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Topic: Small stellated dodecahedron

	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.
		The small stellated dodecahedron is formed by stellating the Platonic dodecahedron (by "stellating " is meant developing the faces contiguous to a specified base so as to form a regular pyramid).
		The great dodecahedron is determined by the intersections of the twelve planes which intersect the Platonic icosahedron in five of its edges; or each face has the same boundaries as the basal sides of five covertical faces of the icosahedron.
	www.1911encyclopedia.org /Polyhedron (2150 words)

	[No title]
		dodecahedron +------------------------------------------------------------ The dodecahedron is a polyhedron with 20 vertices and 12 faces.
		small stellated dodecahedron +------------------------------------------------------------ The small stellated dodecahedron is a polyhedron with 12 vertices and 12 faces.
		small ditrigonal icosidodecahedron +------------------------------------------------------------ The small ditrigonal icosidodecahedron is a polyhedron with 80 vertices and 72 faces.
	www.math.harvard.edu /~knill/sofia/data/polyhedra.txt (2272 words)

	Dodecahedron - LoveToKnow 1911
		The "small stellated dodecahedron," the "great dodecahedron" and the "great stellated dodecahedron" are Kepler-Poinsot solids; and the "truncated" and "snub dodecahedra" are Archimedean solids (see Polyhedron).
		In crystallography, the regular or ordinary dodecahedron is an impossible form since the faces cut the axes in irrational ratios; the "pentagonal dodecahedron" of crystallographers has irregular pentagons for faces, while the geometrical solid, on the other hand, has regular ones.
		The "rhombic dodecahedron," one of the geometrical semiregular solids, is an important crystal form.
	www.1911encyclopedia.org /Dodecahedron (167 words)

	stellation theory
		The regular dodecahedron is surrounded by three layers of bounded cells : 12 golden pentagonal pyramids, then 30 wedges (tetrahedra) which insert themselves between the pyramids, and finally 20 spikes (triangular bi-pyramids) which fit between the wedges.
		The three stellations of the dodecahedron are non convex regular polyhedra (the great icosahedron is a stellation of the icosahedron).
		For the dodecahedron we thus have a (the dodecahedron), b (the 12 pyramids), c (the 30 wedges), and d (the 20 pikes).
	www.ac-noumea.nc /maths/amc/polyhedr/stell_dod_.htm (455 words)

	Dodecahedron Day: Glossary (Site not responding. Last check: 2007-11-01)
		For example, a pentagon and the regular dodecahedron are convex, but a pentagram and the small stellated dodecahedron are nonconvex.
		For example, the convex hull of the small stellated dodecahedron is the regular icosahedron.
		For example, a pentagram and the small stellated dodecahedron are nonconvex.
	websites.quincy.edu /~matskvi/ddd/glossary.html (311 words)

	Geometry Clipart ETC
		Rhombic Dodecahedron A solid consisting of 12 similiar faces, each of which is rhomb, the angle between any two adjacent face being 120 degrees.
		Stellated Dodecahedron A regular solid each face of which formed by stellating a face of the dodecahedron.
		Truncated Dodecahedron A regular dyocaetriacontahedron formed by cutting off the faces of a regular dodecahedron parallel to those of the coxial icosahedron so as to leave the former decagons.
	etc.usf.edu /clipart/galleries/math/geometry.htm (1231 words)

	Acidophilus Related Terms
		The vertex figures for the small and great stellated dodecahedra are the pentagon and triangle respectively, reflecting the absence of internal intersections between those pentagrams which meet at a vertex in those polyhedra.
		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.
		Wenzel Jamnitzer actually found the great stellated dodecahedron and the great dodecahedron in the 1500s, and Paolo Uccello discovered and drew the small stellated dodecahedron in the 1400s.
	www.acidophiluseffects.com /notes/?title=Kepler-Poinsot_solid (651 words)

	The Regular Polyhedra (Site not responding. Last check: 2007-11-01)
		The corners of the dodecahedron become simply the points where the edges of the greater hedgehog intersect one another, and the relationship of the hedgehog to its inner dodecahedron is exactly like the relationship of a pentagram to its central pentagon.
		The twelve pentagrammatic faces of a small stellated dodecahedron may be extended until they meet the extensions of other faces, just as the edges of a dodecahedron are extended to meet other edge-extensions to create the small stellated dodecahedron.
		The great stellated dodecahedron is the only regular polyhedron that is a faceting of the regular dodecahedron; the other three are all facetings of the regular icosahedron.
	members.aol.com /Polycell/regs.html (11220 words)

	An Interactive Creation of Polyhedra Stellations with various Symmetries
		The earliest known author to describe stellations having a lower symmetry than that of the core is Alan Holden in his book, Shape Space and Symmetry [3], where he considered the dodecahedron.
		The complete stellation cell, formed by an array of elementary cells is shown in Fig.15 (stellation e1 from [2]).
		The stellations applet [11] was written to greatly facilitate the process of stellation construction.
	www.mi.sanu.ac.yu /vismath/bulatov (2287 words)

