Klein bottle for sale, inquire within.

December 14, 2001 AD
I'm going to talk of Klein bottles and tesseracts, of cabbages and kings. Or something.

But before that, I need to teach you a bit about four-dimensional space.

4D Space and the Tesseract

"4-dimensional space?" you ask. "Isn't that time?" Only in Einstein's assorted space-time discussions. The idea of dimensions is actually somewhat arbitrary. We work in three, so we think in three. But there can be more.

Imagine a hilly place filled with boulders. You could use latitude-longitude-altitude to find the given location of any boulder, but what about a point on the boulder's surface? I suppose you could give exacting details, but if the boulder rolls a bit, they get thrown off and you need to recalculate from the beginning. Even if the boulder only rolls an inch, the point rolls with the boulder. So its latitude, longitude, AND altitude have all changed much more than just an inch.

It's easier to add three more measurements - one stating how far the point is from the boulder's center, and two more giving the exact angle of the point from some chosen marker on the boulder's surface.

So you have six dimensions right there. Latitude-longitude-altitude-depth-angle1-angle2. Time, if mentioned at all, is the 7th dimension.

You need to think a bit oddly when dealing with 4D space. It's hard to get one's imagination around because it doesn't exist. Or, if it does (and it might), we can't see it. But, with practice and a bit of luck, you can understand it.

Imagine you're perfectly flat. Flatter than paper. Embedded in a 2D universe. Depth means nothing to you. Imagine you're a pious person and live in a cross-shaped house. Conveniently, it's also shaped like a flattened-out cube.

Click image to see where I stole it from. But you don't know it's a cube. In your universe, squares are the big thing. Cubes are funny things math people talk about.

So what happens if some day some bigshot 3-dimensional jerk comes along and folds your house into a cube, you inside it?

You'd never know, though I'm sure you'd be pretty horrified when windows opened onto other rooms and you realized that you'd soon starve to death.

Make no sense? I'll explain. You'd never know it was folded, because the folds happen in three dimensions. You exist in two. However, that doesn't change the fact that it is folded.

Problem is, you'd open a window and it wouldn't open onto the outside flat-universe anymore. it'd open onto another room of your house!

You'd crawl around forever, looking for a way out, but you'd never find it. You'd be caught on the interior of a cube, crawling blindly across the 3D creases and never knowing it. Before long, you'd realize that you couldn't escape. Then you'd bide your time until your air, food or water ran out, unless some 3D being came along and unfolded the cube, granting you escape back into your 2D universe.

Click image to see where I stole it from. Morbid, no?

Well, change the 3's to 4's and the 2's to 3's and the house to a shape like this image, then it could theoretically happen to you. Only worse.

That 3D cross is an unfolded tesseract. (Folded, the same-color sides touch - more on that later.) But before I get to that, I'll explain what a tesseract is.

Start with a point. 0D.

Really lonely, isn't it? Make another. Connect them, and you have a line. (Yes, I know it's really a line segment but you can get stuffed; this is my lesson.) 1D.

Draw another line the same length, as far away from the other as the line is long, then connect them, left end to left end, right end to right end. You have, if you did it right, a square. 2D.

Draw another square a line's length away. Connect it to the first using four lines, to form a cube. 3D.

Now comes the tricky part. In another dimension (maybe one of those freaky Sliders universes, if you like that show), draw a second cube. Connect the first cube's corners to the second cube's corners, through four-dimensional space. That is a hypercube, also called a tesseract. 4D.

Notice how, in connecting the two squares to form a cube, you wound up with six total squares? One on each side of the cube? Well, connecting the two cubes to form a tesseract makes eight total cubes. One is "here", one is "there", and six more have one face - a square - on the cube "here", one face on the cube "there", and their other four sides stretching through the "in-between" space that bridges "here" and "there". If you could swap one of our dimensions (like height) for a cross-section of this "in-between" space, you'd see the other cubes. Or parts of them, anyway. If you "unfolded" a completed tesseract into 3D-space, you'd have eight cubes.

