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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. The star-polyhedron or compound may be constructed by stellating its core. Note that it can also be constructed by faceting its case. See the geom3d:-facet command for more information.
 • 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

 > $\mathrm{with}\left(\mathrm{geom3d}\right):$

Define the 22-nd stellation of an icosahedron with center (1,1,1) radius 2

 > $\mathrm{stellate}\left(\mathrm{i1},\mathrm{icosahedron}\left(i,\mathrm{point}\left(o,1,1,1\right),2\right),22\right)$
 ${\mathrm{i1}}$ (1)
 > $\mathrm{coordinates}\left(\mathrm{center}\left(\mathrm{i1}\right)\right)$
 $\left[{1}{,}{1}{,}{1}\right]$ (2)
 > $\mathrm{form}\left(\mathrm{i1}\right)$
 ${\mathrm{stellated_icosahedron3d}}$ (3)
 > $\mathrm{schlafli}\left(\mathrm{i1}\right)$
 ${\mathrm{stellated}}{}\left(\left[{3}{,}{5}\right]\right)$ (4)

Plotting:

 > $\mathrm{draw}\left(\mathrm{i1},\mathrm{style}=\mathrm{patch},\mathrm{orientation}=\left[-90,145\right],\mathrm{lightmodel}=\mathrm{light4},\mathrm{title}=\mathrm{stellated icosahedron - 22}\right)$