ellis@flairvax.UUCP (Michael Ellis) (06/29/84)
One frequently encounters the notion of `duality' or `reciprocity' in texts
dealing with polyhedra, especially when they are highly regular, as the
platonic and semiregular solids are.
The five platonic solids, for instance, can be grouped into three categories
based on duality as below:
FACES VERTICES EDGES DUAL SYMBOL
(1)tetrahedron 4 6 4 tetrahedron {3,3}
(2)octahedron 8 12 6 cube {4,3}
cube 6 12 8 octahedron {3,4}
(3)icosahedron 20 30 12 dodecahedron {5,3}
dodecahedron 12 30 20 icosahedron {3,5}
Note that the dual object simply switches the numbers of objects in the
tables around. The same notion applies at higher dimensions as well. For
instance, in four dimensions, the platonic equivalents are:
CELLS FACES EDGES VERTICES DUAL SYMBOL
(1)pentatope 5 10 10 5 pentatope {3,3,3}
(2)8-cell 16 32 24 8 tessaract {4,3,3}
tessaract 8 24 32 16 8-cell {3,3,4}
(3)24-cell 24 96 96 24 24-cell {3,4,3}
(4)600-cell 600 1200 720 120 120-cell {5,3,3}
120-cell 120 720 1200 600 600-cell {3,3,5}
Topologically, duality appears to have a clearcut meaning. However, when
one is concerned with geometrical properties (what is the dihedral
angle? what is the ratio of the volume of the dual with the original?)
things are not very clear at all..
Within the scope of regular polytopes, a geometric duality operator can be
satisfactorily defined by the steps below (easily extended to higher
dimensions):
1. Find the sphere that goes thru all the object's vertices
2. Project the centers of all faces (or cells) onto the sphere
3. Connect the new vertices with edges whose projections are
perpendicular with the edges of the original object.
This definition is useless for less regular objects. For example, a very
regular object (called the cuboctahedron), which can be constructed from a
cube (or octahedron) by connecting the midpoints of adjacent edges with new
edges and discarding the original object, will have a dual whose vertices
are not even coplanar, using the above definition of duality!
`Fortunately', the algorithm below (also extendable to higher dimensions):
1. Find the sphere that is tangent to all of the object's edges
2. At each point of tangency, construct a perpendicular (in the
plane tangent to the sphere at that point). Extend the new
edge until it meets another..
..produces `nice' duals, by which I mean:
1. The appropiate edges indeed converge to form vertices
2. The appropriate new vertices are indeed coplanar
3. The duality operator applied on the dual object results
in the original object, EXACTLY (no change in size, orientation..)
..for a restricted set of objects, namely the semiregular solids.
The semiregular polyhedra are composed of regular polygons, of possibly
different kinds, with identical vertices -- like the cuboctahedron, for
instance, with two triangle and two squares at each vertex.
Their duals have identical faces that may be irregular; the dual vertices do
not all necessarily lie on a sphere. Though the duals appear to be `ugly' at
first sight, they frequently have `beautiful' properties. For instance,
applying the above definition of duality to the cuboctahedron, the rhombic
dodecahedron, which packs Euclidean 3-space (and is consequently popular
among crystal structures) is produced.
I have never encountered a satisfactory definition for any arbitrary
polyhedron that produces `nice' duals. Has anyone else? Does/can such a
general geometric duality operator actually exist?
-michaelellis@flairvax.UUCP (Michael Ellis) (06/29/84)
As usual, there were a number of blatant errors in the previous article. Please replace the line: FACES VERTICES EDGES DUAL SYMBOL with FACES EDGES VERTICES DUAL SYMBOL and all references to `8-cell' with `16-cell'. No doubt other gross inaccuracies remain. -michael
little@ubc-vision.CDN (Jim Little) (07/04/84)
Duality is well defined for irregular polyhedra. For a good discussion,
see "Convex Polytopes" by Branko Gruenbaum, John Wiley & Sons, Ltd., 1967,
pp. 46-48. The text is available from their London office. A polytope is
a bounded convex polyhedron.
A polytope can be defined as the intersection of a set of halfspaces;
each half space is given by the equation:
Ax + By + Cz <= 1
For a polytope composed of n planes (the primal polytope), containing the
origin in its interior, one can construct its geometric dual as follows:
1) each plane with equation
Ai x + Bi y + Ci z = 1
is transformed into a point whose coordinates are (Ai,Bi,Ci).
2) each point (s,t,u) is taken into the plane with equation
s x + t y + u z = 1
This transform takes planes into points and takes points into planes.
Consider a vertex p=(s,t,u) of the primal polytope,
lying on a set of faces of the primal. For each face j it is the case that
Aj s + Bj t + Cj u = 1
From this it is clear that in the dual the plane corresponding to the vertex
(s,t,u) contains the point (Aj, Bj, Cj) if and only if the plane (Aj, Bj, Cj)
contains the vertex in the primal.
The condition that the origin does not lie on any of the faces of the
primal polytope is necessary so that the plane equations are well defined.
This transformation can be extended in a straightforward fashion to
d>3 dimensions; the discussion is in terms of R3 only for clarity.
Jim Little
ubc-vision
University of British Columbia