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Topic: Regular polyhedron

	Polyhedron -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-08-08)
		The polyhedron surrounds a bounded volume in three-dimensional space; sometimes this interior volume is considered to be part of the polyhedron.
		Any polyhedron which is vertex-uniform can be deformed slightly to form a vertex-uniform polyhedron with (A polygon with all sides and all angles equal) regular polygons as faces.
		A topological polyhedron is a topological space given along with a specific decomposition into shapes that are topologically equivalent to (additional info and facts about convex polytope) convex polytopes and that are attached to each other in a regular way that needs better description.
	www.absoluteastronomy.com /encyclopedia/p/po/polyhedron.htm (1065 words)

	Regular polytope - Wikipedia, the free encyclopedia
		A regular polygon is a polygon whose edges are all equal and whose angles are all equal.
		A regular polyhedron is a polyhedron whose faces are all congruent regular polygons, and whose vertex figures are all congruent and regular.
		Neither the regular icosahedron nor the regular dodecahedron are amongst them, although one of the forms, called the pyritohedron (named for the group of minerals of which it is typical) is has twelve pentagonal faces, arranged in the same pattern as the faces of the regular dodecahedron.
	en.wikipedia.org /wiki/Regular_polytope (4401 words)

	Regular - Wikipedia, the free encyclopedia
		In the military, a regular unit is a military unit that is part of the regular forces (not militia or reserve).
		In graph theory, a regular graph is a graph such that the all the degrees of the vertices are equal.
		In algebraic geometry, a function is regular on a variety if for every point there exists an open neighborhood so that the function restricts to a rational function whose denominator does not vanish in that neighborhood.
	en.wikipedia.org /wiki/Regular (298 words)

	InterMath / Dictionary / Related Terms
		Octahedron: A regular polyhedron whose eight faces are congruent equilateral triangles.
		Platonic Solids: The five regular polyhedrons, meaning that they are polyhedrons that have faces which are congruent regular polygons and polyhedral angles (angles between faces) that are congruent.
		Regular Polyhedron: A 3-dimensional solid that has congruent regular polygons as faces and has congruent angles between all faces.
	www.intermath-uga.gatech.edu /dictnary/related.asp?termid=170 (110 words)

	Regular Polyhedra
		Regular polyhedra generalize the notion of a regular polygon to three dimensions.
		A regular polyhedron is a polyhedron with congruent faces and identical vertices.
		The dual of the regular octahedron is the cube, and the dual of the cube is the regular octahedron.
	www.math.rutgers.edu /~erowland/polyhedra.html (1011 words)

	The Regular Polyhedra (Site not responding. Last check: 2007-08-08)
		Regular star-polygons were not accepted as regular on a par with the convex regular polygons until Louis Poinsot published his study of regular polygons and polyhedra in 1810.
		Pentagrams and pentagons are conjugate, as are octagons and octagrams and decagons and decagrams.
		This compound is the conjugate of the compound of the regular icosahedron and regular dodecahedron.
	members.aol.com /Polycell/regs.html (11220 words)

	Polyhedrons
		The regular polyhedrons are three dimensional solid or skeleton shaped geometric figures.
		The regular polyhedrons are: Tetrahedron (with 4 equilateral triangle faces), Hexahedron (with 6 square faces), Octahedron (with 8 equilateral triangle faces), Dodecahedron (with 12 regular pentagon faces), and Icosahedron (with 20 equilateral triangle faces).
		If a straight line is extended inward, perpendicular from the center point of every regular polygon face on the surface of the regular polyhedron__ they will intersect at one center-midpoint of the polyhedron.
	www.theofficenet.com /~rad/Polyhedrons.htm (813 words)

	Activity 14
		A regular polyhedron is a polyhedron that is composed of congruent regular polygons with the same vertex code everywhere.
		One regular polyhedron is the dual of another if the polyhedra have the same number of edges but the number of faces of one equals the number of vertices of the other.
		A semi-regular polyhedron is a polyhedron that is composed of several different regular polygons with the same vertex code everywhere.
	homepage.mac.com /efithian/Geometry/Activity-14.html (727 words)

	Geometry of Polyhedrons. (Site not responding. Last check: 2007-08-08)
		Example of a semi-regular polyhedron of 32 faces built of regular hexagons and pentagons.
		Semi-regular polyhedrons built of congruent regular polygons and equilateral triangles.
		Polyhedron of a soccer ball used by UEFA for European Soccer Competitions.
	www.geocities.com /literka/list.htm (353 words)

