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Topic: Platonic solid

	Platonic solid - Wikipedia, the free encyclopedia
		Two-dimensional images of each of the Platonic solids are found within Metatron's Cube, a construct which originates from joining all the centres together from the Flower of Life.
		The Platonic solids may be seen as increasingly better approximations to that sphere.
		In terms of the "variation in altitude" (the ratio between the radius of the circumscribed sphere and the radius of the inscribed sphere), the Platonic solid that best fits the sphere is a tie between the icosahedron and the dodecahedron.
	en.wikipedia.org /wiki/Platonic_solid (1384 words)

	Platonic solid: Encyclopedia topic (Site not responding. Last check: 2007-11-07)
		the Archimedean (Archimedean: in geometry an archimedean solid or semi-regular solid is a semi-regular convex...
		Proposition 13 describes the construction of the tetrahedron, proposition 14 of the octahedron, proposition 15 of the cube, proposition 16 of the icosahedron, and proposition 17 of the dodecahedron.
		(The Archimedean solid (Archimedean solid: in geometry an archimedean solid or semi-regular solid is a semi-regular convex...
	www.absoluteastronomy.com /reference/platonic_solid (1482 words)

	The Particle: Platonic solids
		Although conventional physics refers to solids, liquids, gases, and plasma as the 1st, 2nd, 3rd, and 4th state of matter respectively, it would make more sense to reverse the order, with plasma the first and solids the last, making them ordered in terms of their platonic solid structure complexity.
		Each of the platonic solids is in fact a triangulation of the sphere into polygons.The Euler characteristic is given by F-E+V, where F is the number of polygonal faces, E is the number of edges, and V is the number of vertices in the triangulation.
		The dual of a platonic is the shape formed having its vertices at the centre of each face of the parent platonic.
	www.blazelabs.com /f-p-solids.asp (2442 words)

	Golden Ratio and the Platonic Solids (Site not responding. Last check: 2007-11-07)
		Whereas these later 2 mentioned platonic solids are seen in the 5 pointed star the remaining three platonic solids can be found in the 6 pointed star.
		The dodecahedron is one of the 5 Platonic solids with 12 pentagonal surfaces.
		One of the 5 platonic solids is described by Plato in the "The Phaedo" (110 B.) where he refers to a ball with 12 pentagonal faces and which is the precursor to our modern football, still made with a variable number of pentagons as shown in the adjacent photograph.
	jwilson.coe.uga.edu /EMAT6680/Parveen/platonic_solids.htm (578 words)

	Platonic solid
		Any of the five regular polyhedrons –; solids with regular polygon faces and the same number of faces meeting at each corner – that are possible in three dimensions.
		They are the tetrahedron (a pyramid with triangular faces), the octahedron (an eight-sided figure with triangular faces), the dodecahedron (a 12-sided figure with pentagonal faces), the icosahedron (a 20-sided figure with triangular faces), and the cube.
		A regular solid with hexagonal faces cannot exist because if it did, the sum of the angles of any three hexagonal corners that meet would already equal 360°, so such an object would be planar.
	www.daviddarling.info /encyclopedia/P/Platonic_solid.html (175 words)

	Platonic solid : search word
		A Platonic solid is a convex polyhedron whose faces all use the same regular polygon and such that the same number of faces meet at all its vertices.
		Compare with the Kepler-Poinsot solids, which are not convex, and the Archimedean and Johnson solids, which while made of regular polygons are not themselves regular.
		(the ratio between the radius of the circumscribed sphere and the radius of the inscribed sphere), the Platonic solid that best fits the sphere is a tie between the icosahedron and the dodecahedron.
	www.searchword.org /pl/platonic-solid.html (1263 words)

	Constructing Platonic Solids in the Classroom
		The five Platonic solids are the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron.
		These solids are perfectly symmetrical in that each face of a solid is identical to every other face of the solid, each vertex is identical to every other vertex, and each edge is identical to every other edge.
		Paper patterns of the five platonic solids are available as a postscript file or as two GIF files: plato1 and plato2.
	www.dpgraph.com /janine/mathpage/platonic.html (1586 words)

