Dodecahedron Stellations

By Felicia
Thrash

First,
let’s discuss and find out exactly what a stellation is. **Stellation**
is the process of constructing polyhedron by extending the face planes past the
edges until they intersect. For example, if you extend the edges of a polygon,
they intersect to form a star. In the left diagram below, an 11-sided polygon
has its edges extended to form star polygons. Each possible star is denoted by
a different color. Beyond the outermost star, all the lines diverge, and never
intersect again. Likewise, we can extend the faces and edges of a polyhedron.
If we do this to a dodecahedron, as shown at right, the faces become stars and
we obtain the star polyhedron shown. This process is called *stellation.*

__More about Stellations__

You may ask, “Does
every polyhedron have a stellation?” Well let’s find out.

In the figure
below the edges of the cube
and tetrahedron, once extended, never meet. Therefore, these solids have no
stellations.

If the angle between two faces is greater than 90°
there are several layers of bounded cells which may be assembled to build new
polyhedra.

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*Octahedron*

The only stellation of the octahedron is
Kepler's *Stella Octangula*,
which is also a compound of two tetrahedra.

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*Rhombic Dodecahedron*

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*The rhombic dodecahedron has
only 3 stellations.*

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*Rhombic Dodecahedron*

*1 ^{st} Stellation of the
Rhombic Dodecahedron*

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*2 ^{nd} Stellation of the Rhombic
Dodecahedron*

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*3 ^{rd} and Final Stellation of the
Rhombic Dodecahedron*

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*Icosahedron*

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* Here is view of the astounding 59 stellations of the
icosahedron.*

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*Now that you have been
presented with a little background information on stellations, let’s finally
take a look at the stellations of a dodecahedron.*

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*Dodecahedron*

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Here we see a
dodecahedron face (blue) with the intersections of all other faces indicated.
This is a common way to show the possible stellations of a solid. We see that
there are three distinct groups of cells.

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The
three stellations of the dodecahedron are non-convex regular polyhedra
and are shown above.
The first is the *small stellated dodecahedron*. The small stellated
dodecahedron is formed by placing 12 congruent pyramids on the faces of the
dodecahedron. Next is the *great dodecahedron*.**.**
This is obtained by continuing the star planes of the small stellated
dodecahedron outward until they meet to form the next set of pentagons. These continuations form 30 wedges on
the small stellated dodecahedron. If we extend these pentagons, we get the
stellation on the right, the *great stellated dodecahedron. * These extensions shape 20 spikes onto the great dodecahedron
to form the great stellated dodecahedron. Notice that the great stellated
dodecahedron has the same number of vertices and vertex arrangement as the
dodecahedron.

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More about the stellations
of a dodecahedron

v
*History*

v
*Stellations
of dodecahedron in art and culture*

v
*Duality*