geom3d[stellate]  define a stellation of a given polyhedron

Calling Sequence


stellate(gon, core, n)


Parameters


gon



the name of the stellated polyhedron to be created

core



the core polyhedron

n



nonnegative integer





Description


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The core of a starpolyhedron or compound is the largest convex solid that can be drawn inside it, and the case is the smallest convex solid that can contain it.

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The compound or starpolyhedron may be constructed either by stellating its core, or by faceting its case.

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In order to stellate a polyhedron, one has to extend its faces symmetrically until they again form a polyhedron. To investigate all possibilities, we consider the set of lines in which the plane of a particular face would be cut by all other faces ( sufficiently extended), and try to select regular polygons bounded by sets of these lines.

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Maple currently supports stellation of the five Platonic solids and the two quasiregular polyhedra (the cuboctahedron and the icosidodecahedron).

tetrahedron, cube:

the only lines are the faces itself. Hence, there is only one possible value of n, namely 0.

octahedron:

possible values of n are 0, 1 (the core octahedron and the stella octangula).

dodecahedron:

4 possible values of n: 0 to 3 (the core dodecahedron, the small stellated dodecahedron, the great stellated dodecahedron and the great dodecahedron).

icosahedron:

59 possible values of n: 0 to 58.

cuboctahedron:

5 possible values of n: 0 to 4.

icosidodecahedron:

19 possible values of n: 0 to 18.



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To access the information relating to a stellated polyhedron gon, use the following function calls:

center(gon)

returns the center of the core polyhedron core.

faces(gon)

returns the faces of gon, each face is represented as a list of coordinates of its vertices.

form(gon)

returns the form of gon.

schlafli(gon)

returns the ``Schlafli'' symbol of gon.

vertices(gon)

returns the coordinates of vertices of gon.





Examples


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Define the 22nd stellation of an icosahedron with center (1,1,1) radius 2
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 (1) 
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 (2) 
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 (3) 
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 (4) 
Plotting:
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