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Topic: Small and great stellated dodecahedra
Replies: 4   Last Post: Oct 11, 1995 9:11 AM

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John Conway

Posts: 2,238
Registered: 12/3/04
Re: Polyhedra
Posted: Oct 11, 1995 9:11 AM
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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}.
It's 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

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