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From: ellis@flairvax.UUCP (Michael Ellis)
Newsgroups: net.math
Subject: Geometric Duality
Message-ID: <606@flairvax.UUCP>
Date: Fri, 29-Jun-84 04:43:01 EDT
Article-I.D.: flairvax.606
Posted: Fri Jun 29 04:43:01 1984
Date-Received: Sun, 1-Jul-84 04:53:10 EDT
Organization: Fairchild AI Lab, Palo Alto, CA
Lines: 78

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?

-michael