geom3d

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


•

The core of a starpolyhedron or compound is the largest convex solid that can be drawn inside it. The starpolyhedron 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 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.



•

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 22nd stellation of an icosahedron with center (1,1,1) radius 2
>

$\mathrm{stellate}\left(\mathrm{i1}\,\mathrm{icosahedron}\left(i\,\mathrm{point}\left(o\,1\,1\,1\right)\,2\right)\,22\right)$

>

$\mathrm{coordinates}\left(\mathrm{center}\left(\mathrm{i1}\right)\right)$

$\left[{1}{\,}{1}{\,}{1}\right]$
 (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:
>

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







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