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Topic: Icosahedron

	Stellations of the Icosahedron
		A few of these we have seen elsewhere: the great icosahedron is one of the Kepler-Poinsot polyhedra, and the first stellation in the official list is just the icosahedron itself.
		For the five-colored versions, each of the 20 icosahedral planes is assigned a color according to this five-coloring of the icosahedron, which in turn comes from the most famous of its stellations, the compound of five tetrahedra.
		Exercise: Note that both the shape of the compound of five tetrahedra and the color pattern of the five-colored icosahedron are chiral.
	www.georgehart.com /virtual-polyhedra/stellations-icosahedron-info.html (435 words)

	PlanetMath: icosahedron
		An icosahedron is a polyhedron with twenty faces.
		This is version 3 of icosahedron, born on 2004-06-22, modified 2007-05-02.
		Object id is 5945, canonical name is Icosahedron.
	planetmath.org /encyclopedia/Icosahedron.html (37 words)

	Tetrahedrally Stellated Icosahedron
		From an icosahedron with edge-length of one, the kite-shaped faces have edges of length 1 and 0.6325.
		The dual operation to stellating four faces of the icosahedron is truncating four vertices of the dodecahedron.
		The result is a tetrahedral stellation of the icosahedron which is a dodecahedron.
	www.georgehart.com /virtual-polyhedra/tetrahedrally_stellated_icosahedron.html (682 words)

	The Icosahedron (Site not responding. Last check: )
		The dihedral angle between the two triangles is the same as that between the faces of the icosahedron.
		The icosahedron is the dual of the dodecahedron.
		Connect the centers of adjacent faces of the icosahedron, and the result is a dodecahedron.
	whistleralley.com /polyhedra/icosahedron.htm (260 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: )
		The icosahedron has a large number of stellations, including one of the Kepler-Poinsot polyhedra and some of the regular compounds, which could be discussed here.
		Despite appearances, when an icosahedron is inscribed in a sphere, it occupies less of the sphere's volume (60.54%) than a dodecahedron inscribed in the same sphere (66.49%).
		The symmetry group of the icosahedron is isomorphic to the alternating group on five letters.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=icosahedron (883 words)

	Icosahedron
		An icosahedron [ˌaıkəsə'hiːdrən] noun (plural: -drons, -dra [-drə]) is a polyhedron having 20 faces.
		In geometry, the regular icosahedron is one of the five Platonic solids.
		Despite appearances, when an icosahedron is inscribed in a sphere, it occupies less of the sphere's volume (60.54%) than a dodecahedron inscribed in the same sphere (66.49%).
	www.xasa.com /wiki/en/wikipedia/i/ic/icosahedron.html (374 words)

	Icosahedron Information (Site not responding. Last check: )
		An icosahedron [ˌaıkəsə'hiːdrən] noun (plural: -drons, -dra [-drə]) is a polyhedron having 20 faces, but usually a regular icosahedron is meant, which has faces which are equilateral triangles.
		The icosahedron has a large number of stellations, including one of the Kepler-Poinsot solids and some of the regular compounds, which could be discussed here.
		The symmetry group of the icosahedron is isomorphic to the alternating group on five letters.
	www.bookrags.com /wiki/Icosahedron (815 words)

	Kids.Net.Au - Encyclopedia > Icosahedron
		The icosahedron is one of the five Platonic solids.
		Canonical coordinates for the vertices of an icosahedron centered at the origin are (0,±1,±τ), (±1,±τ,0), (±τ,0,±1), where τ = (1+√5)/2 is the golden mean - note these form three mutually orthogonal golden rectangles.
		The edges of an octahedron can be partitioned in the golden mean so that the resulting vertices define a regular icosahedron, with the five octahedra defining any given icosahedron forming a regular compound.
	www.kids.net.au /encyclopedia-wiki/ic/Icosahedron (195 words)

	Icosahedron
		An icosahedron noun (plural: -drons, -dra [-drə) is a polyhedron having 20 faces, but usually a regular icosahedron is meant, which has faces which are equilateral triangles.
		In geometry, the regular icosahedron is one of the five Platonic solids.
		The icosahedron has a large number of stellations, including one of the Kepler-Poinsot solids and some of the regular compounds, which could be discussed here.
	www.dejavu.org /cgi-bin/get.cgi?ver=93&url=http%3A%2F%2Farticles.gourt.com%2F%3Farticle%3Dicosahedron%26type%3Den (918 words)

