wrog

So there's this Arthur C. Clarke short story, "The Wall of Darkness" (1949). I read it as a kid and found it really haunting. Clarke does Haunting really well.

If you haven't read it already and want to go read it before I completely and totally ruin it, feel free.

(I found the whole thing by searching for "Trilorne" in google, which then gave me a google books hit; we'll see how much longer The Algorithm lets people do that).

But you've already had 70 years, so… onward…

Synopsis

The setting is a weird universe containing a single world with a single sun shining down directly onto its north pole, and then it gets progressively colder as you go south. In the deep south we have this Wall going around the entire planet, built by the Ancients with their Much Better Technology for,… reasons.

Evidently, in Ancient times the sun was warmer or something, the south more accessible, people went there, encountered Something Bad, apparently had sufficient disposable income, deemed this a Problem Worth Solving, and, hence,… Wall.

Fast forward 1000 years, time enough for actual history to get scrambled into legends, and now we have this guy Shervane who badly needs to know what's on the other side, inherits a fortune, and decides to expend it finding out. The rest of the story is about his getting his expedition together, vaccuuming up every engineer he can in order to build The Giant Set of Stairs going up the wall, plus all of the elborate precautions they take in order to keep whatever Monsters might be confined on the far side of the planet from escaping and Destroying Everything.

But it turns out the Ancient Legends are all curiously and uselessly vague. They only say that in the south lies "Madness" and then you have these Möbius strips popping up everywhere in religious writings and whatever else for no reason anyone can discern.

If you already know about the Projective Plane, then you can probably guess where this all is going.

Shervane's old mentor guy figures it out just as Shervane is setting off on the final stage of his journey, activating the winch to get him up the last 20 feet so that he can traverse the top of the wall. More discussion of Möbius strips. Shervane returns a few minutes later. It seems the Ancients got it right after all; now that I've seen it, once is enough and we're done. They blow up the stairs, return home, and everyone lives happily ever after…

… for, … you see,… the Wall only has one side.

End with trademark Arthur C. Clarke Mic Drop.

Making it Work

If a story is well-written enough, you don't stop to think about what might be wrong with it. And, arguably, Clarke was probably not going for full mathematical rigor anyway. The point is the journey, not the destination.

But if you want to be completely anal and have it make sense:

A Projective Plane is what you get when you take a disk (or a hemisphere, topologists won't care) and glue/identify the circular boundary to/with the [single] edge of a Möbius strip. The resulting surface, (*different* from a Klein Bottle, by the way), has no boundary, is not orientable, and is thus not something you can embed in ordinary 3-space.

Another, and for our purposes better, way to visualize it is to take a sphere and identify all pairs of antipodal (opposite) points. Everything less than 90 degrees away from any given point will look completely normal, i.e., you can make a completely sensible map of it and not see anything unusual; it's just that once you get outside of that hemisphere you'll discover that everything is "repeated" backwards on the "far side" of the sphere — this backwards copy being another way to get the Möbius strip effect.

… This all being a place where you can do spherical geometry with the additional advantage that the lines are now essentially half-great-circles, rather than great-circles, and each pair of distinct lines now intersects at exactly one point (i.e., corresponding to an antipodal pair on the sphere) rather than two, as is the case with great-circles on a regular sphere. Which you care about if you're bothered about wanting to satisfy Euclid's axioms and insistent that there be only one intersection point; you can actually do this.

What you lose is orientability: you can parity-reverse anything you want by taking it to the "far" side of the world.

… and this actually would be a not-unreasonable source of madness

E.g., if, in The Time Before the Wall, people journeyed south and kept going in the direction they'd reasonably expect to be further south, they'd actually/equivalently be travelling (as a backwards version of themselves) north from the equator on the "far side" of the world. And after crossing through or circumventing the Fire Lands at the (north/south/only) pole they would eventually arrive back at their hometown, but everything [street signs & layout] would be backwards from their point of view, with Hometown Doctor being confused by Traveler's heart on the wrong side, teeth being backwards from the dental records, and so on.

… and then Traveler dies in a few months due to not being able to digest the left-handed sugars and vitamins (although one might reasonably suppose that in a non-orientable universe like this, sugars/proteins would be less likely to evolve this kind of asymmetry -- not sure how much they even knew about this in 1949.

so, yeah, arguably a public health issue there, though it could just as easily result in thriving international market in left-handed foods if enough people did this

But the main thing Clarke really gets wrong is that Shervane would not, upon completing his (one-way) traversal of the wall, have emerged back at his stairway site, but rather 180 degrees away from it along the equator, in an equally uninhabited, inhospitable place with nobody there waiting for him and likely no way to get down from the wall. He would have had to travel considerably farther to notice anything unusual or figure out what was going on.

