B3D Chapter 3 Summary
I just have a few questions for you regarding chapter 3 that have just been slowly forming in my head over the last couple of days. The first deals with Abbott. As I read about A Square's contact with a sphere from a 3-dimensional world I became curious about this interaction. Was Abbott the first person to conceptualize the interplay of different dimensions? Did anyone before him actually hypothesize about the interaction between 2- and 3-dimensional objects? I was just curious because it seems like such a strange idea to grasp (at least for the time), and I would have thought that someone with a more scientific or mathematical background (here I am talking completely without any knowledge about what type of education Abbott might have actually had) would be more likely to come up with something like this.
[You have, I believe, hit on the most significant contribution Abbott made to the topic of dimensionality. There were some previous writers who challenged readers to imagine the life of a flat creature constrained to live as a shadow on a screen, or on a plane or on the surface of a piece of marble statuary. As far as I know, EAA was the first to concentrate on the interaction among beings from different dimensions. His background included a sort of "Advanced Placement Mathematics" in his secondary school, going through the traditional subject up to and including calculus and differential equations. Mathematics was one of the subjects he would have taken exams in at Cambridge, although he was much more interested in classics and in theology. He had a friend from the school, Howard Candler, who was studying mathematics at Cambridge at the same time and the two of them kept in touch by weekly letters for many years while Candler was mathematics master at the Uppingham School, a hundred miles or so north of London. Candler was interested in theology too, so I suspect that they discussed the dimensional notion when it came up. One possible source of the topic is C. Howard Hinton, the science master at Uppingham, who wrote several articles on the fourth dimension and who wrote his own "An Episode of Flatland" the same year as Abbott wrote his book. Perhaps you would like to find it in the library and see what you think of it?]
Next, I have a basic question about the relationship between slicing a 3-dimensional objcect and the assumptions you can draw with regards to the fourth dimension. With 3-dimensional objects, especially such objects as the torus, can we infer from these objects what possible slicing of hyperobjects will look like? They will obviously be 3-dimensional objects but is there a pattern or are they like something out of an M.C. Escher drawing? What could they tell us about the 4-D object being sliced? This is something that I am fascinated with but am having a hard time actually visualizing.
[Visualization exercises like this take a lot of practice. We'll will get better at it as we imagine what the analogues of a sphere, a cone, or a torus might be in four-space. We started that today in response to Mark's question about the cylinder, and we'll come back to it, in class and/or in the discussions.]
The book mentioned one woman, Alicia Boole Stott, who could visualize slicing sequences of four-dimensional polyhedra. I am curious about her and what she accomplished (especially since she was without any type of training).
[I wonder if there is something on the web somewhere about her. On page 258 of the book "Regular Polytopes" by H. S. M. Coxeter there is a two-page mention of her and her contributions. I'll bet there are other places where she is treated but I don't know of any offhand. Let me know if you have trouble finding the Coxeter reference in the SciLi.]