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geom3d

 duality
 define the dual of a given polyhedron

 Calling Sequence duality(dgon, core, s)

Parameters

 dgon - the name of the reciprocal polyhedron to be created core - the given polyhedron (either a regular solid or a semi-regular 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 p-gonal 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.
 • Regular map: a map is said to be regular, of type $\left[p,q\right]$, if there are p vertices and p edges for each face, q edges and q faces at each vertex, arranged symmetrically in a sense that can be made precise. Thus a regular polyhedron is a special case of a regular map. For each map of type $\left[p,q\right]$, there is a dual map of type $\left[q,p\right]$.
 • Consider the regular polyhedron $\left[p,q\right]$, with its N0 vertices, N1 edges, N2 faces. If we replace each edge by a perpendicular line touching the mid-sphere at the same point, we obtain the N1 edges of the reciprocal polyhedron $\left[q,p\right]$, which has N2 vertices and N0 faces. This process is, in fact, reciprocation with respect to the mid-sphere: the vertices and face-planes of $\left[p,q\right]$ are the poles and polars of the face-planes and vertices of $\left[q,p\right]$. Reciprocation with respect to another concentric sphere would yield a larger or smaller $\left[q,p\right]$.
 • 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 mid-radius of dgon. schlafli(dgon) returns the Schlafli symbol of dgon. vertices(dgon) returns the coordinates of vertices of dgon.

Examples

 > $\mathrm{with}\left(\mathrm{geom3d}\right):$

Define the reciprocal polyhedron of a small stellated dodecahedron with center (0,0,0) radius 1 with respect to its mid-sphere:

 > $\mathrm{SmallStellatedDodecahedron}\left(\mathrm{p6},\mathrm{point}\left(o,0,0,0\right),1.\right)$
 ${\mathrm{p6}}$ (1)
 > $\mathrm{duality}\left(\mathrm{dp6},\mathrm{p6},\mathrm{sphere}\left(\mathrm{s6},\left[o,\mathrm{MidRadius}\left(\mathrm{p6}\right)\right]\right)\right)$
 ${\mathrm{dp6}}$ (2)

Plotting:

 > $\mathrm{draw}\left(\left[\mathrm{p6}\left(\mathrm{color}=\mathrm{red}\right),\mathrm{dp6}\left(\mathrm{color}=\mathrm{green}\right)\right],\mathrm{cutout}=\frac{7}{8},\mathrm{lightmodel}=\mathrm{light4},\mathrm{title}=\mathrm{dual of small stellated dodecahedron},\mathrm{orientation}=\left[0,32\right]\right)$

Define the reciprocal polyhedron of a small rhombiicosidodecahedron with center (0,0,0) radius 1 with respect to its mid-sphere:

 > $\mathrm{SmallRhombiicosidodecahedron}\left(\mathrm{t7},o,1.\right)$
 ${\mathrm{t7}}$ (3)
 > $\mathrm{duality}\left(\mathrm{dt7},\mathrm{t7},\mathrm{sphere}\left(\mathrm{m7},\left[o,\mathrm{MidRadius}\left(\mathrm{t7}\right)\right]\right)\right)$
 ${\mathrm{dt7}}$ (4)

Plotting:

 > $\mathrm{draw}\left(\left[\mathrm{t7}\left(\mathrm{color}=\mathrm{red}\right),\mathrm{dt7}\left(\mathrm{color}=\mathrm{green}\right)\right],\mathrm{cutout}=\frac{7}{8},\mathrm{lightmodel}=\mathrm{light4},\mathrm{title}=\mathrm{dual of small rhombiicosidodecahedron},\mathrm{orientation}=\left[0,32\right]\right)$