Stellating the Octahedron I have a few related questions and I'd be happy to get some help with any one of them.


*

*Is the stellation of a polyhedron generally a 'messy' affair that involves trimming away portions of the enlarged faces? (See below for my implicit assumption, $\text{stellation} \cong \text{dilation},$ that is probably wrong, I see now)


*Are there any polyhedral stellations that come from simply dilating the faces, no trimming necessary?


*Is there a good reference that explicitly defines stellation? I use truncation often but have gotten by with intuition, but this is not the case with stellation.

Context:
I am trying to create a very primitive animation/demonstration that shows the stellation of an octahedron to yield the stella octangula. Unfortunately, it seems that the mental image I have for stellation isn't living up to the real thing. I've always thought that one simply dilates each face, since this is what happens to the star polygons.
Quoting Coxeter (Regular Polytopes, section 6.2 on Polyhedra Stellation), it appears this is not necessarily the case:

In order to stellate a Polyhedron, we have to extend its faces symmetrically until they again form a polyhedron.

At any rate, I started with the stately octahedron, shown here using the software Grapher for OSX (I prefer the faces colored uniformly, but it defaults to checkerboard every time I scale).

Now I start to scale each face about its center; this image shows each face scaled by a factor of $2$. I'm noticing the triangular 'hole' starting to close up in the NorthEast region, for example.

Here it is scaled by a factor of $4$, where the three planes bounding the previous triangular 'hole' have finally met at a point.

And here's a closeup near the three planes meeting at that point, with the 'hole' finally closed.

In the last image I think I can see the stella octangula hiding underneath,

but the there are many portions of the enlarged faces that need to be thrown away.
I only have hands-on experience stellating regular polygons, by extending the edges, to produce star polygons. In that case, it's just a matter of finding the right scale factors for the edges with no trimming required.
My goal was for an expository lecture to non math-majors, but that animation is shaping up to be far too complicated.
 A: Extension is a more general process than dilation. Stellation is often defined as extending the faces or edges of a polyhedron until a new polyhedron is formed. If you extend the edges bounding a regular pentagon until they meet you will obtain a stellated pentagon, sometimes called the pentagram. Dilation is just another form of extension.
Dilating the regular dodecahedron yields the great dodecahedron, which is its second stellation. Dilating the regular icosahedron yields several stellations; the regular compounds of five and ten tetrahedra and also the great icosahedron. Other, less regular polyhedra can also have stellations obtainable via dilation.
However stellation typically creates star faces. The original faces are extended, not trimmed. In the case of the regular octahedron, the Stella Octangula is its only stellation.
There is a process of trimming faces to create star and other polygonal faces inside them. This is called facetting or faceting and is the dual process to stellation. If you stellate a polyhedron, then facetting the dual polyhedron will yield the dual of the stellation. The program Great Stella will display both simultaneously, though it is not available for *nix.
Some references:


*

*Cromwell's Polyhedra (CUP, 1997) has an introductory chapter.

*Coxeter's Regular Polytopes describes stellation from the point of view of regular figures.

*The classic work is Coxeter's co-authored The Fifty-Nine Icosahedra, in which J.C.P. Miller proposed a set of rules to identify distinct stellations.

*That work is less rigorous than is sometimes assumed; a critique may be found in Inchbald, G. "In search of the lost icosahedra", The Mathematical Gazette, Vol 86, No, 506, July 2002, pp.208-215. JSTOR link
The method actually used in The Fifty-Nine Icosahedra differs slightly. The face planes are extended indefinitely, dividing space into many cells. Miller's rules are concerned with which sets of cells form valid stellations. This process does not lend itself to useful dualising.
