What Does "Stellated" Mean?

Date: 03/31/98 at 20:27:34
From: Tornado Team Origami Club
Subject: Definition of "stellated"

Our teacher taught us how to fold 12 squares to make a stellated 
octahedron, and we are now working on a stellated icosahedron. We 
understand the octahedron and the icosahedron, but don't know what 
stellated means. We have looked in our math book, the pre-algebra 
book, the dictionary in our room, the math resource book that belongs 
to our teacher, and have asked our parents, but have not found the 
answer.

Date: 04/01/98 at 05:37:22
From: Doctor Pete
Subject: Re: Definition of 

Hi,

"Stellate," the way you are using the word, is a verb which means "to 
make or form into a star." It is also an adjective meaning "star-like, 
or having a shape of a star." Mathematically speaking, stellation is a 
process which is performed on the faces of solid figures or polyhedra, 
for example, an octahedron or dodecahedron, to make larger, more 
complex solids.

The details are as follows: Take your stellated octahedron. Now, 
imagine an octahedron (or, if you have a paper model of one, that'd be 
even better). It is made of 8 equilateral triangular faces. Consider 
the three faces which "surround" a given face. If you imagine 
extending these three faces along the planes in which they lie, you 
can see that they "cover up" this central face, coming together until 
they form a little pyramid on top, a regular tetrahedron sitting atop 
this central face. So these three faces are now joined by their edges, 
and share a common vertex which is not part of the original 
octahedron. Now, imagine doing this to *all* of the 8 faces, 
essentially building up regular tetrahedra on top of an octahedron. 
You will obtain the stellated octahedron. The symmetrical extension of 
faces along the planes they lie in, so as to form a new closed solid, 
is the process of stellation.

Now, some questions you may want to ask: can you stellate a cube? A 
tetrahedron? What would a stellated dodecahedron or icosahedron look 
like? Can you stellate a stellated solid?

I'd like to write a bit more about this last question. If you took a 
dodecahedron and stellated it, you'd get something called the "small 
stellated dodecahedron." It is composed of 12 faces, each of which is 
a regular pentagram (five-pointed star). Note that although the faces 
extend through each other, I still consider them one whole face, not 
several separate faces. So for the stellated octahedron, there are 
still 8 faces, each of which is a regular triangle, but they pass 
through each other. So we can stellate the dodecahedron further, to 
obtain the "great dodecahedron," which consists of 12 pentagonal 
faces, penetrating each other. This we do by "filling in" the spaces 
between the pentagram points. Finally, we stellate one more time, to 
obtain the "large stellated dodecahedron." Here, we have again 12 
pentagrammic faces, but this time they meet 3 at a vertex, instead of 
5, as in the small stellation. This completes the possible ways of 
stellating a dodecahedron (3). In the icosahedron, there are 59 
stellations of the icosahedron!

If you want to learn more about stellated polyhedra, try to find the 
book "Polyhedron Models" by Magnus J. Wenninger. Another book, by Alan 
Holden, called "Shapes, Space, and Symmetry" (I think) is an excellent 
source as well.

-Doctor Pete, The Math Forum
Check out our web site! http://mathforum.org/dr.math/