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What Does "Stellated" Mean?

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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
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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/
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Associated Topics:
High School Definitions
High School Geometry
High School Polyhedra

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