define a stellation of a given polyhedron - Maple Help

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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

withgeom3d:

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

stellatei1,icosahedroni,pointo,1,1,1,2,22

i1

(1)

coordinatescenteri1

1,1,1

(2)

formi1

stellated_icosahedron3d

(3)

schlaflii1

stellated3,5

(4)

Plotting:

drawi1,style=patch,orientation=90,145,lightmodel=light4,title=`stellated icosahedron - 22`

See Also

geom3d[Archimedean], geom3d[facet], geom3d[polyhedra], geom3d[QuasiRegularPolyhedron], geom3d[RegularPolyhedron]


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