Comments on Visualization

I like it! Where can I get some images of suspension of a glome (he asked naively, as if it were 1994 and Google were not even a wet gleam in someone's cortex)?

-- RonHaleEvans 2009-06-30 07:36 UTC

Turns out Google is not that big a help either. Anyway, is suspension really that useful for visualization? I'm conceiving of (N.B. not imagining) a 5D hypercone with a glome "drawn" on it. That may be coming at it the wrong way around from a reader's perspective. It is pretty interesting, though, and not much bandied about in pop math books, so if you want to include it, please say more.

-- RonHaleEvans 2009-06-30 08:03 UTC

I haven't found any good projections for glome visualization (though some of the partial approaches using latitudes, longitudes, and hyperlongitudes provide some help). If you use the suspension method to go from 1D to 2D you get a diamond-shaped figure. From 2D to 3D you get two cones glued together at their bases. The 5D hypercone would have glomical cross-sections, which is definitely getting into problematical-to-visualize territory. For the suspension method, the 5D hypercone would be a stepping-stone to a 6D hypersphere.

Gluing results in a similar figure, but draws one's focus to the one-to-one correspondence between points on the base of each cone (or hypercone) being glued.

Just wanted to start tossing things out...I'll add some meat and we'll see what looks tasty.

-- Zenoli 2009-06-30 13:44 UTC

Yeah, this is great. I don't mean to dampen your enthusiasm. I was a bad brainstorming partner last night.

In case it's not obvious, my impression was one needs a 5D hypercone to visualize a 4D hypersphere with suspension. Are you saying one only needs a 3D figure to visualize a 4D one? Much better, in that case. Still hard to find the graphics online, unless I overlooked them in my web search last night.

In general, it seems much easier to visualize polytopes than hyperspheres, which doesn't mean the latter isn't worth doing; maybe it's more worth doing.

-- RonHaleEvans 2009-06-30 15:12 UTC

OK, suspension of a circle makes a 3D shape with the topology of a sphere. Suspension of a sphere makes a 4D shape with the topology of a glome. This is interesting. Perhaps we could have a diagram showing the suspension of a circle turning into the suspension of a sphere, much as some texts show a point extending to a line, to a square, to a cube, to a tesseract. On the other hand, I'm not sure burdening the reader with an explanation of why conjoined cones are topologically identical to a sphere, and so on into higher dimensions, will serve our purpose.

I'm probably dwelling on suspensions too much, but they're an operation I wasn't familiar with, and new approaches to explaining this stuff are good, all else being equal.

-- RonHaleEvans 2009-06-30 20:25 UTC

Yes, exactly. The suspensions make a nice progression.

I've been finding glomes particularly interesting lately since they are, as observed, 3-manifolds...that is to say, they are locally Euclidian. This provides some interesting visualization potential as one tries to imagine what it might be like to move about in glomical hypervolume. Tesseracts are more traditional introductions to 4D geometry (and they do make for nifty pictures), but all those overlapping lines can get in the way of other elements of interest.

The suspension and gluing methods are from a paper on the cosmology of Dante's Inferno published in a physics journal...it's referenced in the Wikipedia article about glomes. I chased it down and can send you a copy. It does have a few diagrams of the constructions.

-- Zenoli 2009-07-02 18:12 UTC

I have the Dante paper. It's a good starting place. It has just the kind of progression I was imagining. However, I'm pretty sure we can do better in terms of quality of exposition. It would also be nice to nod to the traditional progression from point to tesseract, even if we focus on glomes.

One nice thing about glomes is that they tie in with the idea of the universe as hypersphere, seeing the back of your head, and other notions from cosmology, which I guess is the kind of moving about you meant.

Incidentally, see the ReadingList for a couple of worthwhile new books on multidimensional geometry.

-- RonHaleEvans 2009-07-04 04:31 UTC