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geom3d[stellate] - define a stellation of a given polyhedron
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Calling Sequence
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stellate(gon, core, n)
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Parameters
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gon
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the name of the stellated polyhedron to be created
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core
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the core polyhedron
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n
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non-negative integer
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Description
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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.
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The compound or star-polyhedron 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 quasi-regular polyhedra (the cuboctahedron and the icosidodecahedron).
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tetrahedron, cube:
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the only lines are the faces itself. Hence, there is only one possible value of n, namely 0.
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octahedron:
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possible values of n are 0, 1 (the core octahedron and the stella octangula).
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dodecahedron:
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4 possible values of n: 0 to 3 (the core dodecahedron, the small stellated dodecahedron, the great stellated dodecahedron and the great dodecahedron).
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icosahedron:
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59 possible values of n: 0 to 58.
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cuboctahedron:
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5 possible values of n: 0 to 4.
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icosidodecahedron:
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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:
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center(gon)
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returns the center of the core polyhedron core.
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faces(gon)
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returns the faces of gon, each face is represented as a list of coordinates of its vertices.
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form(gon)
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returns the form of gon.
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schlafli(gon)
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returns the ``Schlafli'' symbol of gon.
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vertices(gon)
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returns the coordinates of vertices of gon.
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Examples
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>
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Define the 22-nd stellation of an icosahedron with center (1,1,1) radius 2
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![[1, 1, 1]](/support/helpjp/helpview.aspx?si=5964/file04034/math206.png)
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| (3) |
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![stellated([3, 5])](/support/helpjp/helpview.aspx?si=5964/file04034/math220.png)
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Plotting:
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![draw(i1, style = patch, orientation = [-90, 145], lightmodel = light4, title = `stellated icosahedron - 22`)](/support/helpjp/helpview.aspx?si=5964/file04034/math226.png)
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