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geom3d

  

duality

  

define the dual of a given polyhedron

 

Calling Sequence

Parameters

Description

Examples

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 , 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 , there is a dual map of type .

• 

Consider the regular polyhedron , 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 , 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  are the poles and polars of the face-planes and vertices of . Reciprocation with respect to another concentric sphere would yield a larger or smaller .

• 

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

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

(1)

(2)

Plotting:

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

(3)

(4)

Plotting:

See Also

geom3d[Archimedean]

geom3d[polar]

geom3d[pole]

geom3d[RegularPolyhedron]

 


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