with 12 people, there are 66 relationships

Here's another visual tool, the xy chart:

A regular 3 dimensional polygon with 20 vertices is a dodecahedron.  The shape of a 12-sided die.  I'd like to generate an image of a translucent dodecahedron, with internal lines connecting each vertex, representing the relationships between the 20 vertices.

Then I'd like to animate the growth from 2 vertices (a line) to 3 (a triangle) to 4 (a triangular pyramid) all the way up to 20.  

I feel like this would be a great visual tool for this concept.  

These are translucent hedrons, but they lack the internal "relational" segments:

Hi Paul,

I have not read this entire thread. While I do not disagree with your conclusion, I come up with different math. You see a possibility of 3 relationships among 3 people. I see 6 relation ships ( 3x2x1 = 6).

Let’s say we have John, Paul, and. Mary. I see the possibilities as

John/ Paul
John/Mary
Paul/Mary
John, Paul/ Mary
John, Mary/ Paul
Paul,Mary/ John

Paul does not behave the same with Mary as he does with John or when he is not linked with either.  Of course, this just magnifies the problems you pointed out.  For 12 people (12x11x10x9x8x7x6x5x4x3x2x1 = 479,001,600 relationships.