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The Wall of Darkness by Arthur C. Clarke (1949) is a piece of Math fiction, usually quoted in semi-academic circles as illustrating the geometry of the Moebius band.

It deals with a universe with a single sun and planet but no other stars. The world supposedly always stayed light, with only a slight change with the sun dipping toward the horizon a bit in winter. Their planet has an inhospitably hot north, a temperate middle, and an extremely cold south. The south is barren, except for an insurmountable wall that stretches across the world at a point so far south that people can barely reach it during the summer, when things warm up.

There is a rumor that seeing what is on the other side of the wall will make a man go mad. But a curious, wealthy guy named Shervane decides he just has to do it anyway. In a massive project that takes more than 7 years, he has a series of platforms built, and he walks up on the wall, making sure his friend will blow everything up if something horrible happens.

Then he walks away from the sun, which is dimming behind him as he walks, and in front of him another sun appears and grows bright. As he approaches the edge of the wall, he sees his friend (whom he left behind) peering up at him.

Then they blow up the platforms, so that no one else can ever try to breach the wall again, saying it was necessary. He imagines in his mind another him blowing up the platform on the other side, but says of course that is impossible, since he is the only man in the world who knows for sure that the WALL HAS ONLY ONE SIDE.

The geometry of the Moebius strip is explained in the story as well, through a Professor of the protagonist.

But of course, as the author must have known well, the Mobius strip is two dimensional and any model of the universe (in his story) has to be three dimensional.

Now a Klein bottle is usually regarded as a higher dimensional equivalent of the Moebius band. While this in clearly a misconception (the Klein "bottle" is a two dimensional surface, just like the Moebius strip, only it seems to enclose a volume unlike the other), it could well be that the model involved here is the "inside" of the Klein bottle, or, equivlently, that the universe has the Klein bottle for its boundary. But it fails me to recognise how exactly the "wall situation" works with a universe in such a shape.

It has been suggested that the wall is built on the neck of the bottle. But a glance at the figure will show that a person who walks "across the top" of such a wall shall see just the region beyond, not the same side. The property of the Klein bottle (analogous to Moebius band having only one side and not two) is that it has only one of the parities left and right, and only one of inside and outside. For our wall to work in the intended way, it has to "go through" the "surface" connecting the false inside and false outside, which does not make any sense.

It has also been commented that the story is trying to sound more impressive than it actually is. But it could be just that the author intended no more than a mood piece focusing on the philosophy of our existence and our search into "the great unknown."

Can anyone point what the proper geometry of this universe is? Which means to recognise which known mathematical shape - a three dimensional surface - models it accurately.

The summary of the story and the comments given above are copied from here.

EDIT: We allow branched 3D surfaces as well when we say 3D surface, which seems necessary in view of the discussion following Kyle Jones' answer.

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If anyone wants to check it out, the whole story is currently readable on google books starting at books.google.com/books?id=HLz1g95AXDsC&lpg=PP1&pg=PA104 –  Hypnosifl Jul 17 at 11:17

2 Answers 2

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tl;dr The manifold surrounding Shervane's universe is called an Alice handle.

To simplify the explanation and ease visualization, imagine that our hero Shervane is a two-dimensional being who lives on a finite 2D surface, a circular region cut out of a plane. Viewed from above Shervane looks like the letter R as he moves around his world. Looking down we see Shervane sliding toward the edge of his world and oblivion. To save him, we add a long strip of space to the edge of his universe so that when he reaches the old edge he won't soar off into nothingness. We bend the strip around gently in the third dimension so that it connects back to the edge of disk where it started. Shervane's universe is now (roughly) a disk with a cylinder against its edge. Now if Shervane keeps walking he traverses the outside of the cylinder and eventually ends up back where he started except moving in the opposite direction.

Unfortunately, the trip through the third dimension has also flipped Shervane so that he is now his own mirror image. From above he now looks like the letter Я instead of R. His left is everyone else's right and vice versa. To keep Shervane from being flipped, we add a half twist to the strip we stuck to the edge of the disk so that Shervane gets an additional flip before he returns to the disk.

That half twist we added converts the strip we added into a Moebius strip, so now Shervane's universe looks like a disk with its edge tangent to the surface of a Mobius strip, this instead of a disk against a cylinder. To prevent Shervane from walking off the disk in another direction, we must surround the whole disk with these twisted strips. We also must glue the strip edges together so he can't walk off those edges. If you glue the edges of a Moebius strip together, you get the Klein surface. So if Shervane lived in Flatland, in order to make the story work his flat disk would have to have its edges surrounded by and tangent to a unified Klein surface. (If we didn't care about reversing Shervane's left and right, a disk with its entire edge tangent to the surface of torus would suffice.)