	Index: Stellated Polyhedra (80-85) (Site not responding. Last check: 2007-11-01)
		Performing this process of stellation on a dodecahedron yields the small stellated dodecahedron (80), whose faces are pentagrams (five-pointed stars).
		Allowing these faces to expand even further (stellating the stellation) yields the great dodecahedron (not shown), and stellating this figure in turn yields the great stellated dodecahedron (82).
		These are the Kepler-Poinsot polyhedra: the small stellated dodecahedron; its dual, the great dodecahedron; the great stellated dodecahedron; and its dual, the great icosahedron.
	math.arizona.edu /~models/Stellated_Polyhedra/index.html (197 words)

	Star Polytopes
		Stellation and faceting do not give equal results with most polyhedra ot polytopes, but do with regular polygons, since these figures are self-dual.
		The theory of stellation is quite interesting (see a comparison of three primitive operations on convex polyhedra: completion, stellation, and faceting), and of the many devotees of things polyhedral, and things polytopical, many are more or less obsessed with stellation.
		When one moves from the convex polyhedra, to stellations, additonal layers of great subtlety and complexity pose a tremendous challenge, one which might not be fully met within a lifetime.
	home.inreach.com /rtowle/Polytopes/star_polytopes/Star_Polytopes.html (579 words)

	The four regular non-convex polyhedra
		The small stellated dodecahedron can be constructed by putting appropriated five-sided pyramids upon the faces of the given dodecahedron.
		The vertices of the small stellated dodecahedron are also the vertices of an icosahedron.
		The great stellated dodecahedron can be constructed by putting appropriated three-sided pyramids upon the faces of the given icoshedron.
	cage.rug.ac.be /~hs/polyhedra/keplerpoinsot.html (628 words)

	Stellated dodecahedron
		This shows the connection with the triacontrahedron - the small stellated dodecahedron is like a triacontrahedron, with the original icosahedron vertices moved further away from the centre
		When you think this way, you see that the small stellated dodecahedron has twelve identical faces, which meet together in an identical fashion around each of it's twenty vertices.
		So the small stellated dodecahedron is another perfect solid, like the platonic solids.
	robertinventor.com /cubeetc/stellate.htm (371 words)

	Small stellated dodecahedron - Wikipedia, the free encyclopedia
		In geometry, the small stellated dodecahedron is a Kepler-Poinsot solid.
		It is considered the first of three stellations of the dodecahedron.
		It can also be constructed as the first of four stellations of the dodecahedron, and referenced as Wenninger model [W41].
	en.wikipedia.org /wiki/Small_stellated_dodecahedron (150 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		I include instructions for the small and great stellated dodecahedron, made from the same module (30 for each).
		To make the small stellated dodecahedron, assemble the units so that 5 isoceles triangles come together at each point.
		To make the great stellated dodecahedron, assemble the units so that 3 isoceles triangles come together at each point.
	www.cs.utk.edu /~plank/plank/origami/mosely.txt (282 words)

	Stellations of Polyhedra (Site not responding. Last check: 2007-11-01)
		Below are shown stellations of the tetrahedron, cube, octahedron, pentagonal dodecahedron and rhombic dodecahedron.
		Stellations of the icosahedron are much more complex and shown on separate pages.
		This is obtained by continuing the star planes of the small stellated dodecahedron outward until they meet to form the next set of pentagons.
	www.uwgb.edu /dutchs/symmetry/stellate.htm (325 words)

	Kepler-Poinsot Polyhedra (Site not responding. Last check: 2007-11-01)
		In the great stellated dodecahedron and the small stellated dodecahedron, the faces are pentagrams.
		It is easier to see which parts of the exterior belong to which pentagram if you look at a six-colored model of the great stellated dodecahedron and a six-colored model of the small stellated dodecahedron.
		The small stellated dodecahedron has the same vertices and edges as the great icosahedron.
	www.georgehart.com /virtual-polyhedra/kepler-poinsot-info.html (485 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		That these five exhaust the possibilities for convex regular polyhedra was shown by Euclid in the final part of the ``Elements''.
		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).
		The small stellated dodecahedron has five faces meeting at each vertex, and the great stellated dodecahedron has three.
	www.cs.utexas.edu /users/xli/prob/p11/willa.txt (271 words)

	non convex polyhedra 1
		Kepler discovered that twelve stellated pentagons can be assembled along thirty edges, building two new regular non convex polyhedra: the small stellated dodecahedron and the great stellated dodecahedron.
		The sixty visible parts of their faces are golden triangles (isosceles, with one angle of 36°).
		The convex hull of the small stellated dodecahedron is a regular icosahedron; its convex kernel is a regular dodecahedron.
	www.ac-noumea.nc /maths/amc/polyhedr/no_conv1_.htm (219 words)

