Uh-oh. Two days between posts? Par. Three days? Acceptable. Four days? Sliiiippiiiing...
I've been wanting to power my way through this article, where humanity and humousity interact with each other "on their own terms," but it keeps slipping away from me. So fuggit, I will read it on my own time when I am bored (one magical, starry night, when there is nowhere to be and nothing to do, perhaps in another world or another life). So, to get marginally back on track, I will riff on something with which I am very familiar: four-dimensional single-surfaced super-edgeless objects.
Today we talk about Klein bottles! I've even included an artsy-craftsy step-by-step, for the four-dimensionally impaired. But seriously, if you don't know how this works, or have trouble visualizing four-dimensional surfaces (or even uncommon two- and three-dimensional ones), you can follow along with a piece of paper and a pair of scissors to achieve geometrical enlightenment. You will also need either of: A) scotch tape and two different-colored crayons, B) masking tape and two different-colored pencils, or C) a stapler and two different-colored crayons or pencils. A felt-tipped marker will help in any case.
Step One: Cut yourself a long strip of paper. Say, an 11"x3" strip from a standard sheet of printer paper. This will be our fundamental unit of doing things, so maybe cut yourself a few (you'll need at least two, and you might fuck up, so hang on to the 11"x2.5" leftover).
Step Two: Take one of those strips and bend it around, lengthwise, to make a three-inch-wide ring that is eleven inches in circumference. You can probably just visualize this in your head, I bet. Hold it like that, and just imagine taping/stapling it together, OK?
Step Three: You see how it has an obvious inside, an obvious outside, and two obvious edges? Good. Now imagine bending those edges around to meet each other towards the middle, like a donut. If you don't feel like visualizing this, then tape/staple the ends together and actually bend the two edges around to meet each other. Good? Good. You have a donut, or as mathematicians call it, a torus: a smooth, hollow donut. :) This figure, like the ring before it, has an obvious inside and an obvious outside - but because the edges have been, ahem, seamlessly married to each other (in our imaginations, whether or not we actually have scissors and can cut perfectly straight with them), it has no edges of its own.
Step Four: You don't need that torus any more, unless you have trouble imagining the torus on its own. So if you actually made the torus, good for you! You're going to be extra enlightened today. Set it aside. If you visualized the torus, good for you! You're going to be less absolutely enlightened but more precisely so today. Re-use that strip for this step.
Take the strip, bring the ends around to meet each other; but before you connect them, turn one of the ends a half-turn 'round (180°). Now tape/staple them together. Congratulations! You've just made a Möbius strip.
Step Four Sidetrack: The Möbius strip is a geometrically interesting figure, because it has one side and one edge. "Wait," you say, "I took a strip of paper that had two sides and four edges, and now you're telling me that by doing one twist (shout optional) and one edge-join, I've wound up with merely one side and one edge? Preposterous twaddlecock!" That's where you're wrong: we've got tru fax here, instead.
Step Five: Take one of your crayons/pencils, and mark the center of the strip ("either" side) where you've taped/stapled it. Pick a direction, and trace continuously along that direction, adhering fastidiously to the median (i.e. "be careful to stay right in the middle"). You will come back to your tape/staples (you should've used at least two, to prevent hinge-like motion - you used at least two, right? It's not too late! You can always staple moar!), but keep moving right over it and move on. Keep going, and come right back 'round. Yeah, now you're coming back to your starting point, aren't you? Holy carp. You've just made a line along the center of your Möbius strip, lo and behold, exposing its single continuous surface.
Step Five Sidetrack: This is how I visualize spin 1/2 particles. Here's the Wikipedia page on particle spin, but I'ma break it down for ya, don't worry: a particle's spin is simply the inverse of the number of full rotations you have to put it through to get it to look the same as it did when you started - in other words, the number of rotations you need to end up Samey-Samey is "one over x", where x is the particle's spin ("spin 0" being "undefined" in the sense of "any"). So if a particle has "spin 0", then it looks the same no matter how you spin it. If a particle has "spin 1", then it looks the same when you turn it around a full 360° (like the Ace of Spades, to steal Stephen Hawking's metaphor). If a particle has "spin 2", then it looks the same when you turn it around just 180° (like the Queen of Hearts, to again steal Stephen Hawking's metaphor). But if a particle has "spin 1/2", then it looks the same only when you spin it around 720°. Wait, what?! You mean, I turn it all the way around, it doesn't look the same; but I turn it all the way around again, and it does?! What gives?