To make the story work in a three-dimensional universe, Shervane's universe would have to be surrounded by not a Klein surface but rather whatever the 3D analog of Klein surface is. The term for such a manifold is an Alice handle or non-orientable wormhole.

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The wiki article leads to an article about Alice Universes. Which seems to imply that the universe of the story is an Alice Universe. If so, the physics of an Alice Universe is mindblowing - matter are either matter or antimatter depending on direction of travel! –  slebetman Jul 17 at 4:04
    
But would a 3D analog of a Klein surface actually work? A flatlander living on a 2D Klein bottle wouldn't find that there was some boundary where, as they got close to the boundary, they'd come face to face with a mirror duplicate of themselves. From the page at geom.uiuc.edu/~teach95/sos95/big-picture the tiling for a gluing diagram of a klein bottle looks like this: geom.uiuc.edu/~teach95/sos95/big-picture/2k.tile.gif ...nowhere do you see an R facing its own mirror image. –  Hypnosifl Jul 17 at 4:11
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I guess the story doesn't say he met his own mirror image, but rather he walked along the wall for some time and came back to his friend. Still, for that to work in a Klein surface wouldn't he have to walk the whole length of the finite universe (a single tile in a tiling diagram)? I suppose you could imagine that the Sun was actually very small, and the planet itself a walkable size, and a sort of cylindrical shape so it could join up with itself on two faces of a gluing diagram like the one at geom.uiuc.edu/~teach95/sos95/big-picture/3k.glue.gif –  Hypnosifl Jul 17 at 10:38
    
Incidentally, the 3D gluing diagram in my last comment can be referred to as a "1/2-twist torus", see books.google.com/books?id=KhTJZG-U3ssC&lpg=PP1&pg=PA167 ...I also see some other pages on topology that use the phrase "twisted torus" for topologies like this, in 2D or 3D (i.e. take the gluing diagram for a 2-torus or 3-torus and flip or rotate one of the edges or faces). I would guess that an Alice handle falls into this category, since in 2D many authors seem to treat "Klein bottle" and "twisted torus" as synonymous, but I can't find a gluing diagram for an Alice Handle. –  Hypnosifl Jul 17 at 11:08
    
Having read the story at books.google.com/books?id=HLz1g95AXDsC&lpg=PP1&pg=PA104 , I don't actually think this answer is right--the way Clarke describes it, he doesn't walk along the wall parallel to its edge, but rather perpendicular, until he comes to the other edge and sees the mirror image of the stairs (and it's not a long walk, while his planet is huge). Clarke also implies that before the wall, people went mad--implying not just that they looped around the world and found their home with left/right reversed, but that they could actually walk right up to their own mirror images. –  Hypnosifl Jul 17 at 11:41

In both a Moebius Strip and Klein bottle, the "twist" and reconnection are in a higher dimension. The strip is 1D, and the bottle is 2D. It is a mistake to think the bottle goes "through" itself. A true Klein bottle has no hole, as opposed to the physical picture we see.

To make a region of 3D space "have no other side" you need to bend space itself and connect it twisted backward in higher dimensions. If you do that, you find yourself at the origin point eventually without turning around.

You can't make even a Klein Bottle properly, so there is no true representation of this space any more than there is for a tesseract, the 4-D analog of a cube.

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The question wasn't really about whether we can "represent" it in ordinary 3D space, just whether any "known mathematical shape - a three dimensional surface - models it accurately". –  Hypnosifl Jul 16 at 23:16
    
No normal 3D space can do this. Actually even a Mobius Strip isn't an exact analogue because on a strip you do not retrace your path until you reach the origin point. In the case of the wall, space twists you into a 180 degree turn and you end up walking back along your path to the origin. –  Oldcat Jul 16 at 23:56
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A Moebius strip is not 1D. A line is a 1D object. A flat plane is 2D. While a Moebius strip has only one side and one edge, the strip as whole is a 3D object. –  Stan Jul 17 at 0:05
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@Stan, the ideal strip is 2d, twisted in 3d. –  Dima Jul 17 at 1:56
    
@Oldcat - what do you mean by "normal" 3D space though? Are there any topologies of 3D space that are mathematically well-defined but which wouldn't be included in this category? –  Hypnosifl Jul 17 at 2:29

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