From:zare@cco.caltech.edu (Douglas J. Zare)Newsgroups:sci.physics,sci.mathSubject:Re: Physics Thesis: True or False?Date:22 May 1996 01:54:28 GMTOrganization:California Institute of Technology, Pasadena

Followups set to sci.math . Earle D. Jones <ejones@hooked.net> wrote: >[...] >Our natural domain is in 2D. We stretch to visualize well in 3D. >[...] >So--when someone tells me that they can visualize in 4D, I become very >polite and I smile and nod. >[...] It is a skill to visualize well in 3D. Similarly, it is a skill to visualize 4D objects. As I am trying to improve both, and feel I have had some success visualizing certain 3-manifolds immersed in C^2 (I'm working on seeing the CP^2 structure), I am a bit less skeptical. I recommend the following articles and threads from geometry.college though some take the other point of view: http://www.forum.swarthmore.edu/news.archives/geometry.college/article120.html http://forum.swarthmore.edu/~sarah/HTMLthreads/articletocs/4d.visualization.html http://forum.swarthmore.edu/~sarah/HTMLthreads/articletocs/viewing.4d.objects.in.3d.html http://www.forum.swarthmore.edu/news.archives/geometry.college/article211.html Those are mainly about 4D. Coxeter mentioned that in the course of a long walk, he visualized a chain of polytopes, each the vertex figure of the next, starting with a triangular prism and going to the 9th dimension. This chain can't be extended, and he claims to have seen that, too! (The sequence is important in other contexts, and Coxeter was not the first to discover it. Unfortunately, I can't remember its name.) 4D seems within reason, but I would like to know how to develop direct geometric intuition about, say, the E8 or Leech lattices. Douglas Zare http://www.cco.caltech.edu/~zare