Kepler-Poinsot solid

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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)

	POLYHEDRON - LoveToKnow Article on POLYHEDRON
		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.
		Semi-regular Polyhedra.Although this term is frequently given to the Archimedean solids, yet it is a convenient denotation for solids which have all their angles, faces, and edges equal, the faces not being regular polygons.
	www.1911encyclopedia.org /P/PO/POLYHEDRON.htm (3208 words)

	Kepler-Poinsot 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.
		The four regular non-convex polyhedra that exist in addition to the five regular convex polyhedra known as the Platonic solids.
	www.daviddarling.info /encyclopedia/K/Kepler-Poinsot_solids.html (326 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)

	math lessons - Platonic solid
		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.
		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°.
		Plato learned about these solids from his friend Theaetetus.
	www.mathdaily.com /lessons/Platonic_solid (1054 words)

	Dodecahedra
		It is a Kepler-Poinsot solid, and also has the full symmetry of the icosahedral symmetry group.
		It is a Platonic solid and the only convex dodecahedron with all the symmetry axes and mirror planes of the icosahedral symmetry group.
		31] gives a 1960 reference for its first publication, but I have observed that it appears as a pop-up paper model in the 1787 Solid Geometry of John Lodge Cowley.)
	www.georgehart.com /virtual-polyhedra/dodecahedra.html (1228 words)

	Regular Polyhedra
		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.
		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...
	www.cecm.sfu.ca /~hle/polyhedra/regular.html (1058 words)

	iqexpand.com
		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...
		The Platonic solids were admired and adored by the ancient Greek mathematicians and anyone learning...
	regular_polyhedron.iqexpand.com (1235 words)

	Polyhedral Compounds
		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.
		These are all based on the fact that some Platonic solids can be inscribed in others.
	euch3i.chem.emory.edu /proposal/gbms01.uwgb.edu/~dutchs/symmetry/polycpd.htm (731 words)

	The four regular non-convex polyhedra
		The other two were described by Louis Poinsot in 1809 but at least one of them appears on a drawing by the same Jamnitzer.
		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.
	cage.rug.ac.be /~hs/polyhedra/keplerpoinsot.html (628 words)

	Geometric Solids
		There are nine regular solids: the five Platonian, pictured above, and the four polyhedra described by Kepler-Poinsot.
		solid is regular if the spices are the same.
		Cylinder: a solid bounded by two parallel planes which are curved
	homepage.mac.com /montessoriworld/mwei/sensory/sgeosoli.html (398 words)

	Polyhedra
		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.
		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.
	130.44.194.100 /featurecolumn/archive/polyhedra.html (182 words)

	Regular Polyhedra
		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.
		The situation is much different than with regular polygons because there are only five Platonic solids, so we will treat each of them separately.
		We will be interested in calculating the volume and surface areas of these solids.
	www.math.rutgers.edu /~erowland/polyhedra.html (1011 words)

	willa.txt
		Cauchy published a proof in 1813 that the Platonic solids and the Kepler-Poinsot solids are all the possible regular polyhedra.
		In 1809, Louis Poinsot reidentified Kepler's solids, along with two other nonconvex regular polyhedra, the ``great dodecahedron'' and the ``great icosahedron''.
		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).
	www.cs.utexas.edu /users/xli/prob/p11/willa.txt (271 words)

	ICOSAHEDRON - LoveToKnow Article on ICOSAHEDRON
		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).
		CiKOITI., twenty, and ~bpa, a face or base), in geometry, a solid enclosed by twenty faces.
		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)

	Uniform Polyhedra
		If we expand the definition to allow interpenetrating faces, we obtain the four Kepler-Poinsot polyhedra and 53 others, for a total of 75.
		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.
	www.uwgb.edu /dutchs/symmetry/unipol1.htm (253 words)

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

	GRAIL: Tilings and Geometric Ornament: Symmetrohedra
		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.
		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).
	www.cgl.uwaterloo.ca /~csk/washington/tile/symmetro.html (582 words)

	Dual polyhedron - Wikpedia
		So the regular polyhedra — the Platonic solids and Kepler-Poinsot polyhedra — are arranged into dual pairs.
		Duality is defined in terms of polar reciprocation about a given sphere.
	www.bostoncoop.net /~tpryor/wiki/index.php?title=Dual_polyhedron (479 words)

	Talk:Kepler-Poinsot solid - Wikpedia
		Why has this page been redirected to just "Kepler solid" rather than the correct name "Kepler-Poinsot solid"?
		I haven't heard anyone refer to these as just the Kepler solids.
		The term "regular polyhedron" is not defined, and it is not clear whether different polygons may be used as faces.
	bostoncoop.net /~tpryor/wiki/index.php?title=Talk:Kepler-Poinsot_solid (103 words)

	tasmania.ca - Kepler
		Johannes Kepler was born in Weil der Stadt in...
		Kepler Mission: A Search for Terrestrial Planets Kepler Mission: A Search for Terrestrial Planets The Kepler Mission is a space mission designed to detect and characterize hundreds of Eart...
		Kepler Mission to find Earth-size planets in the h...
	www.tasmania.ca /Kepler/reference/search (308 words)

	The Kepler-Poinsot Polyhedra
		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.
		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.
		The 5 Platonic Solids are the convex regular polyhedrons.
	home.comcast.net /~tpgettys/kepler.html (327 words)

	ZEFRANK.COM - message board - numerical order
		One is a Platonic solid (the icosahedron itself), one is a Kepler-Poinsot solid, four are polyhedron compounds, and one is the dual polyhedron of an Archimedean solid.
		This gives fairly small spikes, and results in a solid known as the small triambic icosahedron.
		Note also that the great stellated dodecahedron is not an icosahedron stellation, since the faces of its groups of five triangular pyramids do not lie in the same plane even though they appear very close to it.
	www.zefrank.com /bulletin/showthread.php?t=7931&page=5&pp=15 (352 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.
		Some of the Archimedian solids can be obtained in this way from the Platonic ones.
	www.mathconsult.ch /showroom/unipoly/background.html (308 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.
		Objectives: To develop spatial reasoning, specifically by applying techniques of planar geometry to three-dimensional solids.
	www.math.rutgers.edu /~erowland/polyhedra-project.html (718 words)

	Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 42, No. 1, pp. 1-37, 2001
		These hypermaps are constructed as combinatorial and topological objects, many of them arising as coverings of Platonic solids and Kepler-Poinsot polyhedra (viewed as hypermaps), or their associates.
		Abstract: We classify the regular hypermaps (orientable or non-orientable) whose full automorphism group is isomorphic to the symmetry group of a Platonic solid.
		We also classify the regular hypermaps with automorphism group $A_5$: there are 19 of these, all non-orientable, and 9 of them are maps.
	www.emis.de /journals/BAG/vol.42/no.1/1.html (153 words)

	Publications of the Mathematical Association of New South Wales: PowerPoint Presentations for the Secondary School
		Definitions, Platonic solids (proof that there can only be 5), Kepler’s claim, applicability of Euler’s Formula.
		Some applications of Polyhedra to medicine, crystallography and solid state physics are included.
		The first half of this presentation proves that the conic sections are indeed the same curves that we now treat algebraically and call ellipse, hyperbola and parabola.
	www.mansw.nsw.edu.au /publications/powerpoint-presentations-for-sec-schools.htm (590 words)

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