Visualizing Calabi-Yau spaces?

[Mickey Brausch]

Can you visualize one-dimensional or two-dimensional spaces ?

Mickey Brausch

[pzhon]

[ QUOTE ]

"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.

[/ QUOTE ]
First, the idea that it is easy to visualize in 3D is a common misconception. People usually try to visualize only extremely simple objects in 3D.

Here is a more complicated example. Consider the rectagle with vertices at (0,+-1,+-2). Consider rotations about the x=y=z line.
a) Take two images of the rectangle under rotations. Can they intersect without coinciding?
b) Take two images that do not intersect. Can they be linked?
c) Take 3 images that do not intersect. Can they be pulled apart without passing through each other?

Here is another example.
a) Take 3 mutually skew lines in R^3. How many lines pass through all 3?
b) Take 4 mutually skew lines in R^3. How many lines pass through all 4?

Second, the same techniques most people use to go from 2D information to 3D do not extend to higher dimensions.

There are several "topographic maps" within the visual cortex. These are physical 2D arrays in the brain. They process 2D information such as detecting edges. No analogous 3D topographic maps exist. Your brain is not hardwired to handle complicated 3D information.

By the way, understanding the geometry of R^3 is very different from understanding 3-manifolds, just as being able to understand a piece of paper does not tell you the basics about a Klein bottle. Exercise: How many essentially different simple (non-self-intersecting) loops are there on a Klein bottle?

Once you accept the difficulty of understanding R^3, you may find it easier to understand R^n. Visualization is much more difficult. Very few geometers are able to visualize much in higher dimensions.

[Galwegian]

nice post pzhon

[ QUOTE ]
First, the idea that it is easy to visualize in 3D is a common misconception. People usually try to visualize only extremely simple objects in 3D.

[/ QUOTE ]

I agree totally. I see this a lot with my students, even final year undergrad students. They have not developed their 3d visualisation skills at all. I speculate that one reason why students nowadays have such poor visualisation skills is that they have been made lazy by modern technology (TV, computer graphics) etc. They are used to being presented with information in a 2d way and so they don't practise their 3d visualisation. Perhaps this problem will go away as computer games start to use more realistic graphics - I don't know (only speculating).

I like your problems (I haven't thought about them all yet)

[ QUOTE ]
a) Take 3 mutually skew lines in R^3. How many lines pass through all 3?

[/ QUOTE ]
I think the answer is infinitely many - I suspect that there is a one dimensional space of such lines, ignoring possible singularities.

[ QUOTE ]
b) Take 4 mutually skew lines in R^3. How many lines pass through all 4?


[/ QUOTE ] I think that (generically) the answer is 1 but I don't have rigourous proof yet.

[Dale Dough]

Heh.. I guess we can only process 2D information really well then, because our vision is basically 2D.

Logically extended, does that mean that a 4D being wouldn't be able to/need to process complex 4D info, only 3D?

[MooBot3000]

*Grunch* Read Flatland. Edwin A Abbott.

This is getting more into philosophy. Like can we not visualize more than 3 dims because we've never experienced it or because it's biologically impossible?

[pzhon]

[ QUOTE ]
Consider the rectagle with vertices at (0,+-1,+-2). Consider rotations about the x=y=z line.
a) Take two images of the rectangle under rotations. Can they intersect without coinciding?

[/ QUOTE ]
<font color="white">Yes. At 180 degrees, there is an intersection as (0,1,-1) gets sent to (0,-1,1) and vice versa. There are no other intersections at nontrivial rotations.</font>

[ QUOTE ]
b) Take two images that do not intersect. Can they be linked?

[/ QUOTE ]
<font color="white">No. It suffices to check this for any pair of images, as they are all essentially equivalent. At 120 degrees, the rectangle with vertices (+-2,0,+-1) is not linked with the original. It can be pulled out in the x direction.</font>

[ QUOTE ]
c) Take 3 images that do not intersect. Can they be pulled apart without passing through each other?

