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Selfdual Polytope In geometry , any polyhedron is associated with a second DUAL figure, 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 are also 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 also belong to a symmetric class. Thus, the regular polyhedra – the (convex) Platonic solids and (star) KeplerPoinsot polyhedra KeplerPoinsot polyhedra – form dual pairs, where the regular tetrahedron is selfdual . The dual of an isogonal polyhedron, having equivalent vertices, is one which is isohedral, having equivalent faces [...More...]  "Selfdual Polytope" on: Wikipedia Yahoo 

Pyramid (geometry) In geometry , a PYRAMID is a polyhedron formed by connecting a polygonal base and a point, called the apex . Each base edge and apex form a triangle, called a lateral face. It is a conic solid with polygonal base. A pyramid with an nsided base has n + 1 vertices, n + 1 faces, and 2n edges. All pyramids are selfdual . A RIGHT PYRAMID has its apex directly above the centroid of its base. Nonright pyramids are called OBLIQUE PYRAMIDS. A REGULAR PYRAMID has a regular polygon base and is usually implied to be a right pyramid. When unspecified, a pyramid is usually assumed to be a regular square pyramid , like the physical pyramid structures. A triangle based pyramid is more often called a tetrahedron . Among oblique pyramids, like acute and obtuse triangles , a pyramid can be called acute if its apex is above the interior of the base and obtuse if its apex is above the exterior of the base. A RIGHTANGLED PYRAMID has its apex above an edge or vertex of the base [...More...]  "Pyramid (geometry)" on: Wikipedia Yahoo 

Regular Polygon Regular polygons Edges and vertices n Schläfli symbol {n} Coxeter–Dynkin diagram Symmetry group Symmetry group Dn , order 2n [...More...]  "Regular Polygon" on: Wikipedia Yahoo 

Elongated Pyramid In geometry , the ELONGATED PYRAMIDS are an infinite set of polyhedra, constructed by adjoining an ngonal pyramid to an ngonal prism . Along with the set of pyramids, these figures are topologically selfdual . There are three elongated pyramids that are Johnson solids made from regular triangles and square, and pentagons. Higher forms can be constructed with isosceles triangles. FORMS NAME FACES elongated triangular pyramid (J7) 3+1 triangles, 3 squares elongated square pyramid (J8) 4 triangles, 4+1 squares elongated pentagonal pyramid (J9) 5 triangles, 5 squares, 1 pentagonSEE ALSO * Gyroelongated bipyramid * Elongated bipyramid * Gyroelongated pyramid * Diminished trapezohedron REFERENCES * Norman W. Johnson , "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others. * Victor A. Zalgaller (1969) [...More...]  "Elongated Pyramid" on: Wikipedia Yahoo 

Prism (geometry) In geometry , a PRISM is a polyhedron comprising an nsided polygonal base , a second base which is a translated copy of the first, and n other faces (necessarily all parallelograms ) joining corresponding sides of the two bases. All crosssections parallel to the bases are translations of the bases. Prisms are named for their bases, so a prism with a pentagonal base is called a pentagonal prism. The prisms are a subclass of the prismatoids [...More...]  "Prism (geometry)" on: Wikipedia Yahoo 

Square Pyramid In geometry , a SQUARE PYRAMID is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry. CONTENTS * 1 Johnson solid (J1) * 2 Other square pyramids * 3 Related polyhedra and honeycombs * 3.1 Dual polyhedron * 4 Topology * 5 Examples * 6 References * 7 External links JOHNSON SOLID (J1)If the sides are all equilateral triangles , the pyramid is one of the Johnson solids (J1). The 92 Johnson solids were named and described by Norman Johnson in 1966. A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform (that is, they are not Platonic solids , Archimedean solids , prisms or antiprisms ). They were named by Norman Johnson , who first listed these polyhedra in 1966. The Johnson square pyramid can be characterized by a single edgelength parameter a [...More...]  "Square Pyramid" on: Wikipedia Yahoo 

Reflection Through The Origin In geometry , a POINT REFLECTION or INVERSION IN A POINT (or INVERSION THROUGH A POINT, or CENTRAL INVERSION) is a type of isometry of Euclidean space . An object that is invariant under a point reflection is said to possess POINT SYMMETRY; if it is invariant under point reflection through its center, it is said to possess CENTRAL SYMMETRY or to be CENTRALLY SYMMETRIC . Point reflection can be classified as an affine transformation . Namely, it is an isometric involutive affine transformation, which has exactly one fixed point , which is the point of inversion. It is equivalent to a homothetic transformation with scale factor equal to −1. The point of inversion is also called homothetic center [...More...]  "Reflection Through The Origin" on: Wikipedia Yahoo 

Intersphere 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 polyhedron there is a combinatorially equivalent polyhedron, the CANONICAL POLYHEDRON, that does have a midsphere. The midsphere is socalled because, for polyhedra that have a midsphere, an inscribed sphere (which is tangent to every face of a polyhedron) and a circumscribed sphere (which touches every vertex), the midsphere is in the middle, between the other two spheres. The radius of the midsphere is called the MIDRADIUS. CONTENTS * 1 Examples * 2 Tangent circles * 3 Duality * 4 Canonical polyhedron * 5 Notes * 6 References * 7 External links EXAMPLESThe uniform polyhedra , including the regular , quasiregular and semiregular polyhedra and their duals all have midspheres [...More...]  "Intersphere" on: Wikipedia Yahoo 

