The Mathematician Who Sculpted the Shape of Space | Hacker News Reader

The Mathematician Who Sculpted the Shape of Space

quantamagazine.org

105 pointsdigital5520 comments

I took a differential geometry topics course from Gene Calabi about 12 years ago in grad school. I was surprised to see his name on the upcoming course calendar — I'd heard of him ("Calabi-Yau manifold") but hadn't expected he might still be teaching classes. (And ordinarily he wasn't — he was already long-retired at that point.)

Turned out the reason he was teaching a course that semester was that he was really excited to share a new idea he'd had about how to visualize the complex projective plane. There were two (or three?) students in attendance.

Okay, here's Calabi's visualization of the complex projective plane. The complex projective plane (as everyone knows) is the set of equivalence classes of complex 3-vectors, where two vectors are equivalent if one is a complex scalar multiple of the other. Calabi's visualization: for a given representative, take the real part and imaginary part of that 3-vector separately, now that's a pair of vectors in real 3-space, and those are much easier to visualize. Choose a particular representative for the equivalence class: use up the dilation part of your complex scalar to normalize your 3-vectors, then use up the rotation part to make them orthogonal. Now notice that multiplication by a unit-length complex scalar will rotate the real and imaginary 3-vectors through the plane they span, so we can visualize an element of the complex projective plane as an oriented ellipse in real 3-space.

Actually we have some singular ones — if the real and imaginary parts are the same length (and orthogonal) then instead of ellipses we have circles, and if the real or imaginary part is zero then we have line segments. But we have a 1-parameter family of oriented ellipses (the parameter is the eccentricity) where at each parameter we have a quotient of SO(3) (it's almost the special orthogonal group but not quite) and at one end of the family it reduces to a 2-sphere and at the other end to a real projective plane.

But the main point is instead of having to visualize 3 or 2 complex dimensions (6 or 4 real dimensions) now we get to visualize ellipses (1 real dimension!) inside real 3-space. Much more accessible to geometric intuition.

A slightly lesser-known fact: the complex projective plane double covers the 4-sphere via the conjugation map. i.e. identify two equivalence classes of complex 3-vectors if one is the complex conjugate of the other. The resulting space is the 4-sphere.

Calabi's visualization: take our oriented ellipses in real 3-space (spanned by the real and imaginary part of our complex 3-vector). Complex conjugation means negating the imaginary part, so the result of identifying complex conjugates is throwing out the orientation of the ellipse. So here's a visualization of the 4-sphere: unoriented ellipses in 3-space. There's a 1-parameter family with varying eccentricity; for each eccentricity there's a manifold which is 4-covered by SO(3); the singular sets at each end are a real projective plane.

Nice construction. Took me a little while to figure out that by oriented ellipse you really mean a fixed curve in 3d, not just a fixed direction of the normal to the plane of the ellipse, which was my first reading. That way orientation takes 3 (real) dimensions + 1 dimension for eccentricity, which gives the 4 real dimensions in CP^2.

How accessible it is to intuition is debatable though :P

so would you say that it's basically a panorama of Jones vectors?

Thank you, you write well - being able to transform objects in many contexts is super useful, it unlocks so much creativity.

> where two vectors are equivalent if one is a complex scalar multiple of the other.

I can get an intuitive understanding of equivalence classes of real vectors being equal if they are real scalar multiples of each others, and understand that this is an equivalent case because of the field and linear space definitions/axioms, but I struggle to visualize it, possibly because of the larger dimensionality and the fact that there are two "kind" of dimensions, 2 (complex plane) x 3 (the linear bases).

Are you able to visualize objects like that? Any help?

Ah, I think I was able to think a kind of a visualization myself.

You can think of a complex 3-vector as a triplet of labeled points on the complex plane. Scaling this vector by a complex number is equivalent of rotating and scaling this triplet (0 as origin, so no affine transformations).

Therefore, two vectors in this space are equivalent up to a scalar if you can find a complex number that scales/rotates one triplet to another. I can visualize a rotation/scaling transformation of a point cloud on the complex plane pretty well so I'm kind of satisfied this visualization.

Also, as a neat interpolation between the two vectors, you can think of doing linear interpolation between x := 0 and 1; c1 * c^x where c1 is a complex number component of the first vector and c the complex scalar multiple between the two vectors, rinse and repeat for each component. (Didn't test this in code yet, though) In my mind, this should lead the point cloud of the first vector to spirally slide to form the second vector.

It is harder! Calabi's trick is one attempt at making it easier.

Hmmmmm. If you're ok with the real version... well another way to write down the real version is "lines through the origin in real 3-space" (that's what the equivalence relation turns out to be). How do you feel about "lines through the origin in real n-space"? How about "planes through the origin in real n-space"?

