It was partially in response to your first question that I talked Wednesday about the nature of the collaboration with Julie Strandberg. I hope we will be able to work again together, on this piece or on something new.
Koçak and Bisshopp were indeed impressed. We wrote a long paper on the subject, one that was very well received by the applied math community.
With respect to manifolds, I of course think of them as curves and surfaces generalized, and I'm not sure I have ever formed an idea of a higher-dimensional aggregate that does not come back to familiar geometric experience. On the other hand, I do sometimes deal with configuration spaces that are definitely high-dimensional. I just don't see them as well.
I very much like parallel objects as a way of introducing so many of the curvature concepts we dealt with in Math 106 last semester. I'm not sure how easy it is for most viewers to take the three-dimensional wave fronts into the next dimension, to "stack them up" in a usually inaccessible direction.
I do consider myself fortunate to have been in the right place (Brown) at the right time to get in on the ground floor in the computer imaging business. I get to be considered a pioneer, a status I never anticipated. It's quite exciting to see how things keep changing, and having a long time association with the effort does give a valuable perspective.
Good discussion questions. Perhaps you can put in some links for the benefit of some hypothetical teacher who might want to use such ideas in a class but who is a bit reluctant to lead when she or he doesn't have that much of a notion of where things might lead?