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Topic: Poinsot solid

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Platonic solid

Johannes Kepler

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	Polyhedron - LoveToKnow 1911
		The names of these five solids are: (r) the tetrahedron, enclosed by four equilateral triangles; (2) the cube 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.
		Mensuration of the Platonic Solids.-The mensuration of the regular polyhedra is readily investigated by the methods of elementary geometry, the property that these solids may be inscribed in and circumscribed to concentric spheres being especially useful.
		Two such solids exist: (1) the " rhombic dodecahedron, " formed by truncating the edges of a cube, is bounded by 12 equal rhombs; it is a common crystal form (see Crystallography); and (2) the " semi-regular triacontahedron," which is enclosed by 30 equal rhombs.
	www.1911encyclopedia.org /Polyhedron (2176 words)

	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.
		Thus the 12 faces are extended to pentagrams with the central pentagon inside the solid.
	en.wikipedia.org /wiki/Kepler-Poinsot_solid (672 words)

	Britain.tv Wikipedia - Platonic solid
		The symmetry groups of the Platonic solids are known as polyhedral groups (which are a special class of the point groups in three dimensions).
		Most importantly, the vertices of each solid are all equivalent under the action of the symmetry group, as are the edges and faces.
		The next most regular convex polyhedra after the Platonic solids are the cuboctahedron, which is a rectification of the cube and the octahedron, and the icosidodecahedron, which is a rectification of the dodecahedron and the icosahedron (the rectification of the self-dual tetrahedron is a regular octahedron).
	www.britain.tv /wikipedia.php?title=Platonic_solid (3131 words)

	Platonic solid (Site not responding. Last check: 2007-10-30)
		A Platonic solid is a convex polyhedron whose faces all share the same regular polygon and such that the same number faces meet at all its vertices.
		Compare the Kepler solids which are not convex and the Archimedean and Johnson solids which while made of regular polygons not themselves regular.
		Proposition 13 describes the construction of tetrahedron proposition 14 of the octahedron proposition of the cube proposition 16 of the and proposition 17 of the dodecahedron.
	www.freeglossary.com /Platonic_solid (730 words)

	Kepler solid - Wikipedia
		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.
		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.
		The other two are the greater icosahedron and greater dodecahedron which were described by Louis Poinsot in 1809.
	nostalgia.wikipedia.org /wiki/Poinsot_solid (270 words)

	Body Solid -- Recommendations and Resources (Site not responding. Last check: 2007-10-30)
		Within a solid, atoms/molecules are relatively close together, or "rigid"; however, this does not prevent the solid from becoming deformed or compressed.
		In the solid phase of matter, atoms have a fixed spatial ordering; because all matter has some kinetic energy, the atoms in even the most rigid solid move slightly, but this movement is "invisible".
		They are distinct from the Platonic solids, which are composed of only one type of polygon meeting in identical vertices, and from the Johnson solids, whose regular polygonal faces do not meet in identical vertices.
	www.becomingapediatrician.com /health/20/body-solid.html (2061 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)

	Countertops Solid Surface -- Recommendations and Resources (Site not responding. Last check: 2007-10-30)
		A solid is a state of matter, characterized by a definite volume and a definite shape (i.e.
		Solid state chemistry overlaps both of these fields, but is especially concerned with the synthesis of novel materials.
		As in any strictly convex solid, at least three faces meet at every vertex, and the total of their angles is less than 360 degrees.
	www.becomingapediatrician.com /health/37/countertops-solid-surface.html (2064 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.
		To define a Platonic solid in Maple, one can use the command RegularPolyhedron(gon,[m,n],o,r); where gon is the name of the polyhedron to be defined, [m,n] the Schl&aum;lfli symbol, o the center of the polyhedron, and r the radius of the circum-sphere.
		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...
	www.cecm.sfu.ca /~hle/polyhedra/regular.html (1058 words)

	Uniform polyhedron - Wikipedia, the free encyclopedia
		The Platonic solids date back to the classical Greeks and were studied by Plato, Theaetetus and Euclid.
		Johannes Kepler (1571-1630) was the first to publish the complete list of Archimedean solids after the original work of Archimedes was lost.
		Kepler (1619) discovered two of the regular Kepler-Poinsot solids and Louis Poinsot (1809) discovered the other two.
	en.wikipedia.org /wiki/Uniform_polyhedron (1048 words)

	Convex - Biocrawler (Site not responding. Last check: 2007-10-30)
		For example, a solid cube is convex, but anything that is hollow or has a dent in it is not convex.
		The concepts of convexity and concavity are important in various fields, and in various fields the adjective "convex" has their own specific meanings.
		Some examples of convex subsets of Euclidean 3-space are the Archimedean solids and the Platonic solids.
	www.biocrawler.com /encyclopedia/Convex (509 words)