If we start with "complex lines through the origin in complex 3-space" (the complex projective plane we're trying to understand), and throw away all the complex stuff, then that's "some real 2-planes through the origin in real 6-space", i.e. it's a subset of planes through the origin in real 6-space. It's a proper subset because every complex line is a 2-plane but not every 2-plane in 6-space is a complex line.

So maybe that's a start? It's not a perfect way to visualize it because (a) 2-planes in 6-space aren't exactly easy to visualize either, and (b) okay but which planes in 6-space are the special ones that come from the complex structure that we ignored earlier. But maybe it's something.

This is far outside of my area of competence, but I have always had a hard time understanding/imagining what people mean when they say string theory uses extra dimensions but that those are rolled up or tiny. Like, I know about projective and hyperbolic spaces and understand that a metric can be zero or negative. What I do not understand is how a dimension can be small, like what makes these different from our macro dimensions.

I know that string theory is a research program and that they first started out studying vibrating lines and circles because this avoids the singularity of the point which was a useful idea in quantum gravity but then this didn't work either because reasons so they keeps adding more "micro" dimensions.

Maybe someone can someone give me a drawing of one micro dimensions and one micro dimensions, sort of like the euclidian plane is two macro dimensions.

One possibility is that our world is a "brane" so maybe the bulk of the universe is high dimensional but we are stuck within a milimeter of an 3-dimensional volume embedded in a much bigger space.

- Okay -- what would that imply?

- I dunno

See https://en.wikipedia.org/wiki/Brane_cosmology

A tube has 1 macro dimenson and 1 micro dimension.

https://en.wikipedia.org/wiki/String_theory#Extra_dimensions

https://en.wikipedia.org/wiki/Compactification_(physics)

A motivating example is that the product of manifolds is a manifold in a fairly natural way, and the dimension of the result is the sum of the dimensions of the factors. The factors come in one of two flavors, compact and not compact, with compact roughly meaning finite volume, with the caveat that the topological analog to volume is a little strange and only gives two answers, infinite and finite.

The coordinates of a product of a compact and a non compact manifold are then either constrained in a compact space or not.

Not every manifold is a product in this way though, for an easy example, theres the mobius band (with the boundary removed). This is a "twisted product" of a compact manifold and a non compact manifold, so we have some notion of one small and one large dimension here. even though we dont have explicit projections onto each factor.

The way I've seen it explained in popsci books is like a long fishing line stretched between two points. When you (a large human) look at it you see it as 1 dimensional, you can go either forward or backward. But now imagine you are a tiny bacteria moving along it. You see it as a 2D surface, you can go forward or backward along its length as initially described, but you can also make a 90 degree turn and move around the circumference of the fishing line. The 2nd dimension is there but it's "tiny and curled up".

Not sure how well that analogy tracks the extra tiny dimensions of string theory, but it has stuck with me.

i always thought of these as purely mathematical constructions. i don’t think there is a way to “visualize” the “small” dimensions

Many people give the tube visualization. If you've been playing video games for a long time, you have another option: Consider the Asteroids game field: https://www.youtube.com/watch?v=_TKiRvGfw3Q . This is well known to be topologically a torus. There are no boundaries on this field, yet it is finite in both dimensions.

Note that while Asteroids itself has no equivalent to a "wave" in the game, you could imagine having a wave inside this space. Even though there is no boundary to this space, you can see how any stable wave would have to be related to the size of the space.

Now take the Asteroids playing field, and collapse one of the dimensions to be very small. Now you have a very small spatial dimension, even though it has no boundary. The waves that wrap around that dimension are confined even more tightly than before to certain wavelengths.

This still appeals to human intuitions; your "small" dimension is probably, say, 10 pixels. But you could keep shrinking it until it was microscopic, you just wouldn't be able to fit the game entities in it anymore.

Alternatively, you could expand the playing field into three dimensions, then shrink the third one down, though that requires a bit more geometrical intuition.

Either way you can see that the nature of that coordinate starts to change fairly significantly, and anything moving through it does not move the way you are used to because it wraps around so quickly.

> the tube visualization

But can a tube "have zero Ricci curvature"?

That one is actually not very hard. Imagine the world pacman lives in: It's rolled up with a finite width in both of its two dimensions. Pacman can move left-right and up-down but after a finite length he comes back to where he was.

Now imagine you shrink his world along the up-down axis. And you make it smaller and smaller until pacman just barely fits in. Now he can stricty speaking still move "up" and "down" - but it doesn't matter. He is immediately in the same place. The world has become effectively 1 dimensional as the second dimension has been rolled up so far that it doesn't matter anymore as far as the movement of particles (like pacman) within it is concerned.

Reply on news.ycombinator.com