Click image to see where I stole it from. See this picture of a box within a box? This is a 2D representation of a 3D representation of a 4D tesseract. Know how a tall building or a railroad track looks like its two sides almost meet in the distance? (That good ol' perspective trick.) You're seeing the same effect here. The two cubes are the same size; the smaller one is simply further away through 4D space. Even odder, the six truncated pyramids are also cubes, just cubes with one end 'here' and the other off in the multidimensional distance!

(One site features a Javascript wireframe tesseract to abuse, though I don't think you can get one cube to shrink, just to rotate.)

Now comes the real nightmare. Imagine you're sitting in an eight-cube house that gets folded into a tesseract by some 4D meanies. (Heinlein wrote a story about a tesseract house, called "... And He Built a Crooked House" which does give a great description on how a model tesseract relates to a real one. Click here, the story is linked from that page.)

I'll describe the house, sort of like Heinlein's. (WARNING: I'm making the assumption - which I obviously can't prove - that light, people, and other such things can travel through 4D space without any problems. But it helps the demonstration.)

First level is the porch, a cube with a stairwell leading up. Second level would be another cube and stairwell. Third would be a cube with four other cubes branching off of it and a stairwell leading to the fourth. Fourth would be a single cube.

Everything looks the same once folded. Only the outside exits (doors, windows, etc.) open up onto other rooms.

Hard to imagine how things are connected? I can understand that. Took me a good while to wrap my mind around it. To make it a little simpler, imagine that the four outer cubes of the 3rd level can "swivel" along the main column of cubes. This works great if you have some toy blocks around. You don't need to make the whole tesseract, just have a few cubes to hold up and roll around.

Example 1: You're standing in the central 3rd level room. The bathroom is to your left, the bedroom in front of you. You might notice that the corner to your front left is shared by the bathroom and bedroom. That's important.

You walk into the bedroom, turn left and step through a side door, to the "outside". As you do, the bathroom instantly swivels, like it was on a hinge at the common bathroom-bedroom corner. As you step through, you step through into the bathroom. The bathroom then instantly and invisibly swivels back into place so you can walk through back into the central room and try figure out what the hell's going on.

Just think about what you did. You walked forward into the bedroom and made a 90 degree left turn. Walked forward into the bathroom and made a 90 degree left turn. Walked forward again into the central room. Make another 90 degree left turn and you're right back where you started, facing the bedroom.

You just traced a closed three-sided figure with three right angles. Yep, in a tesseract, a triangle can have three right angles, if it cuts across cube boundaries! If it doesn't, it's a typical maximum-one-right-angle triangle.

Example 2: You're standing in the bedroom, facing that bathroom door that shouldn't be there. You look to your left and see the central room, and on the far side of that, the kitchen. Right where they should be.

You walk forward through the bathroom door. Again, the bathroom swivels and picks you up. This time you keep walking forward, to a door on the far side of the bathroom from the bedroom door. You walk through.

The bathroom swivels again, on its other corner, delivering you to the kitchen.

Undeterred, you keep walking forward. (You haven't turned once.) You open the door on the far side of the kitchen, and the kitchen swivels you over to the fourth room on the 3rd level, the guest bedroom.

You keep walking forward, open the door on the far side of the guest bedroom, and step through.

You're back in the master bedroom again, right on the spot you started from, without turning. You walked in a straight line, and ended up back where you began from.

That's a funny thing about a tesseract - every room/cube is exactly four rooms away from itself, when walking in a straight line room-to-room. An "endless" line would be exactly the length of four rooms.

Even weirder, if you left all four doors open, you can now see your own back in the distance!

Example 3: If you're on the bottom level (the porch), and you open a door in the wall, the 3rd level cube on that side (let's say the main bedroom) swivels down into place in two steps. First it swivels so its floor attaches to the wall of the second level, then swivels again so its outer wall is attached to the wall you're standing at.