	Flatten Icosahedron
		Triangles of an icosahedron of the given radius would be arranged so the triangles pack with no overlap and no gaps in the plane.
		When saving an image of the flattened icosahedron for constructing a paper model it is important to set the Chimera camera to display using orthographic projection instead of the default perspective projection.
		The nearest triangle is the one whose center is closest to the geometric center of the chain (unweighted average of all atom coordinates).
	www.cgl.ucsf.edu /chimera/experimental/flatten_icosahedron/flaticos.html (890 words)

	Truncated icosahedron (Site not responding. Last check: )
		It has the same shape as a football or a 60- carbon fullerene.
		Canonical coordinates for the vertices of a truncated icosahedron centered at the origin are the orthogonal rectangles (0,±1,±3τ), (±1,±3τ,0), (±3τ,0,±1) and the orthogonal bricks/3D-rectangles (±2,±(1+2τ),±τ), (±(1+2τ),±τ,±2), (±τ,±2,±(1+2τ)) along with the ortogonal bricks/3D-rectangles (±1,±(2+τ),±2τ), (±(2+τ),±2τ,±1), (±2τ,±1,±(2+τ)), where τ = (1+√5)/2 is the golden mean.
		(geometry) a polyhedron with twenty faces; the regular icosahedron has regular triangles as faces and is one of the Platonic solids.
	www.serebella.com /encyclopedia/article-Truncated_icosahedron.html (237 words)

	Stellating the icosahedron and faceting the dodecahedron
		Indeed one way to enumerate the stellations of the icosahedron would be to enumerate the facetings of the dodecahedron and then obtain their duals.
		Here is the stellation diagram for the icosahedron as traditionally drawn, with some additional information: I have identified sets of congruent vertices as A to H and sets of congruent edges as m to q (avoiding the letter o).
		The stellation diagram of the icosahedron is reciprocal to the faceting diagram of the dodecahedron.
	www.steelpillow.com /polyhedra/icosa/stelfacet/StelFacet.htm (4982 words)

	Dodecahedron and Icosahedron
		We want six of the edges of the icosahedron to be positioned in the faces of the given cube as indicated on the first drawing.
		The centers of the faces of a regular dodecahedron are the vertices of a regular icosahedron and the centers of the faces of a regular icosahedron are the vertices of a regular dodecahedron.
		The centers of the (twenty) faces of this icosahedron are the vertices of a regular dodecahedron.
	cage.rug.ac.be /~hs/polyhedra/dodeicos.html (483 words)

	Icosahedron -- from Wolfram MathWorld
		There are 43380 distinct nets for the icosahedron, the same number as for the dodecahedron (Bouzette and Vandamme, Hippenmeyer 1979, Buekenhout and Parker 1998).
		The dual polyhedron of an icosahedron with unit edge lengths is the dodecahedron with edge lengths
		In particular, fifteen golden rectangles span the interior of the icosahedron.
	mathworld.wolfram.com /Icosahedron.html (644 words)

	Coloring The Edges of an Icosahedron (Site not responding. Last check: )
		It's possible to color the edges of an icosahedron in five different colors such that the colors of the edges meeting at each vertex are all different.
		If we fix the orientation of the icosahedron, and assign the five colors a,b,c,d,e to the five edges that meet at the "top" vertex, then there are 780 distinct ways of coloring the rest of the edges such that each color adjoins each vertex.
		Each coloring is represented by a string of 30 digits in the range 1 to 5, signifying which of the five colors is applied to the respective edge.
	www.mathpages.com /home/kmath095.htm (428 words)

	Math Forum: World's Largest Icosahedron
		We laid the icosahedron out flat on the ground and planned how we were going to put it up.
		After a number of weekends of work we had our icosahedron built--but the wind came back and snapped the fishing line we had used to hold the vertices together.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /students/showcase/largest.icosa/index.html (576 words)

	icosahedron
		A regular icosahedron has faces that are all equilateral triangles, and is one of the five Platonic solids.
		The length from vertex to opposing vertex of a regular icosahedron is 5
		Chopping off each vertex (corner) of a regular icosahedron reveals the 12 pentagonal and 20 hexagonal faces of the truncated icosahedron, which is one of the 13 Archimedean solids (shapes made from truncating Platonic solids in certain ways).
	www.daviddarling.info /encyclopedia/I/icosahedron.html (158 words)

