B3D: Chapter 6

Alexis Nogelo


While reading the sixth chapter of B3D, a lot of things seemed to click for me. I found it to be a very substantial chapter, with ideas presented in such a way that they truly make sense, probably because all of the visualizing that we have attempted up to this point is sinking in as well.

First of all, I must say that the picture of Professor Banchoff with Salvador Dali is incredibly cool. It gives me such pleasure to imagine the two of them discussing perspective and art and mathematics and hypercubes and fold-outs--it is just a perfect way in which I can consider the link that we have been trying to make in class between art and Mathematics. In fact, this section on perspective did a lot of good work in that direction, as well as helping me comprehend more fully how the more thoroughly we understand our 3-dimensional viewpoints, the easier it becomes to imagine 4-dimensional objects.

I was happy that the description of the the film The Hypercube: Projections and Slicing was included because it helps me to remember what the rotating hypercube looks like. I am not sure that the description would make very much sense had I not actually seen the film, however. I do wonder if Eadweard Muybridge knew that his animation of himself walking would be preserved historically or if he was just messing around with his new equipment. Did he publish the film in any way? His nudity in the film seems artistically done, but perhaps he never intended for us to see the pictures. It just raises questions that always arise when the public has access to unfinished books or personal writings of any kind--it just occurred to me while looking at this early example of animation.

The chapter really got exciting in 'The Polyhedral Torus in the Hypercube' and the Stereographic Projection sections. The Clifford Torus is amazing. I do not quite know what I want to say about it, but I love the pictures of it and I do feel like I understand it when I study the images--I think I am on that brink of understanding something about it, but without any sense of full grasp, if that makes any sense at all. I had looked at these pages when I first got B3D and recognized the cover picture. At that time, however, I had very little tools with which to figure out what is happening. This is a great example of how my thinking has advanced during this semester--it seems almost imperceptible as it happens, but occasionally one notices the jumps. I do not know if I am able to relate this using English, but the change that I can perceive in my own mind is both substantial and enjoyable.

Topics for Discussion:

6.1 The development of the Clifford torus from the Polyhedral torus. Also, the vertices on a hypercube as resting on the hypersphere.

6.2 The central projection of the Clifford torus--why does it look like the familiar torus of revolution?

6.3 The cyclides of Dupin. What is going on with the longitude/latitude switch? Is it the same as the right/left glove switch that would occur if a glove were placed inside of one cubical face of a rotating hypercube?

Prof. Banchoff's response
Alexis Nogelo