	Regular polyhedra
		Polyhedron is a solid (3D) body whose surface consists of a number of polygonal faces subject to two conditions to exclude some "abnormal" cases.
		A polyhedron is said to be regular if all its faces are equal regular polygons and the same number of faces meet at every vertex.
		Regular faces cease to be regular polygons if of course they were regular to start with.
	www.cut-the-knot.com /do_you_know/polyhedra.shtml (1138 words)

	Platonic Solids
		Some of regular and semi-regular solids meet in the nature in the form of crystals, other in the form of viruses, which can be considered with the help of a super-microscope.
		As the simplest regular polygon it is possible to consider an equilateral triangle because it has the least number of line segments, which can limit a part of a plane.
		A polyhedron is called regular one if all its faces are equal (or congruent) among themselves and they are regular polygons.
	www.goldenmuseum.com /0213Solids_engl.html (823 words)

	Pythagoras of Samos - Activity 7
		This is the fourth regular polyhedron - the dodecahedron.
		Firstly we have demonstrated that a regular polyhedron must have at least 4 faces, and therefore at least 3 faces must meet at a vertex.
		The interior angles of the regular hexagon are 120°, so if we fit three of them together at a vertex the angles sum to precisely 360°, and therefore they "fill the plane".
	www.mathgym.com.au /history/pythagoras/Activity7.htm (1582 words)

	PlanetMath: regular polyhedron
		A regular polyhedron is a polyhedron such that
		There are only 5 regular polyhedra, as was first shown by Theaetetus, one of Plato's students.
		This is version 14 of regular polyhedron, born on 2002-02-19, modified 2005-11-08.
	planetmath.org /encyclopedia/Regular4.html (196 words)

	ipedia.com: Icosahedron Article
		In geometry, the regular icosahedron is one of the five Platonic solids.
		It is a convex regular polyhedron composed of twenty triangular faces, with five meeting at each of the twelve vertices.
		A regular polyhedron is used because it can be built from a single basic unit protein that's reused over and over again; that saves space in the viral genome.
	www.ipedia.com /icosahedron.html (344 words)

	Reciprocal Polyhedra
		Regular map: a map is said to be regular, of type {p,q} if there are p vertices and p edges for each face, q edges and q faces at each vertex, arranged symmetrically in a sense that can be made precise.
		In Maple, one can define a duality of a regular polyhedron or Archimedean solid via the command duality(dualp,p,s); where dualp is the name of the reciprocal polyhedron of the given polyhedron p with respect to the sphere s which is concentric with p (i.e., s and p have the same center).
		A given regular polyhedra is closed under duality, i.e., the duality of a regular polyhedron is also a regular polyhedron.
	www.cecm.sfu.ca /~hle/polyhedra/duality.html (540 words)

	Dodecahedron Day: Glossary (Site not responding. Last check: 2007-08-08)
		Technically, a polygon or polyhedron is convex when the segment joining any two of its points lies entirely within it.
		For example, a pentagon and the regular dodecahedron are convex, but a pentagram and the small stellated dodecahedron are nonconvex.
		Technically, a polygon or polyhedron is nonconvex if there is a segment joining two of its points, part of which lies outside the polygon or polyhedron.
	websites.quincy.edu /~matskvi/ddd/glossary.html (311 words)

	ICME POD 2 (Site not responding. Last check: 2007-08-08)
		A regular polygon is a polygon in which all the edges have the same length, and all the angles have the same measure.
		A polyhedron is a 3-dimensional shape whose sides (faces) are polygons, and which has no edges that are not edges of two of its faces.
		A regular polyhedron is a polyhedron in which all the faces are regular polygons which are the same shape and size as each other (congruent), and every corner (vertex) of the polyhedron looks the same as every other vertex.
	weasel.cnrs.humboldt.edu /~spain/pod/pod2t.htm (221 words)

	Exam 3 Review
		Name the five regular polyhedra and give the number of faces, edges and vertices, the type of faces, and how many faces meet at each vertex.
		Determine the number of vertices in a polyhedron given the number and type of faces and the number of faces meeting at each vertex.
		Describe all the regular tilings of a sphere and explain why those are the only ones.
	home.snu.edu /~lturner/MC-GeoTop/Exam3Rev.htm (715 words)

	What is a Dodecahedron?
		Roughly speaking, a polyhedron is a higher dimension version of a polygon - for instance prisms and pyramids are polyhedra.
		The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another with equivalent edges.
		Kepler solid is a regular nonconvex polyhedron, all the faces of which are regular polygons and which has the same number of faces meeting at all its vertices.
	www.francesfarmersrevenge.com /stuff/misc/dodecahedron (418 words)