	The icosahedron
		The icosahedron has a total of twenty faces, twelve vertices and thirty edges.
		There is an axis of symmetry of the icosahedron passing through each pair of opposite vertices.
		Since there are five triangles around each vertex, it is easy to see that each axis of symmetry through a pair of opposite vertices is associated to five rotations through multiples of one fifth of a whole turn.
	nothung.math.uh.edu /~mike/hti/handouts/notes/node35.html (236 words)

	Icosahedron -- LEGO (Site not responding. Last check: )
		So, back to today: when I sat down to finally start clearing off the tabletop, I got distracted by geometry again and decided that I'd use many of the hinges and triangles that I already had made to construct an object I knew would be quite straightforward.
		An icosahedron (for all you non-geeks, that's a regular twenty sided polyhedron; one of the five basic Platonic Solids).
		I've been in a "20-sided mood" of late, constructing other icosahedrons mainly out of wood.
	www.ericharshbarger.org /lego/icosahedron.html (390 words)

	What does icosahedron mean? - Blurtit
		An icosahedron is a geometric figure, a polyhedron with 20 faces.
		When we talk about icosahedron, we are generally referring to the regular icosahedron with equilateral triangles as faces.
		The regular icosahedron is one of the five platonic solids.
	www.blurtit.com /q765222.html (241 words)

	SpheroidTruncatedIcosahedronFilter! \| Ask MetaFilter
		From the wikipedia on the icosahedron, it tells us that given an icosahedron with edges of length a, the radius of an inscribed sphere (meaning, a sphere big enough to fit inside the icosahedron and touch all sides) is equal to ~75.5% of a.
		Understanding that a truncated icosahedron is just lopping off a 5-sided pyramid from the icosahedron vertexes, the surface exposed by this is a 5-sided pentagon.
		That napkin math aside, according to the Wiki on the truncated icosahedron itself, the edge is length 2 where the radius squared is equal to 9*golden ratio + 10.
	ask.metafilter.com /39890/SpheroidTruncatedIcosahedronFilter (1048 words)

	Icosahedron Fractal, aka Ikosahedron Fraktal
		The icosahedron fractal grows in powers of 12, corresponding to the number of vertices of an icosahedron.
		It is rather subtle, but try to see the outer icosahedron shape made by these 12 icosahedra set tip-to-tip.
		This is one of Reimund Albers images of a Stage-2 icosahedron fractal taken from his Quicktime Movie that shows two stages of growth.
	www.public.asu.edu /~starlite/icosahedronfractal.html (399 words)

	The Icosahedron
		All vertices of the icosahedron (as with all 5 of the regular solids) lie upon the surface of a sphere that encloses it.
		If we lay a 3D model of the icosahedron on one of its sides, we can see that a line through the centroid O is perpendicular to that side.
		All we have done is placed the icosahedron on one of its faces, dropping it from 30° along the x axis and 30° back along the negative y axis.
	kjmaclean.com /Geometry/Icosahedron.html (1587 words)

	About Domes
		It seems to be the most useful polyhedron for dome building.Each vertex is the same distance from the centre of this polyhedron and thus each vertex is on the surface of an imaginary sphere.
		The icosa face (or basic triangle of the icosahedron) has been broken up into four triangles.
		It is not just the icosahedron which can be used to make a dome.
	www.geocities.com /geodesicsnz/about.htm (390 words)

	Map Projections: Truncated Icosahedra (Site not responding. Last check: )
		The truncated icosahedron is easily recognized as the underlying shape of most soccer balls.
		("fullerene") are also organized in truncated icosahedrons, nicknamed buckyballs (also in honor of Buckminster Fuller, whose geodesic domes superficially resemble this polyhedron).
		Lots of small faces, many nonparallel edges, tiny narrow tabs, all contribute for making this solid rather demanding to build (and, the further one progresses, the harder it gets).
	www.progonos.com /furuti/MapProj/Normal/ProjPoly/Foldout/TIcosahedron/ticosahedron.html (192 words)

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