Which would have made for a much longer story.

WTF is Up With Trilorne?

The other nitpick is the wacky thing Clarke chose to do with Trilorne (the sun, and yes it's weird that he names the sun but not the planet -- I get that there's only one planet in that universe, but there's also only one sun, so…? )

In his description of what happens while Shervane is crossing the wall, Clarke has the sun behind him shrinking to a point and then disappearing as Shervane continues south. Apparently the wall was completely flat on top and several hundred yards thick, but Clarke also had this weird optical effect that makes it appear as a semi-infinite plane.

This was all totally unnecessary. With the sun shining down directly onto the north/only pole, Shervane crossing the equator would simply cause the sun to set below the horizon behind him and simultaneously rise in front of him.

Of course this then raises the question of how the 3rd dimension would actually work in this universe. But it's not actually hard, in a geometric sense, at least (I don't yet know what happens if you try to do GR here) to have stuff repeat (i.e., identify antipodes) at higher altitudes as well as on the surface of the planet and below.

There'd then be no need for any weird physics in the vicinity of the Wall. In fact there's basically nothing at all special about the equator. If you want to draw a complete map of this world you can choose any great circle you want, map all of the stuff inside of it, and then you're done. The map will then have the property that if you walk off of it in any direction, you'll immediately appear (parity-reversed) on the boundary 180° away from where you left, heading back into the circle.

…though it would still be the case that a wall built on that circular boundary would only have one side.

The only actual singularity would be at the center of the planet immersed in a glob of molton iron. No idea how that would actually work; but nobody would ever go there so we don't need to care…

Hell, Clarke could even have put the sun in the same plane as the equator to vaguely try to give the planet a more ordinary day/night cycle. But I think I know why he didn't.

Seasons?

The real problem, see, is, no matter where you put the sun, it shines onto the entire planet at once, because the hemisphere is the entire planet. Meaning there'd never actually be any night. The sun would always be immediately rising from the horizon opposite from wherever it sets. And if he'd put the sun over the equator, he'd have had to explain why they didn't have day and night right at the beginning and thus given the game away up front.

A more interesting question is how the year works. Which I think Clarke just punted on.

He has this vague waffle about how the sun is moving in a small circle and "winter" is when it gets lower in the sky. But that really doesn't make a whole lot of sense, since if the sun stops being over the pole, then it's over someplace else, that place will get hotter as the pole gets cooler.

Also, from a mechanics point of view, I can't see how the sun will be doing anything other than orbiting the planet. That is, there's clearly gravity, and if we assume it works in the usual way, then we get elliptical orbits. It's perhaps a bit weird having the planet not move, though you can justify this by saying any force on it from the sun gets cancelled by a corresponding force from the antipodal sun (i.e., the duplicate copy on the "other" side of the world). Or we just treat it as a frame-of-reference thing. And then it's: planet attracts sun, inverse square law, ellipse, done.

(Or we just do regular physics on a universe with everything duplicated and rely on the symmetry to maintain itself.)

And since the entire freaking universe is just one big 2-body problem and there's nothing else, you can't use any 3-body shenannigans to get out of this.

Which then means if the sun is starting directly over the pole, it heads due south along some sky meridian, then hits the celestial equator, at which point (= vernal equinox which will be the same place as the autumnal equinox) the (antipodal) sun will be heading north up the "opposite" meridian back to the pole. If the sun takes a year to do this then we have alternating 6-month seasons with polar lands being hot vs. equatorial lands being hot -- Clarke puts all of the civilized areas on this planet at around 45° latitude, so it'd be kind of a tossup which you want to call summer vs. winter (spring and fall being the actual hottest seasons for the 45° folks).

This corresponds to an "axial tilt" of 90°, which is probably the most interesting case. A tilt of 0° has the sun permanently stuck over the equator (boring) and tilts of up to 45° put us back in the more familiar situation where the equatorial regions are tropical/warm and the polar stuff not so much.

But, yeah, just not seeing how the sun can hover over the pole.

If you're thinking it's a tide-lock situation (like the way the moon is locked so that the same side is always facing earth, so the earth is always the same place in the lunar sky if you're on that side and never visible at all otherwise), that won't work. Having the planet's rotation axis keep changing to follow the sun as it moves in its orbit (i.e., so that the sun is in orbit but the pole stays pointed straight at it) would violate conservation of angular momentum. In the case of Earth's moon, the tide lock isn't changing the moon's rotation axis; it's slowing down the rotation rate to 1/month by grinding up the inside of the moon until stuff stops moving in an Earth-radial direction.

The only scenario in which the sun stays over the pole is the one in which the sun is falling straight towards the planet and, well, that's not what we call a long-term viable situation.

Ok, done now.