	Formula Derivations for Polyhedra
		The inradius of a solid is the radius of the inscribed sphere.
		For a general formula for the volume of a Platonic solid, picture a pyramid built on one face, with the center of the Platonic solid as the apex of the pyramid.
		The base area is the area of a single face of the Platonic solid, and the height is the inradius of the Platonic solid.
	whistleralley.com /polyhedra/derivations.htm (1093 words)

	Platonic Solids and Plato's Theory of Everything
		Theaetetus not only proved that these solids exist, and that they are the only perfectly symmetrical solids, he also gave the actual ratios of the edge lengths E to the diameters D of the circumscribing spheres for each of these solids.
		Indeed the same Theaetetus who gave the first complete account of the five "Platonic" solids is also remembered for recognizing the general fact that the square root of any non-square integer is irrational, which is to say, incommensurable with the unit 1.
		In addition, the cube is the only Platonic solid that is NOT an equilibrium configuration for its vertices on the surface of a sphere with respect to an inverse-square repulsion.
	www.mathpages.com /home/kmath096.htm (2221 words)

	Imaging the Imagined: The Platonic Solids & Polyhedra: plane & cool shapes
		The Platonic Solids can be named by an integer pair {n,m} where "n" and "m" are the number of edges on each face and the number of faces that meet at each corner, respectively.
		The polyhedra are {x,3} and {3,y} in the ascending and descending arcs of triangluar pattern, respectively, and the values x and y increase as you move from the tetrahedron, {3,3}.
		For each of the Platonic solids, slice off all of the corners, with planar cuts passing through the center of each face adjacent to each corner.
	www.frontiernet.net /~imaging/polyh.html (372 words)

	Platonic and Archimedean (Site not responding. Last check: 2007-11-07)
		There are five Platonic Solids and thirteen Archimedean Solids, which are convex polyhedra whose faces are all regular polygons.
		The "base case" is a solid with two faces glued back-to-back which is "flattened" to a single polygon.
		The seven other Archimedean solids are the truncated dodecahedron, truncated icosahedron, cuboctahedron, rhombicosidodecahedron, truncated icosidodecahedron, snub cube, and snub dodecahedron.
	mcraefamily.com /MathHelp/GeometrySolidPlatonic.htm (596 words)

	platonic (Site not responding. Last check: 2007-11-07)
		In 3 dimensions, the most symmetrical polyhedra of all are the 'regular polyhedra', also known as the 'Platonic solids'.
		All the faces of a Platonic solid are regular polygons of the same size, and all the vertices look identical.
		For more about the Platonic solids, how fool's gold fooled the Greeks into inventing the regular dodecahedron, and highly symmetric structures in higher dimensions, see week62, week63, week64, and week65 of my weekly column on mathematical physics.
	math.ucr.edu /home/baez/platonic.html (2284 words)

	Platonic solid - (Site not responding. Last check: 2007-11-07)
		The five Platonic solids were all known to the ancient Greeks, and a proof that they are the only regular polyhedra can be found in Euclid's Elements.
		These are, of course, not the true shapes of atoms; but it turns out that they are some of the true shapes of packed atoms and molecules, namely crystals: The mineral salt sodium chloride occurs in cubic crystals, fluorite (calcium fluoride) in octahedra, and pyrite in dodecahedra (see uses below).
		Historically, Johannes Kepler followed the custom of the Renaissance in making mathematical correspondences, and identified the five platonic solids with the five planets – Mercury, Venus, Mars, Jupiter, Saturn which themselves represented the five classical elements.
	psychcentral.com /psypsych/Platonic_solids (1462 words)

