Stellated DodecahedronDate: 12/3/95 at 19:1:49 From: Anonymous Subject: STELLATED DODECAHEDRON I am in a ninth grade honors geometry class and am supposed to make a stellated dodecahedron and write a paper about it and the math used in making it. I have already made the model, but am having trouble finding any information on it for my paper. I have checked all the local libraries and have looked on the Internet. Date: 12/3/95 at 21:26:32 From: Doctor Sarah Subject: Re: STELLATED DODECAHEDRON Hi there - In your Internet search, have you found sites like Tom Gettys' Polyhedral Solids at http://www.teleport.com/~tpgettys/poly.shtml and the Pavilion of Polyhedreality at http://www.li.net/~george/pavilion.html ? Here's a quote from Gettys' page on stellated solids - if you need a footnote reference use this URL: http://www.teleport.com/~tpgettys/stellate.shtml "A solid is said to be convex if the line connecting any two points within the solid is fully contained inside of the solid. If we remove the requirement that our polyhedrons be convex, the realm of the Stellated Solids opens up, offering to us a truly wondrous vision into how space can be divided. In the plane, if you extend the sides of a pentagon, for example, you get the familiar 5-pointed star, or pentagram. The same thing can be done with 3 dimensional solids; the process of extending the faces of a polyhedron until they re-intersect is called "stellating", and the resultant form is called a "stellation" of the parent solid. The Octahedron has eight triangular faces. If the three sides around a face are extended, they form a triangular pyramid over that face. Repeating this process over every face results in the Stellated Octahedron. Now then, consider the five Platonic Solids. The Tetrahedron or Cube have no stellated forms; if you extend their faces they never re- intersect. The Octahedron, however, has the one stellated form mentioned above. The Dodecahedron has three stellations, and it turns out they are quite special. The last Platonic solid, the Icosahedron, has a total of 59 stellated forms! We can also stellate the Archimedian solids, but little is known about most of these. It is here that we unexpectedly meet the boundary of contempory mathematics!" _____________________ You'll find an extensive collection of Internet sites dealing with geometry in the Math Forum's Internet Mathematics Library under mathematics by topic: http://mathforum.org/library/browse/static/topic/geometry.html There is also an archived discussion of stellated polyhedra from our newsgroup geometry.pre-college, at: http://mathforum.org/~sarah/HTMLthreads/articletocs/stellated.polyhedra.html which begins, "I have a question about the small stellated dodecahedron and the great stellated icosahedron. What are the angles in the triangles which form the stellations for each solid?" Answers mention the book POLYHEDRA, A VISUAL APPROACH by Anthony Pugh; one says "(p. 85), in both cases these triangles are golden triangles having base angles of 72 degrees and apex angle of 36 degrees." Here's more from this discussion, a quote from John Conway, the famous Princeton Professor of mathematics: "I think some people may be interested in understanding the names of the Kepler-Poinsot polyhedra. If you prolong the edges of a regular pentagon until they next meet, you get the "stellated pentagon", or pentagram. This has "Schlafli symbol" {5/2}, meaning that it is a regular star-polygon with 5 edges that surround its center twice. In three dimensions, any polyhedron whose faces are all regular pentagons may be stellated in the same way. The dodecahedron {5,3} (meaning that at each vertex we see 3 pentagons) gives rise to the stellated dodecahedron in this way. This polyhedron, sometimes called the small stellated dodecahedron for greater precision, has Schlafli symbol {5/2,5}, meaning that each vertex is surrounded by 5 pentagrams {5/2}, arranged pentagonally. But we can also replace each face of the dodecahedron by the corresponding "great face", namely the regular pentagon {5} whose vertices are the 5 vertices of one of these pentagrams. This results in the "great dodecahedron", whose Schlafli symbol is {5,5/2} since at each vertex we have five pentagons {5} arranged "pentagrammatically". [These pentagons are the regular pentagons you can see in an icosahedron.] We can now stellate the pentagons of a great dodecahedron to produce what can be called either the "stellated great dodecahedron" or (more traditionally), the "great stellated dodecahedron". Both names are equally appropriate, since this can be produced either by "greatening" the faces of the "stellated dodecahedron" {5/2,3} or by "stellating" the faces of the "great dodecahedron" {5,5/2}. Its Schlafli symbol is {5/2,3}, since 3 pentagrams {5/2} meet at each vertex. The dual of {5/2,3} is a polyhedron {3,5/2}, whose faces are "great triangles" {3} lying in the same planes as the faces of the icosahedron {3,5}. It is therefore naturally called the "great icosahedron". You will see from these systematic descriptions that it's better to omit the word "small" from "small stellated dodecahedron", since this is in fact the only "stellated dodecahedron" per se. The polyhedra {5/2,5} and {5/2,3} were discovered by Kepler, and their duals {5,5/2} and {3,5/2} by Poinsot. Cauchy proved that these four are the only finite regular star polyhedra. In four dimensions, the appropriate names involve THREE operations: stellation : replaces edges by longer edges in same lines greatening : replaces the faces by large ones in same planes aggrandizement : replaces the cells by large ones in same 3-spaces. So for instance there is a "great-grand stellated {5,3,3}". John Conway" ___________________ I hope this gives you something to work with. -Doctor Sarah, The Geometry Forum |
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