```From:           zare@cco.caltech.edu (Douglas J. Zare)
Newsgroups:     sci.physics,sci.math
Subject:        Re: Physics Thesis:  True or False?
Date:           22 May 1996 01:54:28 GMT
Organization:   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
```