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	Kepler-Poinsot solid - Wikipedia, the free encyclopedia
		A Kepler solid (also called Kepler-Poinsot solid) is a regular non-convex polyhedron, all the faces of which are identical regular polygons and which has the same number of faces meeting at all its vertices (compare to Platonic solids).
		The Kepler solids were defined by Johannes Kepler in 1619, when he noticed that the stellated dodecahedra (there are two, the great and the small) were composed of "hidden" dodecahedra (with pentagonal faces) that have faces composed of triangles, and thus look like stylized stars.
		Kepler's contribution was in recognizing that they fit the definition of regular solids, even though they were concave rather than convex, as the traditional Platonic solids were.
	en.wikipedia.org /wiki/Kepler_solid (497 words)

	Kepler-Poinsot solids
		The four regular non-convex polyhedra that exist in addition to the five regular convex polyhedra known as the Platonic solids.
		As with the Platonic solids, the Kepler-Poinsot solids have identical regular polygons for all their faces, and the same number of faces meet at each vertex.
		These two polyhedra were described by Johannes Kepler in 1619, and he deserves credit for first understanding them mathematically, though a sixteenth century drawing by the Nuremberg goldsmith Wentzel Jamnitzer (1508-1585) is very similar to the former and a fifteenth century mosaic attributed to the Florentine artist Paolo Uccello (1397-1475) illustrates the latter.
	www.daviddarling.info /encyclopedia/K/Kepler-Poinsot_solids.html (326 words)

	POLYHEDRON - LoveToKnow Article on POLYHEDRON (Site not responding. Last check: 2007-10-09)
		These figures are often termed semi-regular solids, but it is more convenient to restrict this term to solids having all their angles, edges and faces equal, the latter, however, not being regular polygons.
		The names of these five solids are: (1) the tetrahedron, enclosed by four equilateral triangles; (2) the cuba or hexahedron, enclosed by 6 squares; (3) the octahedron, enclosed by 8 equilateral triangles; (4)~ the dodecahedron, enclosed by 12 pentagons; (5) the icosahedron, enclosed by 20 equilateral triangles.
		They bear a relation to the Platonil solids similar to the relation of star polygons to ordinary regular polygons, inasmuch as the centre is multiply enclosed in the former and singly in the latter.
	www.1911encyclopedia.org /P/PO/POLYHEDRON.htm (3208 words)

	Dodecahedra
		It is a Platonic solid and the only convex dodecahedron with all the symmetry axes and mirror planes of the icosahedral symmetry group.
		It is a Kepler-Poinsot solid, and also has the full symmetry of the icosahedral symmetry group.
		The twisted rhombic dodecahedron (or "trapezo-rhombic" dodecahedron) is related to the rhombic dodecahedron by sawing it in half along a hexagonal equator and rotating one part a sixth of a revolution relative to the other part.
	www.georgehart.com /virtual-polyhedra/dodecahedra.html (1228 words)

	math lessons - Platonic solid
		At each vertex of the solid, the total, among the adjacent faces, of the angles between their respective adjacent sides must be less than 360°.
		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.
		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.
	www.mathdaily.com /lessons/Platonic_solid (1054 words)

	Expert About so:Solid
		As the solid is heated the molecules vibrate about their position in the lattice until, at the melting point, the crystal breaks down and the molecules start to flow.
		Finally, let A be the area of a single face, V be the volume of the solid, and the polyhedron edges be of unit length on a side.
		Their response to why it would be solid and why it would be a liquid will allow me to see if they know the properties of solids and liquids.
	www.expertsite.biz /dir/so/solid.htm (1814 words)

	Regular Polyhedra
		The shape of the solid angle is conveniently described in terms of the section by a plane perpendicular to the axis of symmetry through the vertex.
		For, as I have proved next, the solids of the first group must lie beyond the earth's orbit, and those of the second group within...Thus I was led to assign the Cube to Saturn, the Tetrahedron to Jupiter, the Dodecahedron to Mars, the Icosahedron to Venus, and Octahedron to Mercury...
		To emphasize his theory, Kepler envisaged an impressive model of the universe which shows a cube, with a tetrahedron inscribed in it, a dodecahedron inscribed in the tetrahedron, an icosahedron inscribed in the dodecahedron, and finally an octahedron inscribed in the dodecahedron.
	www.cecm.sfu.ca /~hle/polyhedra/regular.html (1058 words)

	Kepler solid (Site not responding. Last check: 2007-10-09)
		A 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.
		Because of this, they are not necessarily topologically equivalent to the sphere as Platonic solids are, and in particular V-E+F=2 may not hold.
		A cutaway view of the greater dodecahedron was used for the 1980s puzzle game Alexander's Star.
	www.sciencedaily.com /encyclopedia/kepler_solid (370 words)

