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polyhedron

 
Dictionary: pol·y·he·dron   (pŏl'ē-hē'drən) pronunciation
n., pl., -drons, or -dra (-drə).
A solid bounded by polygons.

polyhedral pol'y·he'dral adj.

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In Euclidean geometry, a three-dimensional object composed of a finite number of polygonal surfaces (faces). Technically, a polyhedron is the boundary between the interior and exterior of a solid. In general, polyhedrons are named according to number of faces. A tetrahedron has four faces, a pentahedron five, and so on; a cube is a six-sided regular polyhedron (hexahedron) whose faces are squares. The faces meet at line segments called edges, which meet at points called vertices. See also Platonic solid; Euler's formula.

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Sci-Tech Encyclopedia: Polyhedron
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A solid whose boundary consists of a finite number of polygonal faces, that is, planar regions that are bounded by polygons. The sides of the faces are edges of the polyhedron; the vertices of the faces also are vertices of the polyhedron. See also Plane geometry; Polygon.

Most polyhedra met in applied geometry are convex and simply connected. A polyhedron is convex if it passes this test: if any face is placed coincident with a plane, then all other points of the polyhedron lie on the same side of that plane. A more informal test is to imagine enclosing the polyhedron within a stretched elastic membrane; the polyhedron is convex if all points on the boundary are in contact with the membrane. A simply connected polyhedron has a boundary that is topologically equivalent to a sphere: if the boundary were made of some perfectly elastic material, then the boundary could be distorted into a sphere without tearing or piercing the surface. A simply connected polyhedron is said to be eulerian, because the number of faces F, the number of edges E, and the number of vertices V satisfy Euler's formula: F + V = E +2

Polyhedra exist having any number of faces greater than three. Some polyhedra have names that convey the number of faces (but not the shape) of the polyhedron: tetrahedron, 4 faces; pentahedron, 5 faces; hexahedron, 6 faces; octahedron, 8 faces; dodecahedron, 12 faces; and icosahedron, 20 faces.

A polyhedron is regular (or platonic) if all faces are congruent and all dihedral angles (the angles between adjacent faces) are equal. There are only five regular polyhedra: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.


A six- or more-sided object. A group of connected polygons.

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Columbia Encyclopedia: polyhedron
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polyhedron (pŏl'ēhē'drən), closed solid bounded by plane faces; each face of a polyhedron is a polygon. A cube is a polyhedron bounded by six polygons (in this case squares) meeting at right angles. Although regular polygons are possible for any number of sides, there are only five possible regular polyhedrons, having congruent faces, each a regular polygon and meeting at equal angles. The five regular polyhedrons are also known as the Platonic solids, although they were known to the Greeks before the time of Plato. They are the tetrahedron, bounded by four equilateral triangles; the hexahedron, or cube, bounded by six squares; the octahedron, bounded by eight equilateral triangles; the dodecahedron, bounded by twelve regular pentagons; and the icosahedron, bounded by twenty equilateral triangles. The 18th-century Swiss mathematician Leonhard Euler showed that for any simple polyhedron, i.e., a polyhedron containing no holes, the sum of the number of vertices V and the number of faces F is equal to the number of edges E plus 2, or V+F=E+2.


Veterinary Dictionary: polyhedral
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Having many sides or surfaces.

Wikipedia: Polyhedron (magazine)
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Polyhedron was a magazine which started out as the official publication of the RPGA (Role Playing Gamers Association). Publication began in the year 1981, and the target audience was players of the Dungeons & Dragons roleplaying game. Articles were written by gamers for other gamers in the style of the Dragon magazine, and information was included on RPGA membership and events. The magazine was quarterly at first and became bimonthly in 1983. For several years it was available only to RPGA members; for some, joining the RPGA essentially amounted to a subscription to Polyhedron. Polyhedron was produced by RPGA members (some of whom were professionals in the game industry) for RPGA members.[1]

In 2002, Paizo Publishing acquired publishing rights and merged the Polyhedron magazine with the sister publication Dungeon to form a single magazine (issue 90 of Dungeon and issue 149 of Polyhedron were one and the same magazine, and this dual numbering continued throughout this period). This ended the association of Polyhedron with the RPGA. It also marked a major change in the magazine's focus, from a primarily Dungeons & Dragons-oriented magazine similar to Dragon to a general d20 system magazine that often featured entirely new, simple role-playing games based on this system, along with support for non-D&D d20 games such as d20 Modern. Eventually another formerly separate magazine, the Living Greyhawk Journal, briefly became a section in Polyhedron as well.

Though this version of Polyhedron had many vocal supporters, sales were poor, a situation many blamed on putting two magazines with distinct target audiences together in one somewhat higher-priced package. The Polyhedron section was removed from Dungeon as part of a major revamp of the latter magazine in 2004 and Polyhedron is no longer published in any form.

For a short time in 2006 and 2007, Wizards of the Coast used the name "Polyhedron" for a Dungeons & Dragons email newsletter. As far as can be determined at this time, the newsletter consisted of advertisements for the company's Web site and does not seem to have been particularly popular.

References


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Some good "polyhedron" pages on the web:


Math
mathworld.wolfram.com
 
 
 
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