There's only one problem. It can't reorient itself. It's upside down. (Notice that picture above? Where the black-on-cyan arrow is pointing up on one cube and down on the other? It's representing the same thing.) When you open the door, you'll be confronted with an upside-down version of the 3rd level bedroom. Assuming gravity works by pulling you towards the thing that was the "floor" in the unfolded tesseract, step through and you'll fall to the 3rd level floor - what looks to you to be the ceiling! Look back at the way you came, and now that room looks upside-down.

Deciding to get to the bottom of this, you charge forward, away from the porch. The cube swivels back into place and you walk into the central 3rd level room.

You charge forward, into the kitchen, You open the far door and jump through....

Completing another straight-line four room circuit. The kitchen swivels down to the porch, and you fly out, and fall to the porch floor. Which, from the kitchen's point of view, is still the ceiling!

Example 4: Take my word on this one, since the swiveling is almost impossible to visualize.

You charge up the stairs from the porch. Second level, third level, fourth level. You open the fourth level ceiling hatch - which should lead to the roof - and climb through.

Back into the porch. Another four-room circuit, the top level ceiling attaches to the bottom level floor.

Example 5: How's this for an insane four-room circuit? You climb through a trapdoor in the main bedroom floor. It swivels down, and the floor attaches to the wall of the second level. Gravity and vertigo do a serious number on you, since from your point of view, you'd jump down through the floor and wind up coming out of a wall!

You recover, and walk to the wall opposite. You open the door and walk through.

But it's a hard walk, since you're climbing up through a trapdoor in the kitchen, which has swiveled down to meet you!

Undeterred, you find a trapdoor in the ceiling of the kitchen and climb through. And out of a wall in the fourth level. Run to the opposite wall, open the door there, jump through, and you're falling down into the main bedroom, hopefully onto the bed!

Trapped, until someone 4D comes along and unfolds your house for you. If not though, at least you could suicide creatively. How many people can shoot themselves in the back of the head?

Of course, you might live in a tesseract and not know it. Some theories about the nature of the universe only work if the universe is a 4D (or 5D, or even 10D!) figure, some of these figures being tesseracts or other similar objects. The universe might actually follow the paradox of being boundless yet limited!

Of course, it could be worse. We could be living inside a Klein bottle.

Klein Bottles: How to hold nothing in something.

A Klein bottle is a 3d Möbius band (a.k.a. a Möbius strip). In fact, it's two Möbius bands attached to each other at their edges.

A Möbius band is, I'm told, what happens when you twist a 2-dimensional object (a paper or cloth strip) through 3-dimensional space (where we are) so it forms a 1-dimensional object (the Möbius band) which has only one side and one edge. (As one site I saw pointed out - the three-arrow "recycled" symbol is a Möbius band!)

Click either image to see... ... where I stole them from. A Klein bottle, then, is what happens when you twist a 3-dimensional object (a tube) through 4-dimensional space so it forms a 2-dimensional object (the Klein bottle) which has no volume - a real Klein bottle couldn't hold liquid, it'd eventually spill out from the side. Not over the side - from the side.

I could try describing one, but it's nigh impossible. You'd need to see an image, just like these two. Or, this really nice one.

Ah, hell, I'll give it a shot for blind people and text browsers.

Okay, take a tube. Not a normal tube, but one that's really wide on one end and small on the other. Bend the fat end inward (if you bent it outward, you'd have a tuba-shaped tube, or something).

Now comes the fun part. Take the small end and force it through the side of the fat end, so the small end is just matching up with the inward-pointing fat end. Glue ends together. You now have a Klein bottle. Sort of.

You can't make a real 4D Klein bottle like you can a 3D Möbius band. You can make a reasonable 3D copy, though, and even sell it. (These are 'immersed' in our universe, because they're imperfect 3D copies. A real Klein bottle is 'embedded' in 4D, just like you're embedded in 3D now.)

Another way of describing it is like this:

Bend a piece of paper's north end so it's touching its south end. You get a cylinder.

Bend a piece of paper's east end so it's touching its west end, but give it a half-twist first. You get a Möbius band.

Do both, and you get a Klein bottle. So, a Klein bottle is a Möbius band that started life as a cylinder. But don't try it - it won't work, since you can't distort the paper through 4D space.