Well, imagine a section of a Möbius strip - say, the section you've taped/stapled. Now pretend you're a "one-dimensional line running from edge-to-edge" (hold on to that thought, call it "the spork reference") across where you've taped/stapled. Travel all the way around the strip, until you get to the tape again. You're back at the same "part" of the strip, right? The part where the tape is? OK, so that's like 360° around it, but it still doesn't look "the same", because you're on the opposite "side". So keep going another 360°, and you're back where you started, having traveled 720° around the same circle but not seeing it the same way twice until you went around twice. Crazy, huh?
Step Five Sidetrack Two: OK, you remember "the spork reference" in the last paragraph? It was about that "edge to edge" bit. Take your felt marker, or your other-colored pencil/crayon, and mark out the edge from wherever you taped/stapled your Möbius strip. I mean the very edge of the paper, which is why a felt marker is so handy for this, but really anything will do so long as you keep in mind that you're on the edge and not on a side. Trace that edge until you run out of edge (i.e. until you come back to where you started).
One edge, innit? Crazy, innit? One continuous face, one continuous edge. For extra credit, you can follow the project here. It will deliver extra credit enlightenment. Or at least confusion (which is, in the Buddhist scheme of things, enlightenment by any other name - you just have to come back 'round to it until you understand it, blowing your mind in a different way each time).
Step Six: Take your Möbius strip - if you marked it up too much or diced it up in some previous step and/or sidetrack, don't worry, just make another one with your leftovers from Step One. :) Take the two "apparently opposite but really identical" edges, like from where you taped/stapled it end-to-end, and bend them together like you did with the torus in Step Three. Now, you can't do this completely, since you yourself only have three spatial dimensions to work with. But if you had four spatial dimensions to work with, then you could do this, and then you'd have a Klein bottle: a Möbius strip folded in upon itself like a torus.
This shape is particularly interesting - it had one face and one edge to begin with, and now still has one face but had its one and only edge married to itself, giving it... not zero edges, but minus one edges. It is "a bottle that pours into itself," with no well-defined inside or outside.
We're done with steps, I'm just rapping from here on out. Look at that original picture:
So there's your "bottle pouring into itself", up there: imagine liquid in the "reservoir" (the "fattest part," traveling up the "neck" (the upper skinny part), down through the "stem" (the mid-to-bottom skinny part), then "spilling/pouring" "out" along its "outside". Except if you follow that topographical "outside", you'll see that it goes along the "other" side of the "neck" and "stem" back into the interior of the "reservoir" - it only "doesn't" because of how it's blown. Four-dimensional-topography-speaking, it ought to do that.
The "so-called" Klein bottle shown above actually has a "well-defined" volume, insofar as it can be measured in two distinct paradigmatic ways. First, you can imagine it in that orientation, its "reservoir" filled with liquid. It would have an airtight seal where the "stem" penetrates the "reservoir", which would be impenetrable from the "reservoir" side (remember: spin 1/2 and sides and stuff) but permeable from the "outside of the 'stem' side" of it (because 4D geometry). Except we're working in 3D only, so it's impermeable from the "outside of the 'stem' side," as you see there. So you could fill up the "reservoir", all the way up the "neck" until it bends back down against the horizontal (as determined by gravity) to become the "stem".
Or, if you wanted to do even better, you could turn that 3D shape upside down and submerge it in water, rotating it until all the air bubbles flowed out, lifting it up with the "reservoir" upside-down to the orientation displayed above, filling both the "neck" and "stem" until the stem yields to the outside of the "reservoir". That is, while not "well-defined," at least a definite volume. But that just shows the limitations of a 3D Klein bottle: a true Klein bottle is in 4D and has no such definition because it is not subject to such 3D limitations - it pours into itself and has no definite volume because it has one surface and minus one edges.
Contemplate that for a bit.