thoughtsThoughts on Topology

I've been doing a lot of thinking about topology today, and I thought I'd share my musings and deductions with you. Of course, I have absolutely no credentials or education in this stuff, only what I pick up in encyclopedia articles and such. Still, though, it's been in my brain, so it goes on the website.

Anyway, I suppose I'd better give a little explanation of topology first, for those few of you who know less about it than me. Now, I'm not talking about mapping out the contours of rocks. I'm talking about mapping out the shape of the universe.

A piece of paper!

See, imagine we're on this flat two-dimensional sheet of paper here. We can wander around all over the place, and every part of the paper is the same as every other part. This sheet of paper is our universe. When we reach the edge of the paper, that's the edge of the universe, and nothing can go beyond that. Or, alternately, everything can go beyond that and spill out into the oblivion beyond. But, for the purposes of this illustration, let's say nothing can go beyond the edge of the universe. Thing is, most physicists today are pretty sure that our universe has no edge.

Now, we could fix this problem by extending the paper out in all directions, making it an infinite flat sheet of paper, but most physicists don't think we have an infinite universe, either. How can we have a finite universe with no edge, you ask?

Globe!

Well, let's take this globe here, and wipe off the continents, making it a big spherical sheet of white paper. Now the paper has no edge, if you head off in any one direction, you eventually wind up where you started. No edge, finite size. All it took was another dimension to bend the universe around in.

And, of course, if the globe is large enough, it acts on a local level just like you would expect a flat sheet of paper to act. That's why we took so long as a species to determine that the world was round. Any curve, viewed closely enough, looks like a straight line.

In a similar way, our species is now trying to determine the true shape of the universe. Sure, it looks flat. It looks like an infinite expanse of regular ol' 3D space. But somewhere in the stars there has to be a clue, some view of the same constellation from different angles, that shows the contours of the hyperdimensional shape we live in.

Now, say we cut open that globe and spead it out as a sheet of paper again, with little notations on how it goes back together, just so we can look at space from a more convenient perspective. I'm doing this just so we can contrast the 3D sphere with another type of shape, the torus. Remember the arcade game Asteroids? Or, if you're a little younger, any of the early Final Fantasy type games where you get an airship? You had a rectangular map, and when you disappeared off the top, you reappeared at the bottom, and when you disappeared off the left, you reappeared on the right.

Taurus

Now, imagine you've got a rectangular sheet of paper, and you roll it up, gluing the top edge to the bottom edge. You now have a cylindrical tube. Imagine that tube is stretchy enough so that you can curl it up and glue the right edge to the left edge. What you now have is a torus, like a donut or inner tube. This is probably the easiest shape to deal with, if we're talking about 2D.

If you had a little tiny universe you could crawl into that was in the shape of a 3D torus (let's say each dimension is only about two metres long before it loops), you would look like an infinite array of yous. You'd look up and see the soles of your feet hovering just over your head, you'd look down and see your scalp, you'd look right and see your left shoulder and the back of your head, turned away.... multiplied out to infinity. Of course, you'd be tempted to think "hey, there's a copy of me over there!", but of course, it's not a copy. It's you. You're just seeing yourself as the light curves around the world, so to speak.

We clear so far?

Okay, that's the basics. On to the stuff I was pondering today when I should have been working.

You can't drink from a klein bottle.  Okay, you can, but just not here.

You all know what a Möbius strip is, right? Take a rectangular strip of paper and give it a little half turn before gluing it into a loop. Now you have a piece of paper with only one side and only one edge. (Go ahead and try it, if you don't believe me.) Now, if we could do that again with the remaining edge (we can't, but pretend we can) we wind up with what's called a Klein bottle. You can't really properly represent it in 3D terms that make sense to the eye. That curvy arm part that goes around to form the hole in the bottom.... it's not supposed to intersect the side wall, there.

The issue with this sort of universe.... well, look at those squares there. Half of them have their As and Bs pointed the opposite way. If you travelled from one square to an adjacent square, you'd come back to where you started, but it would all be backwards. The signs would all read from right to left, and, from their point of view, you'd now be left-handed and dyslexic.

Or, at least, that's what I thought. Now, after a lot of sketchy little diagrams, I'm pretty sure that that sort of thing only works in continuums with an even number of dimensions. In 3D, when you move to the next cube over, it's just rotated 180 degrees, nothing you can't fix with a quick pirouette. Or, at any rate, I'm pretty sure about that.

Pirouette!

The other thing I was thinking about was the idea of a sort of pointy half-torus shape, which works out great in 2D. When you approach one of the points, you just see a ring of yourself getting closer and closer together, until at last you can't reach the point of the universe because you're pushing in against yourself.

pointy

Six spheres converging on a point in space

At least, again, that's what I thought. But of course, that doesn't really work out in 3D. I had thought you'd converge on a single point in the universe, looking like a sphere of yourself, until eventually your fingertips are thrusting their widths against each other in a little geodesic dome thing, probably one of the Platonic solids. Problem is, I was thinking of this with a bunch of spheres. When you add body parts and so on, you begin to see that the rotations are all screwed up. I couldn't get it to be consistent.

I had forgotten the tea cup = donut illustration they always use when talking about topology, that two shapes without any holes through them are essentially the same thing.

That is one heck of a poorly made donut.

So now for the pointy universe idea, I have this torus that's pinched on two ends, like so. The filament between the two halves of the universe is infinitessimally thin, too thin for even light to pass through. For inhabitants of the universe, it looks like a big black wall.

Of course, as you approach this wall, you start seeing a 2D array of yourself closing in. Your feet get closer and closer to your head, your shoulders push tighter and tighter against each other, you reach out your hand but the fingers appear to bend toward each other so that you can't quite touch the elusive (because it's nonexistent) wall.

Not quite as cool as the converging sphere of selves thing, but it would still be awesome to experience. The experience of standing on your own shoulders would no longer be the exclusive province of the contortionist.


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