Visualizing Calabi-Yau spaces?

[Matt R.]

OK, so essentally Calabi-Yau spaces are multi-dimensional manifolds. There is a very precise mathematical definition, but I don't understand it so I won't bother mentioning it.

In string theory, it is hypothesized that there are extra spatial dimensions that take the form of a 6-dimensional calabi-yau manifold. I think sometimes (in M-theory?) it is 7-dimensional. But I'm not exactly sure.

What I am wondering, is how do you visualize such a structure? The 2-D and 3-D renditions are very unsatisfying, because well, they are in 3-dimensions -- not 6 or 7. They look like a giant blob with ridges on it (wikipedia has a picture).

I understand that you are simply saying spacetime at its most fundamental level is described using 11 (10, 9, whatever) coordinates instead of 4. Which means that these extra dimensions we don't see are simply extra coordinates. Is it *possible* to visualize what a 10-dimensional structure would look like in a concrete way? Any way I can think of can still adequately be described in 3-dimensions (like the 3-D rendition on wikipedia). Is the "correct" way to visualize such a structure simply to realize that it is described using extra coordinates that are unseen, and thus cannot be "seen" in traditional ways (for instance, you can't "see" the fourth dimension in space-time, but you know it's there intuitively).

The more I think about it, the more I believe that these calabi-yau spaces are simply mathematical techniques to understand string theory. They don't really physically "exist" in the sense where we are even capable of visualizing them correctly.

Anyone have a unique way of picturing what a 6-dimensional structure would look like (that can't be described in a 3-D coordinate system)? Maybe I just haven't opened my mind up yet to how higher-dimensional objects "appear".

[ChromePony]

I'm really not sure that we are capable of visualizing in more than 3 dimensions, everything we come up with is some sort of 3-D extension or fabrication. These shapes are certainly primarily mathematical in nature, but just because we cannot see them does not mean they do not exist in reality. There are plenty of examples of things being discovered in science before they could actually be 'seen' quarks come to mind as one example. I guess its all a matter of how you want to look at it, but I think its not a matter of opening your mind so much as it is a basic human inability to see beyond 3-D...at least for now.

[Ben Young]

I would think it would be impossible to visualize this without understanding the precise mathematical definition.

[Matt R.]

Ben,
There is a precise mathematical definition for a 4, 5, 6 dimensional circle or sphere that I understand. I still can't visualize it though. The definition for a calabi-yau shape is really esoteric... at least the wikipedia entry is. I'm more interested in visualization "techniques" for higher dimensional structures, rather than specifically a calabi-yau manifold.

Chrome,
Yes, I agree that they do "exist", just maybe not a visualizable (is that a word? it is now!) sense. You can expand the size of a quark in your mind, for example, to picture what it may look like if you were to see a very large one with your own eyes. But what would you see if you expanded a 6 dimensional object? Can a 6-dimensional object even exist beyond a mathematical abstraction? Like you said, I think the answer is yes, but we certainly can't experience it with our eyes. I just really want to be able to picture the abstraction I guess, and it is bothering me that I cannot. The analogies to a 1-D world (like in the book Flatland, which I think are the same ones used in Elegant Universe) aren't helping me much, because it is easy to "see" a 2nd dimension expand to create a 2-D world as we live in a 3-D world. Going from 3-D to 4-D just isn't happening in my head. The only analogy I can come up with is verbal... instead of an up/down and left/right (edit -- ok, I just verbally created a 2-D world, oops. But you get the idea), there is up/down, left/right, and bloop/blarp or whatever name you want to come up with for the new back/forth directions. But even then, all the objects within the universe would then be 4-D instead of 3-D, and of course I cannot see this either.

I guess even a 1-D world is difficult to picture beyond a theoretical sense -- after all there is no such thing as an infinitely thin line in our 3-D world. At least this is more concrete though. I guess I'll just chalk up the 6-D shape as something that is impossible to see.

[baumer]

Visualizing anything more than three dimensions is pretty tough.

Imagine an X-axis, then imagine the Y-axis which intercepts the X-axis at a right angle.
Then imagine the Z-axis intercepting both the X and Y at a right angle.
The 4th spatial dimension would, of course, be measured on the "wtf-axis" which is at right angles to the X,Y, and Z axes.

I just can't seem to find out where that line is!

[BluffTHIS!]

[ QUOTE ]
I'm really not sure that we are capable of visualizing in more than 3 dimensions, everything we come up with is some sort of 3-D extension or fabrication. These shapes are certainly primarily mathematical in nature, but just because we cannot see them does not mean they do not exist in reality. There are plenty of examples of things being discovered in science before they could actually be 'seen' quarks come to mind as one example.

[/ QUOTE ]


The difference with quarks is that they are 3-D objects themselves in our 3-D world. So the real question here in addition to visualizing an extra-dimensional object if same does in fact exist, is that will such dimensions past the 4th dimension commonly taken to be time, have 3-D effects which can be observed and recognized as same. That is, can we ever prove an observed effect to be that of an extradimensional physical reality? If not, then our 3-D world is Flatland where we either confuse some such extra-dimensional effects with other 3-D ones, or just can't ever explain them properly.

[Metric]

I don't think anyone realistically tries to visualize what they look like. Higher dimensional manifolds (usually in the form of Lie groups) have been used in theoretical physics for a long time, but I've never ever seen an appeal to intuition with regard to these things -- the reasoning is always about very defininte algebraic properties.

[SNOWBALL]

FWIW, Brian Greene says he can't visualize them either.

[Galwegian]

It is possible to visualise some 4 (and even 5) dimensional objects - there are vatious techniques used in mathematics for this.

An example, nobody disputes that is is possible to generate a convincing 2 dimensional image of a 3d object. Painters do this all the time. You can use the same idea to visulaise 4d objects. You imagine what it would look like if you drew a 3D image of it. In mathematics such pictures are sometimes called Schlegel diagrams. You often see drawings of 4d hypercubes in pop science books. This is an example of a Schlegel diagram.

[Matt R.]

I just searched for a rotating 4D hypercube on google. It looks [censored] crazy, and I'm not quite sure what I'm looking at yet.