geom3d[duality] - define the dual of a given polyhedron
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Calling Sequence
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duality(dgon, gon, s)
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Parameters
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dgon
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the name of the reciprocal polyhedron to be created
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core
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the given polyhedron (either a regular solid or a semi-regular solid)
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s
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a sphere which is concentric with the given polyhedron, or a radius of the sphere concentric with the given polyhedron.
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Description
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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.
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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 p-gonal face of the given map, the dual map will have a vertex where p edges (and p faces) come together.
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Duality is a symmetric relation: a map is the dual of its dual.
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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.
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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:
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center(dgon)
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returns the center of dgon.
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faces(dgon)
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returns the faces of dgon, each face is represented
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as a list of coordinates of its vertices.
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form(dgon)
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returns the form of dgon.
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radius(dgon)
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returns the mid-radius of dgon.
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schlafli(dgon)
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returns the ``Schlafli'' symbol of dgon.
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vertices(dgon)
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returns the coordinates of vertices of dgon.
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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 mid-sphere:
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Plotting:
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Define the reciprocal polyhedron of a small rhombiicosidodecahedron with center (0,0,0) radius 1 with respect to its mid-sphere:
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Plotting:
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