geom3d[duality]  define the dual of a given polyhedron

Calling Sequence


duality(dgon, gon, s)


Parameters


dgon



the name of the reciprocal polyhedron to be created

core



the given polyhedron (either a regular solid or a semiregular solid)

s



a sphere which is concentric with the given polyhedron, or a radius of the sphere concentric with the given polyhedron.





Description


•

The edges and vertices of a polyhedron constitute a special case of a graph, which is a set of N0 points or nodes, joined in pairs by N1 segments or branches. Hence, the essential property of a polyhedron is that its faces together form a single unbounded surface. The edges are merely curves drawn on the surface, which come together in sets of three or more at the vertices. In other words, a polyhedron with N2 faces, N1 edges, and N0 vertices may be regarded as a map, i.e., as the partition of an unbounded surface into N2 polygonal regions by means of N1 simple curves joining pairs of N0 points.

•

From a given map, one may derive a second, called the dual map, on the same surface. This second map has N2 vertices, one in the interior of each face of the given map; N1 edges, one crossing each edge of the given map; and N0 faces, one surrounding each vertex of the given map. Corresponding to a pgonal face of the given map, the dual map will have a vertex where p edges (and p faces) come together.

•

Duality is a symmetric relation: a map is the dual of its dual.

•

This process of reciprocation can evidently be applied to any figure which has a recognizable ``center''. It agrees with the topological duality that one defines for maps. The thirteen Archimedean solids hence are included in this case, i.e., for each Archimedean solid, there exists a reciprocal polyhedron.

•

For a given regular solid, its dual is also a regular solid. To access information of the dual of an Archimedean solid, use the following function calls:

center(dgon)

returns the center of dgon.

faces(dgon)

returns the faces of dgon, each face is represented


as a list of coordinates of its vertices.

form(dgon)

returns the form of dgon.

radius(dgon)

returns the midradius of dgon.

schlafli(dgon)

returns the ``Schlafli'' symbol of dgon.

vertices(dgon)

returns the coordinates of vertices of dgon.





Examples


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Define the reciprocal polyhedron of a small stellated dodecahedron with center (0,0,0) radius 1 with respect to its midsphere:
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 (1) 
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 (2) 
Plotting:
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Define the reciprocal polyhedron of a small rhombiicosidodecahedron with center (0,0,0) radius 1 with respect to its midsphere:
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 (3) 
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 (4) 
Plotting:
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