Hugh
thank you for your very interesting, stimulating essay. You show an amazing breadth of knowledge in the area. I even saved your essay on my machine for the future reference. I also looked at your site and read all the posts on this blog -- very interesting! I'm giving you a high rate it deserves.
I answered the 2 questions you asked in my blog and here wanted to discuss an aspect you bring up in your essay. It has to do with the S3 hypersphere. I know that topologists call it a 4-sphere, emphasizing the 4-dimensionality of the object as a whole, while mathematicians and physicists call it a 3-sphere, being mainly interested in its 3-dimensional surface. Thus you wrote, "Reimann, Gauss, and Clifford believed the shape of the cosmos was a 3-sphere" (and give an interesting reference [12]). It so happens that I too believe that the universe is a 4-sphere ..lol.. I insist on the topological definition of the same object, and for good reasons! Here is why:
...well, I know that the contest comes to the close and maybe you won't have the time to discuss it now.. and so to keep it short:
1. we live in 4 spatial dimensions, while being aware of only 3. I discussed how this can be explained in my last year essay (did not do too good of a job, I'm afraid).
2. we --well, the nuclei of the atoms that comprise us-- are 4-dimensional objects. (I know a big surprise here, which however is explained in the same model). I found excellent references in your essay for all this, but why call a 4D proton a 'black hole'? (this is a rhetorical question pending my looking up the reference you provided).
Of course, my model is not original (I did not know this at first, but already knew it by the time I wrote the last year essay). The novelty I tried to bring was in explaining these... facts of life? in simple terms and vivid analogies so that anyone can get to see it and so agree with the undoubtedly shocking at the first glance statements 1. and 2.
The top-down model of the universe I tried to convey in my last year essay comes down to a 4-sphere (in topological definition) the 3D surface of which is the 'visible universe', i.e. this 3D surface is what EMR is confined to. This makes this surface akin to a 3D screen from which we get all the info. Just like in the Plato's cave, the real 4D objects (ex. protons) exist in.. 'a large extra dimension' speaking in modern terms.. and only cast 'a shadow' or projection on the 3D screen.
For example, this model explains why a nucleus looks so small -- because it 'sits' in an extra dimension and only touches the 3D surface. There are many other implications of this model, explaining some very intriguing cosmological observations.
I wonder, would it be possible, using your Landscape Test, to *prove* that matter is actually 4-dimensional and that we live in a 4D universe, crawling on its 3D surface?
Also, with your amazing knowledge and expertise in this area, where did you see such an explicit description of such a 4D universe model?
The other *proof* I was looking for is that 4-space is unique among all N-spaces (N>2) in the sense that it has the highest degree of all conceivable symmetries. Because this would serve as yet another rationale why our universe is 4D.
Thank you very much again for inviting me to read your very interesting essay. I understand how difficult it was for you to cut it down to 9 pages, after the first 30-page draft -- and yet to took a risk with the last section. Why?
-Marina