B3D Chapter 5 Summary

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In chapter 5, many different topics are covered, but there are two main items that I would like to discuss.  While reading the chapter, the beginnings of the chapter helped to pave a smooth path towards some of the more abstract material in the latter portions of the chapter.  And it is these latter portions which I would like to take a look at.

First, I would like to examine a little more closely, the folding hypercube.  Alright, with the composition of a similar object in 2-D would be a wire square.  Would there be any unique properties to, say, a wire hypercube?  In general, would there be any new properties to any similar object that, instead of being represented in the (n-1)-dimension, it was placed in the (n-2)-dimension?

Also, I would like to know a little more about duality.  Why is this the first time I have ever seen this term used in mathematics?  It seems like such an obvious link yet, I can't say that I remember ever hearing of it in any type of mathematical sense.  I was interested in the uses of duality with respect to n-dimensions.  While the duals of regular polyhedra in 3-D is easily portrayed, as the dimension increases, what kind of math is used to solve for these duals?  Who discovered this principle of duality for polyhedra?