Visualizing Calabi-Yau spaces?

[cambraceres]

The entire edifice of the new physics is reposed on an unvisualizable foundation. Quantum Theory is almost completely unvisualizable, relativity is not much better, even though there are plenty of pretty spaceship analogies to "help" one understand the physical law. These analogies, the one about the flatlanders included, are not helpful. What authors are doing when they use these tools is to attempt to elucidate the physical constitution of an entity to a rationality that is not properly equipped to handle it.

This makes it sound as though I say that these entities have physical form, and I am not convinced of this. I can not reconcile a view of these special types of existents that holds them to have intrinsic properties, implying physical form. I believe these extra dimensions are not valid in the same sense the self evident 3 are. I mean to say 3, not 4, there are 3 self evident dimensions. The other dimensions are not defined, because our minds do not have the ability to handle them. We define the proper amount of dimensions, or modes of understanding, or whatever you wish to call it. We define them in terms of our own minds. We can handle 3 dimensions because our minds, based on boolean logic, define the world in these terms. If our pathology were radically different, well then our universe would be too. We are the validation for external forms, they exist in the form we define them. I'm flirting with an open advocacy of the view that reality is observer created.

We do not see entities in terms of higher dimensions because we can't with our current methods and abilities of observation and cognition. If we could recognize them, they would appear, something is said to exist in terms of our own understanding of it.

Cam

[Galwegian]

Here's a little thought experiment that might help. Imagine that you have an ordinary 3d cube made of glass (say it's roughly the size of a standard Rubik cube). Imagine also that the edges of this cube are pinted black. Now, if you hold this cube close to your eye, you will see all the edges through one face of the cube. Try to imagine the pattern of edges that you would see in this case - it can easily be drawn on a sheet of paper foe example.

Now the pictures that you commonly see of a 4d hypercube are what it looks like if you can imagine looking at it through one of its transparent faces. Of course, each face of the 4d hypercube is itself a 3d cube - just as each face of a 3d cube is a square. So as you look thorugh one face, you will see the "back face" as another 3d cube inside the one you are looking through. These two cubes will be connected by edges.

If you think about this for a while it should start to make some sense (at least it does to me - I think!)

[Matt R.]

Wow, that kind of did it for me. It's just very strange because I still can't visualize it all at once. It's more "seeing" what happens when I look at it from different angles. So, I guess I'm not really seeing a full 4D figure -- just what a 4D figure's representation looks like in a 3D world. Sort of like what BluffThis was saying.

I think I understand it better now. OK, moving on: how do you visualize a 6-D Calabi-Yau manifold? [img]/images/graemlins/grin.gif[/img]

[Galwegian]

Ok - I think 6D is very hard to visualise in 3d. I won't say impossible because I'm sure that some people (not me!) do have geometric intuition about these things. My impression is that is 5d is about the highest dimension that anyone can have real geometric insight into. At least in mathematics most of the interesting results in higher dimensions appear to be proven using mostly algebraic rather than geometric methods (as far as I know).

As regards Calabi-Yau manifolds in 6d, I'm not expert enough to offer any specific insight to that situation. I do know that the definition involves some highly technical algebra (Chern classes etc..) so it is probably extremely difficult to get good intuition for these objects. However, that is just the nature of the beast. If it was easy to visulaise 6d objects then we would probably be able to understand the universe a lot better than we do now.

[cambraceres]

Visualization becomes a crutch as complexity increases, eventually, the algebraic definition becomes the only valid elucidation of a mathematical entity defined in higher dimensions. Analogies don't do any real good for you.

Cam

[Galwegian]

On the contrary, analogy becomes crucial when trying to prove results about higer dimensional spaces. Analogy with the appropriate 2d or 3d situation is the guide for the algebra.
In other words, without some geometric intuition, one has no idea where the algebra should go. Of course, finding the corrct analogy is the key.

[Dale Dough]

I gave this some thought a while back.

"Visualizing" higher-dimensional objects really shouldn't be that difficult. It progresses logically from 2D and 3D objects, which are common in our world.

Our brains should be capable of such visualizations easily. Probably, if you were able to somehow trick a baby into 'seeing' 4D using impulses into his brain, he would develop a system to deal with that 'world' easily.

I wish it were possible to do that now, just for fun. We can't create real objects obviously, but I think it would make for wicked computer games.

[chezlaw]

[ QUOTE ]
On the contrary, analogy becomes crucial when trying to prove results about higer dimensional spaces. Analogy with the appropriate 2d or 3d situation is the guide for the algebra.
In other words, without some geometric intuition, one has no idea where the algebra should go. Of course, finding the corrct analogy is the key.

[/ QUOTE ]
I strongly disagree. I have no visualisation ability at all (its fairly common) and struggled a bit at first in 3D compared to some others though I soon caught up. Once we started abstract algebra and higher geometries it was obvious that my lack of visualization skills gave me a head start over others - they soon caught up as well.

Those who visualise well way over-estimate how neccessary it is.

chez

[Galwegian]

I'm guessing from your post chez, that you are an undergrad mathematics student. I will agree that not much geometric intuition is required for a typical first or second course in abstract algebra. However I think that when it comes to looking for new results, there has to be some other intuition to guide the algebra.

Its only my opinion but I think that mathematics in the second half of the 20th century was hampered by an obsession with abstract algebra (category theory being the worst example of this obsession). It is only recently that mathematics has started to return to its geometric roots. People are starting to think again more about geometry - Thurston's/Hamilton/Perelman proof of Poincare, Gromov's work in geometric group theory. I think that abstract algebra is a powerful tool but it is ultimately not much use unless there is some geometric or physical intuition to back it up. Of course that's only my opinion.

[chezlaw]

[ QUOTE ]
I'm guessing from your post chez, that you are an undergrad mathematics student. I will agree that not much geometric intuition is required for a typical first or second course in abstract algebra. However I think that when it comes to looking for new results, there has to be some other intuition to guide the algebra.

Its only my opinion but I think that mathematics in the second half of the 20th century was hampered by an obsession with abstract algebra (category theory being the worst example of this obsession). It is only recently that mathematics has started to return to its geometric roots. People are starting to think again more about geometry - Thurston's/Hamilton/Perelman proof of Poincare, Gromov's work in geometric group theory. I think that abstract algebra is a powerful tool but it is ultimately not much use unless there is some geometric or physical intuition to back it up. Of course that's only my opinion.

[/ QUOTE ]
I did a maths degree. Postgrad I did AI and logic which don't help this debate much.

I also studied physics and found theoretical physics no problem. Couldn't do applied electronics but that's an inability to look at circuit diagrams without nausea rather than a visualization problem.

All my experience tells me visualization whilst very useful is not vital and that those who visualise can't understand how those who dont, think. Intuitions are vital, I'd agree but I have plenty of those.

I used to get into those debates a lot. The smug group at the back (as we were fondly known) found a schism between them and me when discussing many problems. They would talk pictures and describe twisting and turning and I would talk in terms of symmetries, invariances and abstractions. Wierd looks and misunderstandings were only discounted because we all generally ended up in the same place.

I have my mathematical/reasoning limitations but I doubt visualization is an issue. I could be wrong.

chez