	Kepler-Poinsot solid -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-09)
		The Kepler solids were defined by (German astronomer who first stated laws of planetary motion (1571-1630)) Johannes Kepler in 1619, when he noticed that the stellated dodecahedra (there are two, the great and the small) were composed of "hidden" dodecahedra (with pentagonal faces) that have faces composed of triangles, and thus look like stylized stars.
		Kepler's contribution was in recognizing that they fit the definition of regular solids, even though they were concave rather than convex, as the traditional (Any one of five solids whose faces are congruent regular polygons and whose polyhedral angles are all congruent) Platonic solids were.
		The other two are the great icosahedron and great dodecahedron which were described by (Click link for more info and facts about Louis Poinsot) Louis Poinsot in 1809.
	www.absoluteastronomy.com /encyclopedia/k/ke/kepler-poinsot_solid1.htm (318 words)

	The four regular non-convex polyhedra
		A polyhedron, considered as a solid is convex if and only if the line segment between any two points of the polyhedron belongs entirely to the solid.
		Two of them were described by Johannes Kepler in 1619 as being regular, although the objects themselves certainly were known earlier.
		The other two were described by Louis Poinsot in 1809 but at least one of them appears on a drawing by the same Jamnitzer.
	cage.rug.ac.be /~hs/polyhedra/keplerpoinsot.html (628 words)

	iqexpand.com (Site not responding. Last check: 2007-10-09)
		The Platonic solids were admired and adored by the ancient Greek mathematicians and anyone learning...
		The Platonic Solids and one odd-ball Polyhedron The kinship among the polyhedra is reflected in the...
		In particular, KeplerÂ’s hedgehogs have the face-planes of the...
	regular_polyhedron.iqexpand.com (1235 words)

	Platonic solid (Site not responding. Last check: 2007-10-09)
		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.findterm.net /pl/platonic-solid.html (1185 words)

	Kepler-Poinsot Polyhedra
		As in the Platonic solids, these solids have identical regular polygons for all their faces, and the same number of faces meet at each vertex.
		These two polyhedra were described by Johannes Kepler in 1619, and he deserves credit for first understanding them mathematically, but a 16th century drawing by Jamnitzer is very similar to the former and a 15th century mosaic attributed to Uccello illustrates the latter.
		To emphasize that these polyhedra are made of large convex faces, it helps to look at a five-color model of the great icosahedron and a six-color model of the great dodecahedron.
	www.georgehart.com /virtual-polyhedra/kepler-poinsot-info.html (485 words)

	Regular Polyhedra
		We will be interested in calculating the volume and surface areas of these solids.
		The situation is much different than with regular polygons because there are only five Platonic solids, so we will treat each of them separately.
		A similar relation exists for any number of dimensions.) Whereas it is fairly straightforward to find the volume of the first three Platonic solids without splitting them up into pyramids, computing that of the icosahedron is much easier by splitting it up.
	www.math.rutgers.edu /~erowland/polyhedra.html (1011 words)

	ICOSAHEDRON - LoveToKnow Article on ICOSAHEDRON (Site not responding. Last check: 2007-10-09)
		CiKOITI., twenty, and ~bpa, a face or base), in geometry, a solid enclosed by twenty faces.
		The regular icosahedron is one of the Platonic solids; the great icosahedron is a Kepler-Poinsot solid; and the truncated icosahedron is an Archimedean solid (see POLYHEDRON).
		In crystallography the icosahedron is a possible form, but it has not been observed; it is closely simulated by a combination of the octahedron and pentagonal dodecahedron, which has twenty triangular faces, but only eight are equilateral, the remaining twelve beiIig isosceles (see CRYSTALLOGRAPHY).
	54.1911encyclopedia.org /I/IC/ICOSAHEDRON.htm (413 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		The five convex regular polyhedra are known as the Platonic solids, and have been known at least since the fourth century BCE.
		In 1619, Johannes Kepler showed that two stellations of the dodecahedron, the ``small stellated dodecahedron'' and the ``great stellated dodecahedron'', could each be regarded as a regular solid with twelve intersecting faces, each of which is a regular pentagram (a five-pointed star, which is a nonconvex regular pentagon).
		Cauchy published a proof in 1813 that the Platonic solids and the Kepler-Poinsot solids are all the possible regular polyhedra.
	www.cs.utexas.edu /users/xli/prob/p11/willa.txt (271 words)

	Geometric Solids
		Apex and apices or apexes (plural) - the vertex of an angle.
		solid is regular if the spices are the same.
		There are nine regular solids: the five Platonian, pictured above, and the four polyhedra described by Kepler-Poinsot.
	homepage.mac.com /montessoriworld/mwei/sensory/sgeosoli.html (398 words)

	Kepler solid (Site not responding. Last check: 2007-10-09)
		The Kepler solids were defined by Johannes Kepler in 1619, when he noticed that the stellated dodecahedrons (there are two, a greater and a lesser) were composed of "hidden" dodecadrons (with pentagonal faces) that have faces composed of triangles, and thus look like stylized stars.
		Wentzel Jamnitzer[?] actually found the great stellated dodecahedron and the great dodecahedron in the 1500s, and Paolo Uccello discovered and drew the lesser stellated dodecadron in the 1400s.
		The other two are the greater icosahedron and greater dodecahedron which were described by Louis Poinsot[?] in 1809.
	www.city-search.org /ke/kepler-solid.html (608 words)

