171 (number)
171 (one hundred [and] seventyone) is the natural number following 170 (number), 170 and preceding 172 (number), 172. In mathematics 171 is a triangular number and a Jacobsthal number. There are 171 transitive relations on three labeled elements, and 171 combinatorially distinct ways of subdividing a cuboid by flat cuts into a mesh of tetrahedron, tetrahedra, without adding extra vertices. The diagonals of a regular decagon meet at 171 points, including both crossings and the vertices of the decagon. There are 171 Face (geometry), faces and Edge (geometry), edges in the 57cell, an Abstract polytope#The 11cell and the 57cell, abstract Uniform 4polytope, 4polytope with hemiRegular dodecahedron, dodecahedral Polyhedron, cells that is its own dual polytope. See also * The year AD 171 or 171 BC * List of highways numbered 171 * References {{DEFAULTSORT:171 (Number) Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Natural Number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal numbers'', and numbers used for ordering are called ''ordinal numbers''. Natural numbers are sometimes used as labels, known as '' nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports jersey numbers). Some definitions, including the standard ISO 800002, begin the natural numbers with , corresponding to the nonnegative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural numbers form a set. Many other number sets are built by succ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Edge (geometry)
In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higherdimensional polytope. In a polygon, an edge is a line segment on the boundary, and is often called a polygon side. In a polyhedron or more generally a polytope, an edge is a line segment where two faces (or polyhedron sides) meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal. Relation to edges in graphs In graph theory, an edge is an abstract object connecting two graph vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment. However, any polyhedron can be represented by its skeleton or edgeskeleton, a graph whose vertices are the geometric vertices of the polyhedron and whose edges correspond to the geometric edges. Conversely, the graphs that are skeletons of threedimensional polyhedra can be characterized by Steinitz's the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Dual Polytope
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all can also be constructed as geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a corresponding symmetry class. For example, the regular polyhedrathe (convex) Platonic solids and (star) Kepler–Poinsot polyhedraform dual pairs, where the regular tetrahedron is selfdual. The dual of an isogonal polyhedron (one in which any two vertices are equivalent under symmetries of the polyhedron) is an isohedral polyhedron (one in which any two faces are equivalent .., and vice v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a threedimensional 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 the same plane. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a 3dimensional example of a polytope, a more general concept in any number of dimensions. Definition Convex polyhedra are welldefined, with several equivalent standard definitions. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of "polyhedron" have been given within particular contexts,. some more rigorous than others, and there is not universal agreement over which of these to choose. Some of these definitions exclude shapes that have often been counted as polyhedra (such as the selfcrossing polyhedra) or include shapes that are often not considered as valid po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Regular Dodecahedron
A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals (60 face diagonals, 100 space diagonals). It is represented by the Schläfli symbol . Dimensions If the edge length of a regular dodecahedron is a, the radius of a circumscribed sphere (one that touches the regular dodecahedron at all vertices) is :r_u = a\frac \left(1 + \sqrt\right) \approx 1.401\,258\,538 \cdot a and the radius of an inscribed sphere (tangent to each of the regular dodecahedron's faces) is :r_i = a\frac \sqrt \approx 1.113\,516\,364 \cdot a while the midradius, which touches the middle of each edge, is :r_m = a\frac \left(3 +\sqrt\right) \approx 1.309\,016\,994 \cdot a These quantities may also be expressed as :r_u = a\, \frac \phi :r_i = a\, \frac :r_m = a\, \frac where ''ϕ'' is the golde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Uniform 4polytope
In geometry, a uniform 4polytope (or uniform polychoron) is a 4dimensional polytope which is vertextransitive and whose cells are uniform polyhedra, and faces are regular polygons. There are 47 nonprismatic convex uniform 4polytopes. There are two infinite sets of convex prismatic forms, along with 17 cases arising as prisms of the convex uniform polyhedra. There are also an unknown number of nonconvex star forms. History of discovery * Convex Regular polytopes: ** 1852: Ludwig Schläfli proved in his manuscript ''Theorie der vielfachen Kontinuität'' that there are exactly 6 regular polytopes in 4 dimensions and only 3 in 5 or more dimensions. * Regular star 4polytopes ( star polyhedron cells and/or vertex figures) ** 1852: Ludwig Schläfli also found 4 of the 10 regular star 4polytopes, discounting 6 with cells or vertex figures and . ** 1883: Edmund Hess completed the list of 10 of the nonconvex regular 4polytopes, in his book (in German) ''Einleitung in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Abstract Polytope
In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines. A geometric polytope is said to be a ''realization'' of an abstract polytope in some real Ndimensional space, typically Euclidean. This abstract definition allows more general combinatorial structures than traditional definitions of a polytope, thus allowing new objects that have no counterpart in traditional theory. Introductory concepts Traditional versus abstract polytopes In Euclidean geometry, two shapes that are not similar can nonetheless share a common structure. For example a square and a trapezoid both comprise an alternating chain of four vertices and four sides, which makes them quadrilaterals. They are said to be isomorphic or “structure preserving”. This common structure may be represented in an underlying abstract polytope, a purely algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

57cell
In mathematics, the 57cell (pentacontakaiheptachoron) is a selfdual abstract regular 4polytope ( fourdimensional polytope). Its 57 cells are hemidodecahedra. It also has 57 vertices, 171 edges and 171 twodimensional faces. The symmetry order is 3420, from the product of the number of cells (57) and the symmetry of each cell (60). The symmetry abstract structure is the projective special linear group, L2(19). It has Schläfli symbol with 5 hemidodecahedral cells around each edge. It was discovered by . Perkel graph The vertices and edges form the Perkel graph, the unique distanceregular graph with intersection array , discovered by . See also * 11cell – abstract regular polytope with hemiicosahedral cells. * 120cell – regular 4polytope with dodecahedral cells * Order5 dodecahedral honeycomb  regular hyperbolic honeycomb with same Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tesse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Face (geometry)
In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object; a threedimensional solid bounded exclusively by faces is a ''polyhedron''. In more technical treatments of the geometry of polyhedra and higherdimensional polytopes, the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions).. Polygonal face In elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include polyhedron side and Euclidean plane ''tile''. For example, any of the six squares that bound a cube is a face of the cube. Sometimes "face" is also used to refer to the 2dimensional features of a 4polytope. With this meaning, the 4dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells. Number of polygonal faces of a polyhedron Any convex polyhedron's surface has Euler characteristic :V  E + F = 2, where ''V'' is the num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

170 (number)
170 (one hundred ndseventy) is the natural number following 169 and preceding 171. In mathematics 170 is the smallest ''n'' for which φ(''n'') and σ(''n'') are both square (64 and 324 respectively). But 170 is never a solution for φ(''x''), making it a nontotient. Nor is it ever a solution to ''x''  φ(''x''), making it a noncototient. 170 is a repdigit in base 4 (2222) and base 16 (AA), as well as in bases 33, 84, and 169. It is also a sphenic number. 170 is the largest integer for which its factorial can be stored in IEEE 754 doubleprecision floatingpoint format. This is probably why it is also the largest factorial that Google's builtin calculator will calculate, returning the answer as 170! = 7.25741562 × 10306. There are 170 different cyclic Gilbreath permutations on 12 elements, and therefore there are 170 different real periodic points of order 12 on the Mandelbrot set The Mandelbrot set () is the set of complex numbers c for which the function ... [...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 tensided polygon or 10gon.. The total sum of the interior angles of a simple decagon is 1440°. A selfintersecting ''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] 

Tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the threedimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets. For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and anot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 