A 2D surface that follows this logic is a sphere

@Slarty not quite, on a sphere if you walk off one point you appear on the opposite point, whereas I take this explanation to be you re-enter as the point you left from.

A sphere has no edges. If you "walk off" it you're no longer on the sphere and you might re-enter at any point.

@Slarty, nope. on a sphere you can take 2 90 degree turns and end up in the same place. You can't do that on a looped square.

I.e. you cannot mash this shape into a sphere while maintaining the looping: oi64.tinypic.com/scz33p.jpg

A looped square (if similar to old computer games like Asteroids) is equivalent to a torus. One which "turns you around" next to the edge is . . . I'm not sure. That's not really topology but some other kind of magic

@Nacht Yes true about the 90 degree turns, but I can't find any reference to the topology of a "looped square"?

@a4android but what about the SIDE edges of the strip?

Well, this is the surface that follows this logic. By the way, in King's Quest games, this mechanic was called "magical law of containment".

@a4android oooh - looking at that makes my head hurt ;o) it looks like a torus shape to me, or does it have a twist in it somehow? Are we talking a klein bottle type topography or something different again?

Being "inexplicably turned around" so that you're at the same point but pointing the opposite direction is not something you can do with topology (at least not for a nice nonsingular space).

@Slarty You are right only from a topological way, but not from a metric point of view. We can discard Earth being flat and bounded by taking some simple measures.

@a4android, does that mean that if you cross the edge, you come back as a mirrored version of yourself? That's the only way to keep it mathematically consistent, but would be weird as hell for the people living there.

@ArturoTorresSánchez Good point. I assume people never reach the edge. Perhaps they are, but the space might have a double twist to reverse their mirror image.

@JustinThyme The space is the equivalent of a Moebius strip. It's not actually a Moebius strip. Sorry no sides.

@Slarty The topology of the space in Clarke's story is explained by analogy with a Moebius strip. It could easily be another topology like a Klein bottle's.

@DanStaley The phrase "inexplicably turned around" is a natural language description of how this seems to happen. It shouldn't confused with how it actually happens. Although if you reach a point where you were now pointing in the opposite direction, inexplicably, that could be one way of working this. Nice idea. Thanks for suggesting it.

You can stitch two halves of each side of a square but then you end up with four singularities where everythingcan be seen twice, one in the middle of each edge. Perhaps people built special buildings around these spots?

The adaptation of this solution depends on weather or not you ever WANT your citizens to come to the edge. It seems to me that this solution just produces a flat sphere. You walk in one direction and end up in the same place, just as if you started walking around a sphere and ended up in the same place. I wonder what the difference would be, except that technically I suppose in this world if you could see far enough straight ahead (in any direction) you would see yourself. Only it would be delayed, because of the speed of the light. Could produce interesting plot twists. But if you looked UP?

But it does beg the question be asked, 'Exactly HOW would you create a map of this place? If every straight road led back to itself? And how would you determine the compass points (directions)? What would the star map look like, for astrological navigation (even on land)? I beg to differ with some responses - I can not see how taking two 90 degree turns wouldn't cause you to end up back in the same place.In fact, taking a single 90 degree turn ANYWHERE would get you back to the same place where you turned. Taking ANY degree of turn ANY place gets you back to the same place. Like a globe.

A torus is not the only way to arrange this. The surface can also be metrically flat with the topology of a klein bottle (moebius twist in one dimension, but not the other) or a projective plane (with moebius twists in both dimensions).

@JustinThyme You raise interesting points. I had assumed, as does Clarke, this spatial property exists as a shell around this world. Thus, preventing anyone reaching its edge. Space inside this outer shell would be normal, flat space. Looking up could be interesting even so. One small correction: it's celestial navigation not astrological navigation: "Cast a horoscope to plot our course home." Mmm. That's a novel idea indeed.

@LoganR.Kearsley Those are interesting alternatives. The topological argument was always by analogy, even in Clarke's story, so it's precise configuration was such to make the story work. The idea seemed fit for purpose for the OP's flat world.

@ a4android Actually, I did mean the star map for astrological navigation. I have a difficult time imagining what the zodiac signs would look like, and how do events 'enter' and 'leave' the various signs. 'Cancer ascending'? But yes, as you point out, celestial navigation would be tricky too. How WOULD this planet move with respect to the Universe?

@Pere I can't believe nobody has mentioned the Poincaré half-plane: you would not be "turned around" but you could walk endlessly never reaching the edge.