	Foundations and Structure of Mathematics 1 "The Platonic Solids" webpage
		In fact, Plato associated four of the Platonic solids, the tetrahedron, octahedron, icosahedron, and cube, with the four Greek elements: fire, air, water, and earth.
		After abandoning the Platonic solids, he was successful in describing planetary orbits with his three laws of planetary motion.
		Vertices of the graphs correspond to vertices of the solids, edges of the graphs correspond to edges of the solids, and regions determined by the graphs correspond to faces of the solids.
	www.etsu.edu /math/gardner/5025/platonic/platonic.htm (1141 words)

	VB Helper: Tutorial: Drawing Platonic Solids
		The Platonic solids were defined by the Greek mathematician and philosopher Plato (427-347 BC).
		The Platonic solids include the tetrahedron (4 triangular faces), cube or hexahedron (6 square faces), octahedron (8 triangular faces), dodecahedron (12 pentagonal faces), and the icosahedron (20 triangular faces).
		For the Platonic solids, the result is another Platonic solid.
	www.vb-helper.com /tutorial_platonic_solids.html (2137 words)

	Platonic solid fractals and their complements (Site not responding. Last check: 2007-11-07)
		Except for the special case of the cube, each fractal is created by positioning reduced versions of the platonic solid along the line from the center to each vertex of the solid.
		The exact reduction and the distance along the line is chosen so the final form is connected and the overall shape is the same as the seeding platonic solid.
		In addition the complement of each fractal is shown (in blue), that is, the fractal subtracted from the original platonic solid.
	astronomy.swin.edu.au /~pbourke/fractals/platonic (141 words)

	Platonic hydrocarbons - Wikipedia, the free encyclopedia
		Platonic hydrocarbons are the molecular representation of platonic solid geometries with vertices replaced by carbon atoms and with edges replaced by chemical bonds.
		Not all platonic solids have a molecular counterpart:
		Note that with increasing number of carbon atoms in the frame, the geometry will eventually reflect a sphere.
	en.wikipedia.org /wiki/Platonic_hydrocarbons (120 words)

	polyhedra (Site not responding. Last check: 2007-11-07)
		If we consider any platonic solid and "join" the center ponts of sides, we get a new platonic solid (see the lower table).
		The animations presented in the following table show that it is possible to construct the dual of a given platonic solid by truncating it successively.
		In the model formed by a tetrahedron and its dual (which is also a tetrahedron) presented in the duality table, if we enlarge the interior tetrahedron so that the edges of both tetrahedron are at the same distance of the common center, we obtain a composed polyhedron - the stella octangula.
	www.atractor.pt /mat/Polied/poliedros-e.htm (309 words)

	PlanetMath: regular polyhedron (Site not responding. Last check: 2007-11-07)
		The dihedral angle between any two faces is always the same.
		These polyhedra are also known as Platonic solids, since Plato described them in his work.
		There are only 5 regular polyhedra, as was first shown by Theaetetus, one of Plato's students.
	planetmath.org /encyclopedia/PlatonicSolid.html (203 words)

	Platonic (Site not responding. Last check: 2007-11-07)
		Platonic portrays a rotating, translucent Platonic solid (tetrahedron, cube, octahedron, dodecahedron or icosahedron), with the faces rendered opaque where they lie inside a second, independently rotating Platonic solid (not shown).
		The second solid is always either the same as, or dual to, the first.
		One polyhedron is dual to another if there is a face in place of every vertex, and a vertex in place of every face; the cube and the octahedron are duals, as are the dodecahedron and icosehedron, while the tetrahedron is its own dual.
	gregegan.customer.netspace.net.au /APPLETS/24/24.html (98 words)

	The regular polyhedra: Platonic solids (Site not responding. Last check: 2007-11-07)
		A regular polyhedron or platonic solid is a polyhedron all of whose faces are congruent regular polygons, and where the same number of faces meet at every vertex.
		Perhaps the best known example of a platonic solid is the cube whose faces are six congruent squares.
		These solids have star vertices and faces which are regular pentagons and equilateral triangles respectively.
	nothung.math.uh.edu /~mike/hti/handouts/handout4/node2.html (168 words)

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