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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
non-negative integer
Description
The core of a star-polyhedron 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.
The compound or star-polyhedron may be constructed either by stellating its core, or by faceting its case.
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.
Maple currently supports stellation of the five Platonic solids and the two quasi-regular 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.
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
Define the 22-nd stellation of an icosahedron with center (1,1,1) radius 2
Plotting:
See Also
geom3d[Archimedean], geom3d[polyhedra], geom3d[QuasiRegularPolyhedron], geom3d[RegularPolyhedron]
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