	The Kepler Solids
		Both are stellated dodecahedra; they are called "small" and "great" to distinguish them.
		The small stellated dodecahedron (on the left above) has twelve pentagonal pyramids built on the faces of a dodecahedron.
		Its construction is similar to that of the great stellated dodecahedron.
	britton.disted.camosun.bc.ca /jbkeplersolids.htm (565 words)

	The Math Worksheet Site.com -- Small Stellated Dodecahedron (Site not responding. Last check: 2007-11-01)
		This is a pattern for making a paper Small Stellated Dodecahedron.
		A Small Stellated Dodecahedron is a stellation of a dodecahedron.
		Instructions for building a Small Stellated Dodecahedron can be found here.
	themathworksheetsite.com /subscr/small_stellated_dodecahedron.html (145 words)

	Dodecahedron Day: The Small Stellated Dodecahedron (Site not responding. Last check: 2007-11-01)
		The small stellated dodecahedron is one of the
		Question: The twelve outer vertices of the small stellated dodecahedron are the vertices of which Platonic solid?
		This small model should be attempted by experienced model builders only.
	websites.quincy.edu /~matskvi/ddd/models/ssd.html (166 words)

	Amazon.com: "small stellated dodecahedron": Key Phrase page (Site not responding. Last check: 2007-11-01)
		Then there will also be thirty sphenoids or wedge- shaped pieces which convert the small stellated dodecahedron into the great...
		The quantity v refers to a convex vertex solid e f v small stellated dodecahedron...
		In the great stellated dodecahedron and the small stellated dodecahedron, the faces are pentagrams (five-pointed stars).
	www.amazon.com /phrase/small-stellated-dodecahedron (542 words)

	Cool math 4 kids .com - Polehedra ( Geometry ) - Gallery of Polyhedrons (Site not responding. Last check: 2007-11-01)
		Just like the small stellated dodecahedron, the great stellated dodecahedron is simply 12 pentagrams intersected in a special way.
		You should be able to find pentagrams that are red, green, etc. Once again, like the small stellated dodecahedron, there are two pentagrams of each color.
		The stellated dodecahedron is made up of 20 tips with 3 isosceles triangles in each tips for a total of 60 triangles.
	www.coolmath4kids.com /poly_stelicosa.html (230 words)

	Stellation of Small Stellated Truncated Dodecahedron
		All the faces have 5-fold rotational symmetry, being parallel to the faces of the dodecahedron.
		The model is a stellation of the small stellated truncated dodecahedron, pictured here, which should probably really be called the truncated small stellated dodecahedron.
		This shows one of the two types of faces, from the pentagon plane of the small stellated truncated dodecahedron.
	web.aanet.com.au /robertw/StelSSTD.html (336 words)

	The Stellated 120-Cell (Site not responding. Last check: 2007-11-01)
		Mount each of the trihedral rosettes onto the small stellated dodecahedron in such a way that the long reds connect to three of the vertices of the small stellated dodecahedron and the medium yellow connects (quite naturally) to a vertex of the dodecahedron from step 1.
		After attaching all 20, you should see a large Y3 rhombic triacontahedron (30 rhombii) bounding all the pieces.
		At this point you should notice a similarity to one of the final stages of the Zome model of the 600-cell.
	homepages.wmich.edu /~drichter/stellated120cell03.htm (199 words)

	polyhedra
		There are only five platonic solids - tetrahedron, cube, octahedron, icosahedron and dodecahedron.
		Johannes Kepler, in 1619, found two polyhedra which are simultaneously regular and not convex - the small stellated dodecahedron and the big stellated dodecahedron.
		The previous table points to a certain distribution of the 5 regular polyhedra in 3 classes: Tetrahedron (dual of itself), Cube and Octahedron, Dodecahedron and Icosahedron.
	www.atractor.pt /mat/Polied/poliedros-e.htm (309 words)

	Kepler-Poinsot Solids (Site not responding. Last check: 2007-11-01)
		The Kepler-Poinsot solids are stellations of the dodecahedron and icosahedron.
		In each case, faces of a dodecahedron are extended outward into a star.
		The small stellated dodecahedron has twelve five-sided pyramids built on the faces of a dodecahedron.
	www.uwgb.edu /dutchs/symmetry/kpsolid.htm (522 words)

	Augmenting the small stellated dodecahedron
		Each pentagrammic face of the small stellated dodecahedron can be excavated with a pentagrammic antiprism completing a cycle around the axis.
		A further augmentation of the small stellated dodecahedron can be obtained by excavating each pentagrammic face with a great retrosnub icosidodecahedron - vertex (
		Another augmentation of the small stellated dodecahedron can be obtained by excavating each pentagrammic face with a great icosidodecahedron - vertex (
	web.ukonline.co.uk /polyhedra/uniform/augmented/39.html (302 words)

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