[/ QUOTE ]
<font color="white">Not necessarily. That was the motivating example for this exercise. I happen to know that (0,+-1,+-2), (+-2,0,+-1), (+-1,+-2,0) give 3 rectangles arranged in a Borromean rings configuration. No pair of rectangles is linked, but you can't separate all 3.

If you take three rotations that are all within some 180 degree interval, then they can be separated.</font>

[ QUOTE ]
Here is another example.
a) Take 3 mutually skew lines in R^3. How many lines pass through all 3?

[/ QUOTE ]
<font color="white">Generically, there is one line passing through a point and two lines. From the perspective of a point, the lines generically either appear to cross, or one line will cross the antipodal image of the other. The latter means the point is on a line segment connecting the two lines.

It's a bit easier to see this if you think of the celestial RP^2 = S^2/{x~-x} instead of the celestial sphere S^2, since then lines correspond to lines rather than semicircles, and any pair of lines intersects or coincides in RP^2.

There are infinitely many lines passing through all three lines.
</font>

[ QUOTE ]
b) Take 4 mutually skew lines in R^3. How many lines pass through all 4?

[/ QUOTE ]
<font color="white">Perhaps surprisingly, the answer is not 0, 1, or infinity. Two complex lines intersect 4 lines in general position. The two complex lines can both be real, but they might both not be real. So, the answer is "either 0 or 2."

To see the possibility of 2 lines intersecting all 4, you can let the configuration be singular. Let two pairs of the lines intersect. Each of those pairs intersects in a point and also defines a plane containing the lines. The two lines intersecting all 4 are the line connecting the intersections of the pairs and the line formed by the intersection of the planes formed by the intersecting pairs.

Another approach is to see that the answer should be the same as in projective 3-space, and you can move one line to the plane at infinity, say the horizontal line at infinity. Lines which intersect this are precisely the lines that are parallel to the xy-plane. So, consider the intersections of planes parallel to the xy-plane with the other 3 lines. You get 3 points, which are the vertices of a triangle, and which move with velocities v1, v2, and v3 as you increase z. The three points are in the same line when the triangle has 0 signed area. When z &gt;&gt; 0, the triangle is roughly in proportion with {zv1, zv2, zv3}, and when z&lt;&lt;0, the triangle is roughly in proportion with {-zv1, -zv2, -zv3}. These triangles are just rotated 180 degrees from each other, and have the same sign of their signed area. Generically, either the sign stays nonzero, e.g., ((1,0)+(z,0)) and its +-120 degree rotations, or it crosses 0 and then back.
</font>

[bearly]

like wow. how would you establish that another is imagining what you are describing? i believe a famous philosopher called these "private language games"..........b

[pzhon]

[ QUOTE ]
how would you establish that another is imagining what you are describing?

[/ QUOTE ]
Success means you develop the intuition to answer questions and prove statements, among other things. These answers and proofs can be verified by people with less intuition.

[ QUOTE ]
i believe a famous philosopher called these "private language games"

[/ QUOTE ]
I believe philosophers call very different things private language games.

[Meromorphic]

A mathematician and an engineer attend a lecture by a physicist. The topic concerns Kaluza-Klein theories involving physical processes that occur in spaces with dimensions of 9, 12 and even higher. The mathematician is sitting, clearly enjoying the lecture, while the engineer is frowning and looking generally confused and puzzled. By the end the engineer has a terrible headache. At the end, the mathematician comments about the wonderful lecture.

The engineer says : "How do you understand this stuff?"

Mathematician : "I just visualize the process."

Engineer : "How can you visualize something that occurs in 9-dimensional space?"

Mathematician : "Easy, first visualize it in N-dimensional space, then let N go to 9."

[bearly]

and i believe your "intuition" has about the same status as "discernment" in the evangelical/pentacostal churches. you will decide who has this intuition (you have invented). as to your last remark: glib, but vapid..............b