
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 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.

•

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 midsphere 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 midsphere: the vertices and faceplanes of $\left[p\,q\right]$ are the poles and polars of the faceplanes 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 midradius 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 midsphere:
>

$\mathrm{SmallStellatedDodecahedron}\left(\mathrm{p6}\,\mathrm{point}\left(o\,0\,0\,0\right)\,1.\right)$

>

$\mathrm{duality}\left(\mathrm{dp6}\,\mathrm{p6}\,\mathrm{sphere}\left(\mathrm{s6}\,\left[o\,\mathrm{MidRadius}\left(\mathrm{p6}\right)\right]\right)\right)$

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 midsphere:
>

$\mathrm{SmallRhombiicosidodecahedron}\left(\mathrm{t7}\,o\,1.\right)$

>

$\mathrm{duality}\left(\mathrm{dt7}\,\mathrm{t7}\,\mathrm{sphere}\left(\mathrm{m7}\,\left[o\,\mathrm{MidRadius}\left(\mathrm{t7}\right)\right]\right)\right)$

Plotting:
>

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



