Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.



Re: Polyhedra
Posted:
Oct 11, 1995 9:11 AM


I think some people may be interested in understanding the names of the KeplerPoinsot 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 starpolygon 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 3spaces.
So for instance there is a "greatgrand stellated {5,3,3}".
John Conway



