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### Stellated Dodecahedron

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Date: 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.
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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:

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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Associated Topics:
High School Geometry
High School Polyhedra

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