	Uniform Polyhedra
		The solids above are derived from the rhombicuboctahedron and rhombicosidodecahedron by faceting, or removing parts of the solid bounded by planes within the solid.
		The solids above are derived by faceting the cube and dodecahedron to produce 8/3 and 10/3 faces.
		The two solids above have the same vertices and edges as the preceding two pairs, but the 8/3 and 10/3 faces have been faceted to result in intricate rosettes.
	www.uwgb.edu /dutchs/symmetry/unipol1.htm (253 words)

	Gematria - Free Encyclopedia of Thelema
		One example of Gematria is that there are twenty-two solid figures that are composed of regular polygons.
		There are five Platonic solids, four Kepler-Poinsot solids, and thirteen Archimedean solids.
		The art of Gematria is knowing which solid is associated with which letter.
	www.egnu.org /thelema/index.php/Gematria (354 words)

	Polyhedral Compounds (Site not responding. Last check: 2007-10-09)
		Its vertices are the vertices ofd a cube, its edges are the face diagonals of a cube, and the solid common to both tetrahedra is an octahedron.
		In addition to combining polyhedra and their duals, there are several regular compounds of polyhedra that result in solids with greater symmetry than the component polyhedra.
		The icosahedron and octahedron are duals of the dodecahedron and cube.
	euch3i.chem.emory.edu /proposal/gbms01.uwgb.edu/~dutchs/symmetry/polycpd.htm (731 words)

	Polyhedra (Site not responding. Last check: 2007-10-09)
		All mathematicians are familiar with the Platonic solids: the tetrahedron, the cube, the octahedron.
		These are the five convex solids all of whose faces are identical regular polygons.
		On the other hand, if the faces are required to be identical regular polygons, but the solid is not required to be convex, we obtain the four Kepler-Poinsot polyhedra.
	130.44.194.100 /featurecolumn/archive/polyhedra.html (182 words)

	Kepler-Poinsot solid - Encyclopedia Glossary Meaning Explanation Kepler-Poinsot solid (Site not responding. Last check: 2007-10-09)
		Kepler-Poinsot solid - Encyclopedia Glossary Meaning Explanation Kepler-Poinsot solid.
		Here you will find more informations about Kepler-Poinsot solid.
		The orginal Kepler-Poinsot solid article can be editet
	www.encyclopedia-glossary.com /en/Kepler-Poinsot-solid.html (392 words)

	Regular Polyhedra Project
		Summary: In Section I, students derive the volumes and surface areas of the five Platonic solids in terms of side length.
		In Section II, students apply Euler's formula to polyhedra, count the number of diagonals in each of the Platonic solids, and find the dual polyhedra of the Platonic solids.
		A diagonal is a line segment drawn between two vertices of a polyhedron that are not adjacent to a common face.
	www.math.rutgers.edu /~erowland/polyhedra-project.html (718 words)

	The Kepler-Poinsot Polyhedra (Site not responding. Last check: 2007-10-09)
		The 5 Platonic Solids are the convex regular polyhedrons.
		It was Johann Kepler who, in 1619, first realized that 12 pentagrams can be joined in pairs along their edges in two different ways that result in regular solids.
		It was not until 1811 that the French mathematician Augustin Cauchy showed that the Kepler-Poinsot solids are stellated forms of the dodecahedron or the icosahedron.
	home.comcast.net /~tpgettys/kepler.html (327 words)

	Talk:Kepler-Poinsot solid - Wikpedia (Site not responding. Last check: 2007-10-09)
		The term "regular polyhedron" is not defined, and it is not clear whether different polygons may be used as faces.
		I haven't heard anyone refer to these as just the Kepler solids.
		I think it should be changed, but it seems so odd that maybe there is a reason for it.
	bostoncoop.net /~tpryor/wiki/index.php?title=Talk:Kepler-Poinsot_solid (103 words)

	About the Uniform Polyhedra
		There are 18 convex uniform polyhedra, namely the five Platonic solids and thirteen Archimedean solids.
		If we drop the condition that the solids be convex we get many more shapes.
		Instead, we will find a way to compute the coordinates of any of the uniform polyhedra directly from their so-called Wythoff symbols.
	www.mathconsult.ch /showroom/unipoly/background.html (308 words)

	GRAIL: Tilings and Geometric Ornament: Symmetrohedra
		Note that Symmetrohedra can't be used to represent all 18 Platonic and Archimedean solids.
		The way we defined them, the resulting polyhedra will have all the symmetries of one of the polyhedral groups generated by reflections, and so we can't represent the two snub Archimedeans (though we could with a suitable extension to our notation).
		Name: Alternate Bowtie Icosahedron Notes: This solid is a "near miss" in the sense that it's almost a Johnson solid, except that some of the faces are not quite regular.
	www.cgl.uwaterloo.ca /~csk/washington/tile/symmetro.html (582 words)

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