Here's what makes it four-dimensional. Notice how it looks like the small end is pushing through a hole in the side of the fat end? It isn't. There's no hole. (If you tried to make a Klein bottle, you'd need to drill a hole.)

The bottle's "pipe" twists through four-dimensional space. It passes through the same 3D space that the fat end does, but not the same 4D space. When the small end gets to where the fat end should be, the fat end isn't there. It's a few dimensions over. Color is used in those two images to show the fourth dimension.

Another way of looking at it is to try re-flatten a Möbius band into a flat piece of paper, without tearing or disconnecting it. It doesn't work. If the band can't rotate through 3D space, up away from the flat plane of "standard" paper, it can't be a Möbius band. By the same token, a Klein bottle that can't twist through 4D space isn't a true Klein bottle.

So, let's say you had a real 4D Klein bottle. You could fill it with water, right?


If you poured water into it (at least, as I see it), it would spiral through the 4D piping and eventually come out on the outside of the small pipe!

Klein bottles, being 2D objects, lack something you need a third dimension for - volume. A Möbius band is one-sided because it's not a true 2D object. A Klein bottle lacks volume because it isn't a true 3D object. So there's no "inside" for a Klein bottle. It does have an interesting trait, though. Run soda pop through it and you'd likely end up with diet soda when it came out.

Why? Because a Klein bottle is "non-orientable" and doesn't quite work the same as standard "orientable" reality. If you don't go through it properly, you'll come out the wrong way around. It'll be like falling through a mirror. Left becomes right, right becomes left.

But why is your soda suddenly diet? Because life on earth isn't symmetrical. Water is symmetrical. The three atoms (H-O-H) are the same three atoms the other way around (H-O-H). But our molecules stick out in odd places, and so do the molecules of everything we eat.

We don't break stuff we eat down to atoms - if we could, we could eat almost anything and it'd be digestible and never hurt us - it'd be ripped into simple atoms before hurting us. Bacteria (botulism, salmonella, etc.) would be torn apart instantly.

Instead, our bodies break things into smaller molecules that we can cope with and build new cells with. But these molecules still have the odd bits sticking out just like ours do. Our bodies can't use atoms that stick out the wrong way - imagine a kid trying to play with Lego where half the pieces have bumps on top while the other half have their bumps on the sides. If he was lucky, he could manage something that looked like an M.C. Escher painting redone by Salvador Dali.

Your body wouldn't fare that well. It'd just ditch the useless molecules, and you'd have a very nasty case of diarrhea. You also still be hungry. Eat all the food you want, and never gain an ounce.

If you were lucky, it'd be the food that got reversed by a trip through a Klein bottle. Then you could just eat some normal food and maybe keep the Kleined stuff around for when you're hungry but on a diet.

But if it was the other way around and you fell through the bottle? It'd be interesting to see how long you could subsist on nothing but water. (Or how long you could go until you found and travelled through the bottle again, making left back into left and right back into right.)

The 4D world sure is full of pitfalls for us 3D folks, isn't it?

It's their fault.

(Major Spellbreaker spoilers below. If you've never heard of Infocom or Spellbreaker, then don't worry about it.)

Want to know where I first heard of these things? From games on my old C64 (and later on some emulators:)... the old Infocom text adventures. In Spellbreaker, sixteen magic cubes are merged into the sixteen corners of a tesseract in an attempt to rebuild the universe in the image of the tesseract's owner. (Maybe a nod to the idea that the universe is nothing but a giant tesseract?)

Trinity - a game whose front cover runs chills down my spine - contained a pergola that was shaped like a Klein bottle. You needed it when a clockwise-threaded gnomon (also the place where I learned that word) needed to be fit into a counter-clockwise-threaded sundial. (Good game, but I have a warning - with spoilers - for you if you wish to play it.)

The tesseract and Klein bottle are touched on briefly in hint books for those games.

Like I said before, get nostalgic and read a game. It's good for you.

The Archon

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Run along home.
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