	The regular polyhedra: Platonic solids (Site not responding. Last check: 2007-10-30)
		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.
		These solids, known as stellated dodecahedra are star shaped and have faces which are regular pentagrams.
		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)

	Gematria - Biocrawler (Site not responding. Last check: 2007-10-30)
		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.biocrawler.com /encyclopedia/Gematria (448 words)

	Plastic (Site not responding. Last check: 2007-10-30)
		Solid rockets used during World War II used nitrocellulose explosives for propellants but it was and dangerous to make such rockets very
		This new solid fuel more slowly and evenly than nitrocellulose explosives was much less dangerous to store and though it tended to flow slowly out the rocket in storage and the rockets it had to be stockpiled nose-down.
		Such solid fuels could be into large uniform blocks that had no or other defects that would cause nonuniform Ultimately all large military rockets and missiles use synthetic rubber based solid fuels and would also play a significant part in civilian space effort.
	www.freeglossary.com /Plastic (4671 words)

	[No title] (Site not responding. Last check: 2007-10-30)
		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)

	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)

	Solid Bluetooth
		Solid-state laser - A solid-state laser is a laser that uses a gain medium that is a solid, rather than a liquid such as dye lasers or a gas such as gas lasers.
		Semiconductor-based lasers are also in the solid state, but are generally considered separately from solid-state lasers (see semiconductor laser).
		Kepler-Poinsot solid - 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).
	www.qmiinc.com /114/3.html (1003 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.
	www.montessoriworld.org /sensory/sgeosoli.html (398 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.
		One of them appears on a 16th century drawing by Jamnitzer and the other on a 15th century mosaic on the floor of the San Marco in Venice.
		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)

	Dodecahedron Day: Glossary (Site not responding. Last check: 2007-10-30)
		For example, a truncated icosahedron is an Archimedean solid.
		For example, the convex hull of the small stellated dodecahedron is the regular icosahedron.
		Platonic solid: one of the five convex regular polyhedra.
	websites.quincy.edu /~matskvi/ddd/glossary.html (311 words)

	Archimedean Solid (Site not responding. Last check: 2007-10-30)
		Elongated Square Gyrobicupola by their symmetry group: the Archimedean solids have a spherical symmetry, while the others have ``dihedral'' symmetry.
		Condition (2) requires that the sum of interior angles at a vertex must be equal to a full rotation for the figure to lie in the plane, and less than a full rotation for a solid figure to be convex.
		Circumradii of these solids are given in the entries for the
	www.math.sdu.edu.cn /mathency/math/a/a316.htm (557 words)

	Archimedean Solids
		The Platonic, Kepler-Poinsot solids are uniform, so are the right regular prisms and antiprisms of suitable height - namely, when their lateral faces are squares and equilaterals, respectively.
		In the case when two regular polyhedra, {p,q} and {q,p}, are reciprocal with respect to their common mid-sphere, the solid region interior to both polyhedra forms another polyhedron, say {p/q}, which has N1 vertices, namely the mid-edge points of either {p,q} or {q,p}.
		In Maple, one can define an Archimedean solid by using the command Archimedean(gon,sch,o,r); where gon is the name of the polyhedron to be defined, sch the Schläfli symbol (Maple's Schläfli), o the center of the polyhedron, and r the radius of the circum-sphere.
	oldweb.cecm.sfu.ca /~hle/polyhedra/archimedean.html (666 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)

	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.
		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.
		Together, the Platonic solids and these Kepler-Poinsot polyhedra form the set of 9 regular polyhedra.
	www.georgehart.com /virtual-polyhedra/kepler-poinsot-info.html (485 words)

	Icosahedron - LoveToKnow 1911
		eKovt, twenty, and g (3pa, 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 being isosceles (see Crystallography).
	www.1911encyclopedia.org /Icosahedron (122 words)

	Kepler-Poinsot Solids (Site not responding. Last check: 2007-10-30)
		A natural extension is to extend the faces of a three-dimensional solid (polyhedron) until they meet.
		The Kepler-Poinsot solids are stellations of the dodecahedron and icosahedron.
		The solid has the edges and vertices of an icosahedron, but instead of triangular faces has triangular dimples.
	www.uwgb.edu /dutchs/symmetry/kpsolid.htm (522 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)

	[No title]
		Three kinds of bounds are to be considered in any solid body; namely points, lines and surfaces, or with the names specifically used for this purpose: solid angles, edges and faces.
		Tunnels through solids were analyzed by means of non-separating curves (a closed curve in a surface such that the surface remains in one piece) and cavities in the solids were thought to be resulting disconnected surfaces.
		August Ferdinand Möbius defined a polyhedron as a system of polygons arranged in such a way that the sides of exactly two polygons meet at every edge and it is possible to travel from the interior of one polygon to the interior of any other without passing through a vertex.
	www.me.metu.edu.tr /kiper/Polyhedra/Euler_dosyalar/rightside.htm (1597 words)

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