If you’re into finite yet boundaryless landscapes, check out the Zoom Quilt.
I think the topology here is S2xS1. Via Crooked Timber.
Update: This would be a good opportunity to point to the website of reader James Harrison. He’s a sculptor who recently completed a commission in London called Hands of Cantor, exploring notions of topology and infinity.
That is so cool.
But, I thought it is R2xS1, since at each point along S1, we have a finite and compact 2d picture with a boundary (the pictures do not look like they are wrapped around)…
Any good field theorist identifies spatial infinity as a matter of course. If you wanted to compactify each picture as a two-torus I wouldn’t complain.
I think it’s actually just an S^1 x S^1. Imagine you’re on the inside of a hollow cylinder, periodically identified, watching the walls of the cylinder. There’s no reason to have more than two dimensions.
You are probably correct about the topology, in the higher (or hire) dimension sense…
Eh? For all we know, the topology could be trivial. Everything could be happening on the surface of a small planetoid. One is free to use polar coordinates in flat space after all, even if Lubos Motl does believe that there is something sacred about Cartesian coordinates…..or maybe this computer can’t run this thing properly?
Amazing. I spent almost an hour on Friday looking for that thing, and I couldn’t remember what it was called. Thanks!
The hollow cylinder description seems no good. You’re not moving laterally to different parts of the scene, you’re penetrating *into* it; hence the name “Zoom Quilt.”
Admittedly my topology is rusty (well, actually, it was never very good to begin with), but it seems more like this world is the inside of a torus. Each snapshot along the zoom axis is a two-dimensional Euclidean scene, and the ends of the zoom axis are identified.
Sean, can you comment on where we have gone right or wrong?
Think of it as a bunch of S3s connected by wormholes, i.e., it’s the connected sum S3 # S3 # S3 … where we eventually reconnect with the first one. This is homeomorphic to an S3 connected with itself. This is accomplished by removing two 3-balls and identifying the S2 boundaries. It’s not hard to see that removing two 3-balls from an S3 gives S2 x I where I is an interval. Identifying the two S2s then gives us the topology S2 x S1 as Sean said.
Compactifications other than S3 are left as an exercise for the reader :).
Saucy Wench said:
Admittedly my topology is rusty (well, actually, it was never very good to begin with), but it seems more like this world is the inside of a torus. Each snapshot along the zoom axis is a two-dimensional Euclidean scene, and the ends of the zoom axis are identified.
Um, isn’t that what I said? A cylinder, periodically identified, is a torus. Or when you say the “inside”, are you filling in the interior?
I still don’t see why everyone thinks you need a three-dimensional topology, unless you’re trying to situate yourself as an observer relative to the image. In that case, I would say a disc times S^1 is the most reasonable thing to imagine. But, the image itself is two-dimensional, and it has the topology of a torus.
Think of it this way: you clearly move along some S^1 while viewing the image. Each incremental movement brings in a new thin shell of the image, that is, an S^1, while the middle part recedes into the distance.
The trouble with all of your descriptions is that you’re imagining the image as a two-dimensional function of the observer’s coordinates. But you really should be identifying the parts that are the same as you move from z to z + delta z. The only new part is an S^1.
Really, it’s S^1 x S^1.
It looks more like a local S^3 x S^2 x S^1 now, but perhaps there could be something half-flat about it.
Or when you say the “inside”, are you filling in the interior?
Yes, the cross section of the torus is each flat scene. It can’t be on the walls of a cylinder; it doesn’t wrap laterally. You don’t slide each scene to the left or right to get to the next scene. You move into it.
It also can’t merely be two-dimensional. Each snapshot alone is two-dimensional, and then we penetrate each snapshot along the third “zooming” dimension.
Each of your “thin shells” is two-dimensional, so they can’t be S^1’s.
Yes, the cross section of the torus is each flat scene. It can’t be on the walls of a cylinder; it doesn’t wrap laterally. You don’t slide each scene to the left or right to get to the next scene. You move into it.
But of course that isn’t what I was saying. Apparently I’m not very comprehensible.
Imagine you’re in the center of a cylinder. For concreteness, say you’re in Euclidean 3-space and the cylinder is x^2 + y^2 = 1, and you are moving along the z-axis in the direction of increasing z and the identification z ~ z + 10 is made. Further imagine that each band z in [n, n+1) is painted a different color (mod 10). From your starting point at the origin, looking in the positive z direction, you see, say, a large red annulus, with a smaller orange annulus inside, etc. As you move forward the red annulus “zooms” out of sight and the orange annulus fills most of your field of vision, etc. At some point the red band appears as a small disc in the center of your field of vision, gradually expanding to a large annulus as you reach z = 10.
Now do you see what I mean? The image has the topology of a torus, but if you’re trying to position yourself as an observer then you must postulate that you are “inside” the torus. Only then is a third dimension needed. Of course the image you see doesn’t have quite the perspective one would imagine in the scenario I describe, but the topology is the same, it’s just projected onto a 2d square in a slightly different way than your visual system would project it.
Now, this is a really stupid and trivial thing to be spending so much energy discussing, so I don’t think I will comment further.
I think we agree anyway and were getting our signals crossed. 🙂