	Pseudo Rhombicuboctahedra
		The pseudo-rhombicuboctahedron is not classified as a semi-regular polyhedron, because the essence (and beauty) of the semi-regular polyhedra is not about local properties of each vertex, but the symmetry operations under which which the entire object appears unchanged.
		He discovered a 14th semiregular polyhedron [figure of pseudo-rhombicuboctahedron] which differs from the one shown in [figure of rhombicuboctahedron] only in that the upper part, consisting of 5 squares and 4 equilateral triangles, is rotated through an angle of pi/4.
		Notice that its "equators" are regular octagons, so one half can be rotated 45 degrees relative to the other, giving the dual to the pseudo rhombicuboctahedron, which I suppose must be called the pseudo trapezoidal icositetrahedron.
	www.georgehart.com /virtual-polyhedra/pseudo-rhombicuboctahedra.html (691 words)

	How Many Regular Polyhedrons Are There In This or Any Universe?
		Regular polygons are thus convex polygons whose vertex angles are all equal (or congruent) and whose sides are likewise all congruent.
		If a polyhedron has faces which are regular and congruent polygons -- all of them -- and if at each vertex exactly the same number of faces meet, then we have a"regular" polyhedron.
		Three regular hexagons have a vertex angle sum of exactly 360 degrees, and they won't fold into a trihedral angle because there is no gap.
	www.iit.edu /~smile/ma8606.html (1460 words)

	Regular Polyhedra
		A polyhedron is said to be regular if its faces are regular polygons and its corners are regular solid angles; it then necessarily has equal faces and equal angles.
		Schläfli in 1852 invented the symbol {p,q} for the regular polyhedron whose faces are p-gons, q meeting at each vertex, or the polyhedron with face {p} and vertex figure {q}.
		The possibility of further regular figures was first envisaged by Bredwardin, a fourteenth-century Englishman, who extended the sides of an ordinary polygon to form a star-polygon.
	www.cecm.sfu.ca /~hle/polyhedra/regular.html (1058 words)

	SparkNotes: Surfaces: Terms
		Cube - A regular polyhedron with six faces, all of which are congruent squares.
		Dodecahedron - A twelve-faced regular polyhedron whose faces are all congruent regular pentagons.
		Regular Pyramid - A pyramid whose base is a regular polygon and whose altitude intersects the plane of its base at the center of the base.
	www.sparknotes.com /math/geometry1/geometricsurfaces/terms.html (827 words)

	Regular Polyhedra Project
		Draw a regular tetrahedron, and show that it is a regular polyhedron.
		From the top left they are the regular tetrahedron (four faces), the cube (six), the regular octahedron (eight), the regular dodecahedron (twelve), and the regular icosahedron (twenty).
		After this transformation, the vertices of the polyhedron correspond to intersection points in the plane, edges correspond to segments, and faces correspond to regions (including the exterior "region", which also began as one of the faces of the polyhedron).
	www.math.rutgers.edu /~erowland/polyhedra-project.html (718 words)

	[No title]
		A regular polyhedron is a polyhedron in which all faces are the same regular polygon, and for which the configurations at each vertex are the same.
		A "semi-regular" polyhedron allows two different regular polygons as faces, still with the condition that there be only one vertex configurations.
		A regular polyhedron is a > polyhedron in which all faces are the same regular polygon, and for > which the configurations at each vertex are the same.
	www.math.niu.edu /~rusin/known-math/99/soccer (877 words)

	The Pleasures of Mathematics (Site not responding. Last check: 2007-08-08)
		A regular polyhedron has faces that are regular polygons, connected at their edges.
		In a convex polyhedron, the edges are all on one side of a plane containing the vertex.
		That there are only five regular polyhedra has been known for more than 2500 years, but the fact is much less important than the reasoning used to establish it.
	www.du.edu /~etuttle/classics/pleasure.htm (587 words)

	The Geometry Page
		A regular polyhedron is defined as a finite polyhedron composed of a single type of regular polygon such that each element (vertex, edge and face) is surrounded identically.
		Regular polyhedra are often represented with a notation called Schläfli symbols which consist of two numbers between curly braces.
		Perhaps these classes have an analogous relationship to the general class of infinite regular polyhedra as the prisms and antiprisms have to the finite regular polyhedra.
	www.superliminal.com /geometry/ogeometry.htm (1812 words)

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