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.

•

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:
>

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


