron In geometry, the trigyrate rhombicosidodecahedron is one of the Johnson solids (). It contains 20 triangles, 30 squares and 12 pentagons. It is also a canonical polyhedron. It can be constructed as a rhombicosidodecahedron with three pentag ... () where three cupolae are rotated. External links * World of Polyhedra - metabigyrate rhombicosidodecahedron(interactive rotatable wireframe applet) Johnson solids {{Polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three edges and three 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- collinear, determine a unique triangle and simultaneously, a unique 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 Euclid's Elements. The names used for modern classification a ... [...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 w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagon
In geometry, a pentagon (from the Greek language, 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 polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ''regular pentagon'' (or ''star polygon, star pentagon'') is called a pentagram. Regular pentagons A ''regular polygon, regular pentagon'' has Schläfli symbol and interior angles of 108°. A ''regular polygon, 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 polygon, 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 cir ... [...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 avo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Midsphere
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 ... [...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 geom ... [...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 faces, 12 regular pentagonal faces, 60 vertices, and 120 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 topological rhombicosidodecahedron: Prominently its 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 '' expanded'' or '' cantellated'' dodecahedron or icosahedron, from truncation operations on either uniform polyhedron. Dimensions For a r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagonal Cupola
In geometry, the pentagonal cupola is one of the Johnson solids (). It can be obtained as a slice of the rhombicosidodecahedron. The pentagonal cupola consists of 5 equilateral triangles, 5 squares, 1 pentagon, and 1 decagon. Formulae The following formulae for volume, surface area and circumradius can be used if all faces are regular, with edge length ''a'': Stephen Wolfram,Pentagonal cupola from Wolfram Alpha. Retrieved April 11, 2020. :V=\left(\frac\left(5+4\sqrt\right)\right)a^3\approx2.32405a^3, :A=\left(\frac\left(20+5\sqrt+\sqrt\right)\right)a^2\approx16.57975a^2, :R=\left(\frac\sqrt\right)a\approx2.23295a. The height of the pentagonal cupola is :h = \sqrta \approx 0.52573a. Related polyhedra Dual polyhedron The dual of the pentagonal cupola has 10 triangular faces and 5 kite faces: Other convex cupolae Crossed pentagrammic cupola In geometry, the crossed pentagrammic cupola is one of the nonconvex Johnson solid isomorphs, being topologically iden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree (angle)
A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of a plane angle in which one full rotation is 360 degrees. It is not an SI unit—the SI unit of angular measure is the radian—but it is mentioned in the SI brochure as an accepted unit. Because a full rotation equals 2 radians, one degree is equivalent to radians. History The original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year. Ancient astronomers noticed that the sun, which follows through the ecliptic path over the course of the year, seems to advance in its path by approximately one degree each day. Some ancient calendars, such as the Persian calendar and the Babylonian calendar, used 360 days for a year. The use of a calendar with 360 days may be related to the use of sexagesimal numbers. Anothe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
