

Ask HN: Can you explain Calabi-Yau spaces to me? - zackattack

I have a basic understanding of college level maths (analysis, algebra, discrete math) but the Wikipedia and Mathworld articles are too difficult for me (they use terminology I don't understand, making parsing very difficult). The simple-English Wikipedia article didn't meet my needs.<p>Thanks.
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anigbrowl
I'm not sure if you want to actually do math with C-Y spaces, or just 'get a
feel for them'. In the former case, all I can think of is to obtain a copy of
Mathematica (I think the demo will run for a week or a month) and then
download some of the example notebooks and tinker around...although my own
attempts at this proved fruitless :)

If you just want to get a conceptual insight, try the excellent Wikipedia
article on manifolds, which is fairly non-technical and introduces the concept
of what a manifold is and why you would want to use one as a mathematical
tool.

<http://en.wikipedia.org/wiki/Manifold>

In its barest essence a manifold is a way of reductively mapping a complex
geometry. Knowing how to construct one, you can then deconstruct a complex
multivariate problem (eg the mathematics of 10-dimensional strings) into
simpler geometric problems, using the mathematics of manifolds to escape the
hideous logical problems resulting from boundary conditions.

Consider an unbounded plane, ie normal cartesian space. You can roam around
forever in the X-Y dimensions, boring. Now, suppose it's bounded; for example,
mapped onto a finite sphere. This is one case of a 2-dimensional manifold
(assuming a perfectly smooth sphere). It turns out that geometry still works
perfectly well even though it's not flat. Georg Riemann was out to prove
Euclid's parallel postulate by doing a _reductio ad absurdum_ , and showing
that if parallel lines could meet then all of geometry would cease to make
sense. To his surprise, it didn't, and he found himself the unexpected
inventor of spherical geometry.

Now consider a different kind of two-dimensional manifold: the world of Pac
Man. Pacman lives in a rectangular world, but he can exit one side of the
rectangle and re-enter at the opposite. On some maps he can also exit at the
top and re-enter at the bottom.

But this isn't mappable to a sphere. Rather, imagine a cross shape, like a +.
Please think of the arms of the cross as having thickness, rather than just 2
lines; better yet, draw a cross consisting of 5 squares on a sheet of paper
and cut it out. Draw a line around the middle square and put a little pacman
in it. Now, take the top and bottom squares and tape their outer opposite
edges together. Do the same with the left and right squares. You get something
resembling two cylinders at right angles to each other. Pacman lives on the
little square in the middle where he eats pills and is chased by ghosts. He
can leave one side of the square and travel via the curved path on the paper
to the other side; he does so by literally traveling _through another
dimension_ \- that is, he teleports (oooh!). It's important to remember that
although a direct path along the curved loop of the paper is 2x the distance
across pacman's rectangular world, as far as pacman is concerned it's actually
zero. When he exits on one side, he doesn't disappear from the game screen for
2x the time it normally takes him to traverse the maze in a straight path, he
reappears instantly on the other side. You could represent this in a simpler
fashion by using a torus (donut shape), which is yet another type of
2-dimensional manifold, one on which pacman slides around on the surface.

In fact, this would be a much better way to represent it (because it gets us
out of the annoying zero-length weirdness), but I offer the idea of the linked
cylinder because it's worth considering that if you turn it upside down, there
is an equivalently-sized opposite space. That's equally true of the donut (the
inside has the same configuration as the outside) but less obvious. Pacman can
never enter this opposite space unless he finds a way to go around the edge of
the paper, which is another dimensional shift again, over and above his
opposite-edge wanderings. Is there an anti-pacman there, called namcap? Maybe
- or alternatively, you could imagine that when pacman eats a power pill, the
ghosts become oppositely charged and are suddenly vulnerable to him -
something we might not have thought of if we were looking at the game taking
place on the outer skin of a torus.

So...manifolds are a set of handy mathematical tools for messing around with
contiguous multi-dimensional spaces. String theory somewhat resembles a
10-dimensional game of pacman where both pacman and the ghosts are travelling
at varying speed and can't eat each other except when their speeds are
matched. When you see pictures of C-Y manifolds they're only slices of the
manifold from one perspective. We can see pacman's whole world at one because
he lives in 2 dimensions and we have the advantage of living in 3. To pacman
we would seem like beings of infinitely variable shape and number, as our
bodies could be sliced from an infinite variety of angles. So if your hand
intersected his 2-d world, it would appear as a circle, followed by 3 circles,
followed by 5, which would then merge together into a sort of oval (going from
your fingers towards your wrist). Similarly, the cool C-Y shapes on the math
pages are 3-dimensional slices of a 10-dimensional conceptual space, the
overall shape of which we cannot meaningfully speak about except through
mathematics, and which are impossible for us to really appreciate.

This is, of course, 1% mathematics and 99% bullshit. Not responsible for lost
property left in other dimensions.

~~~
zackattack
Excellent, thank you. But in paragraph 8, you do mean that "worth considering
that if you turn it inside out" instead of "worth considering that if you turn
it upside down", correct?

~~~
anigbrowl
Not really...I was imagining holding it so the center square with pacman was
flat in front of one, similar to the old tabletop arcade machine.
Topologically I guess this is inside out, but with the open loops upside down
seemed more appropriate.

Edit: oops, I just realized it reads as if I were still talking about the
torus, in which case inside out would be correct (but there would be no
edges).

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hga
I'm afraid not, but here's a funny Amazon.com review that roughly shows where
they fit into string theory: "The string theorists were scammed!"
[http://www.amazon.com/review/R2H7GVX4BUQQ68/ref=cm_cr_rdp_pe...](http://www.amazon.com/review/R2H7GVX4BUQQ68/ref=cm_cr_rdp_perm)

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ilkhd2
try Mathoverlow, probably...

~~~
zackattack
I just bought the John Derbyshire book "Unknown Quantity."

~~~
anigbrowl
Derbyshire is a weird and interesting person.
<http://en.wikipedia.org/wiki/John_Derbyshire> and follow the links - a self-
professed racist and authoritarian, married to a woman of a different
ethnicity and who is also a former illegal alien. I am unable to decide
whether he is a genius with a wicked sense of humor or a total crackpot.