Uniform Polyhedron A UNIFORM POLYHEDRON is a polyhedron which has regular polygons as faces and is vertextransitive (transitive on its vertices , isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent . Uniform polyhedra may be regular (if also face and edge transitive), quasiregular (if edge transitive but not face transitive) or semiregular (if neither edge nor face transitive). The faces and vertices need not be convex , so many of the uniform polyhedra are also star polyhedra . There are two infinite classes of uniform polyhedra together with 75 others [...More...]  "Uniform Polyhedron" on: Wikipedia Yahoo 

Vertex Figure In geometry , a VERTEX FIGURE, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. CONTENTS* 1 Definitions – theme and variations * 1.1 As a flat slice * 1.2 As a spherical polygon * 1.3 As the set of connected vertices * 1.4 Abstract definition * 2 General properties * 3 Dorman Luke construction * 4 Regular polytopes * 5 An example vertex figure of a honeycomb * 6 Edge figure * 7 See also * 8 References * 8.1 Notes * 8.2 Bibliography * 9 External links DEFINITIONS – THEME AND VARIATIONSTake some vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points. 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 [...More...]  "Vertex Figure" on: Wikipedia Yahoo 

Cuboctahedron In geometry , a CUBOCTAHEDRON is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices , with 2 triangles and 2 squares meeting at each, and 24 identical edges , each separating a triangle from a square. As such, it is a quasiregular polyhedron , i.e. an Archimedean solid Archimedean solid that is not only vertextransitive but also edgetransitive . Its dual polyhedron is the rhombic dodecahedron . The cuboctahedron was probably known to Plato Plato : Heron 's Definitiones quotes Archimedes Archimedes as saying that Plato Plato knew of a solid made of 8 triangles and 6 squares [...More...]  "Cuboctahedron" on: Wikipedia Yahoo 

Rhombic Dodecahedron In geometry , the RHOMBIC DODECAHEDRON is a convex polyhedron with 12 congruent rhombic faces . It has 24 edges , and 14 vertices of two types. It is a Catalan solid , and the dual polyhedron of the cuboctahedron . This animation shows the construction of a rhombic dodecahedron from a cube, by inverting the centerfacepyramids of a cube. CONTENTS * 1 Properties * 2 Dimensions * 3 Area and volume * 4 Orthogonal projections * 5 Cartesian coordinates * 6 Topologically equivalent forms * 6.1 Parallelohedron * 6.1.1 Bilinski dodecahedron * 6.2 Trapezoidal dodecahedron * 7 Related polyhedra * 7.1 Stellations * 8 Related polytopes * 9 See also * 10 References * 11 Further reading * 12 External links * 12.1 Computer models * 12.2 Paper projects * 12.3 Practical applications PROPERTIESThe rhombic dodecahedron is a zonohedron . Its polyhedral dual is the cuboctahedron [...More...]  "Rhombic Dodecahedron" on: Wikipedia Yahoo 

Pentagonal Pyramid In geometry , a PENTAGONAL PYRAMID is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point (the vertex). Like any pyramid , it is selfdual . The regular pentagonal pyramid has a base that is a regular pentagon and lateral faces that are equilateral triangles . It is one of the Johnson solids (J2). Its height H, from the midpoint of the pentagonal face to the apex, (as a function of a, where a is the side length), can be computed as: H = 5 5 10 a 0.5257 a . {displaystyle H={sqrt {frac {5{sqrt {5}}}{10}}},aapprox 0.5257,a.} Its surface area, A, can be computed as the area of pentagonal base plus five times the area of one triangle: A = ( 25 + 10 5 4 + 5 3 4 ) a 2 3.8855 a 2 [...More...]  "Pentagonal Pyramid" on: Wikipedia Yahoo 

Hexagonal Pyramid In geometry , a HEXAGONAL PYRAMID is a pyramid with a hexagonal base upon which are erected six triangular faces that meet at a point (the apex). Like any pyramid , it is selfdual . A RIGHT HEXAGONAL PYRAMID with a regular hexagon base has C6v symmetry . A right regular pyramid is one which has a regular polygon as its base and whose apex is "above" the center of the base, so that the apex, the center of the base and any other vertex form a right triangle [...More...]  "Hexagonal Pyramid" on: Wikipedia Yahoo 

Regular Polytope In mathematics , a REGULAR POLYTOPE is a polytope whose symmetry group acts transitively on its flags , thus giving it the highest degree of symmetry. All its elements or jfaces (for all 0 ≤ j ≤ n, where n is the dimension of the polytope) — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension ≤ n. Regular polytopes are the generalized analog in any number of dimensions of regular polygons (for example, the square or the regular pentagon) and regular polyhedra (for example, the cube ). The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both nonmathematicians and mathematicians. Classically, a regular polytope in n dimensions may be defined as having regular facets and regular vertex figures . These two conditions are sufficient to ensure that all faces are alike and all vertices are alike [...More...]  "Regular Polytope" on: Wikipedia Yahoo 

Honeycomb (geometry) In geometry , a HONEYCOMB is a space filling or close packing of polyhedral or higherdimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as nhoneycomb for a honeycomb of ndimensional space. Honeycombs are usually constructed in ordinary Euclidean ("flat") space. They may also be constructed in nonEuclidean spaces , such as hyperbolic honeycombs . Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. It is possible to fill the plane with polygons which do not meet at their corners, for example using rectangles , as in a brick wall pattern: this is not a proper tiling because corners lie part way along the edge of a neighbouring polygon. Similarly, in a proper honeycomb, there must be no edges or vertices lying part way along the face of a neighbouring cell [...More...]  "Honeycomb (geometry)" on: Wikipedia Yahoo 