Planet with infinite overlapping surface on an axis, problems of the concept

Question

I’m beginning to build a world for a story I want to write. The world consists of a planet, roughly earthsized, with a couple of natural satellites (moons), and roughly same orbit and a similar star. The difference is, if you travelled in the east-west axis you could advance infinitely without ever reaching the same point. You could tell that you’ve completed a “lap” by watching at the stars. But if you went to the poles you would find… A bizarre nexus that I’m yet struggling to find a good description for. Thing is, you could travel several “laps” in a single day if you are near the poles, or could take you more than one lifetime to do it on the equator, depending on your technologic level, access to vehicles and geography.

This “overlapping” where infinite places occupy the same space is limited to the planet. In fact, every “lap” receives light from the same sun and sees the same moons and stars.

As you can see I have the concept pretty much tied up, but I struggle with several caveats. I am aware that there is not a definitive “correct” answer as this is just outside the realms of real physics, but I guess we can find a workable solution.

A) How would this be perceived from space?

I have a civilization that uses the moons as “repeaters” to send back radio waves and communicate with people on all laps… And a single Sun is lighting every lap… So waves coming from outside might affect all laps simultaneously… maybe? While this “lap” propagation does not occur if the wave is originated inside the atmosphere… maybe? Just making wild guesses here.

That makes me very confused, as from the space you might get an infinite amount of reflected light from the Sun… But we don’t want to blind and fry everything around our planet, so maybe I should go for another explanation here…

B) How does one ENTER into the planet’s atmosphere, and where exactly?

My guess right now is that solid matter can’t enter into the planet as it would be dispersed at an infinite number of laps, thus getting fractioned in infinitely small infinite fragments. So, magic aside, space travel would be out of the question (you can’t get back…). But maybe there is another concept that works better here…

Added info:

There is magic in the world, but the kind of magic accessible to sentient beings is not “world changing” or vastly defies physics. It’s mostly things that affect their own bodies, so no pocket dimensions, teleportations or things like that. The world is infinite, but I’m not giving it infinite options. The planet will be earthlike almost always. Maybe there are some areas inhabited by different kind of organisms, or an especially volcano-ey and the atmosphere is full of other particles, etc… But I’m not going to have areas where everything is made out of new “elements”, or the earth is made of chocolate, oceans are made of flubber and deserts are made of cinnamon…

This also extends to civilizations. As per the story I want to tell, there is one civilization capable of “getting into space”, but it’s more of a steampunk kind of technology level than ours… We are going to assume that steampunk civilization is simply the most technologically advanced in the world, that they are the first ones to get to that level and the world (which has been crafted by another party) is too young to have reached any more than that. The reason for this is, if I let there be more people that can throw junk into space, then in infinite possibilities there would be an infinite number of civilizations that can do it, and then space (that is out of the infinite physics-breaking world) would be full of infinite junk. Also, in some of those infinite civilizations there might be infinite maniacs crazy and powerful enough to destroy the planet…

You see? Infinite gives me headaches. But I have to keep it infinite for the story to work. And no, so big that seems infinite to us doesn’t work either.

I’ve been talking of “laps” but to clarify… There is not a “breaking point” of what is considered one lap or the next one, there is not a magic “line” following any meridian, so every person or group defines the start and the end of a lap from their point of origin (I guess)

There are similar questions about infinite worlds in this SE (likte this one, where one of the answers suggests a world exactly like the one I'm planning), but I think my question has not been answered before. If it has been, please notify, and know that I am sorry for not having been able to find it before making my question. Also, I didn't know wich tags would work better for this... If you have any suggestion in that regard, please comment :)

If you need any more info or clarification, just ask me in the comments and I will try to fill the gaps.

Thank you.

Answer 1

Several simple solutions to your singularity:

1. There's some kind of force that pushes things away from it. The closer you get, the harder it is to advance further, requiring infinite (and therefore unachievable) force to hit the center.

2. There's time dilation near the singularity, black hole style. You can walk in, but it takes infinite time (relative to the rest of the world) to get there, so nobody's ever done it.

3. There's a black-hole style vortex at the singularity. Anything near the vortex gets disintegrated and either feeds whatever energy source warps space near the planet or gets ejected into space like relativistic jets, preventing people from trying to come in from above/below the poles.

In all cases, the basic answer is "we have no idea what's at the singularity", with a simple explanation of why we have no idea. It also puts a maximum limit on how fast you can traverse laps. The vortex and repulsion force ideas limit speed based on how close you can get and survive. With the time dilation idea, geometry speeds your laps per day as you get close to the pole, while time dilation slows it, and there's some point where those effects combine give you maximum speed.

• A. How is it perceived from space? Well, that depends. If every lap emanates stuff like normal Earth, then it would be perceived as an infinitely bright, infinitely dense region. Of course, infinite density means your entire world is essentially trapped inside a black hole, so it looks like a black hole to everyone else.

  Alternately, there could be some kind of selective process by which stuff emanates. Anything trying to escape gets dispersed and only an infinitesimal amount makes it through per lap (as above, perhaps the rest is used as fuel for the warping), with a finite total radiance. From space, it would therefore look like pure static (any one point on the globe has light coming from every lap). Because there are infinite laps, the infinite static would average to a constant color (though perhaps the poles, with lower populations, would tend to be darker).

• B. How does one enter the atmosphere? If it's a black hole, you don't enter it; you just get sucked in and ripped apart. Because of time dilation in the black hole, your planet sees a finite time pass for an infinite amount of external time. All the light in the universe therefore converges onto the planet. If there's infinite light, the sky is just a constant, solid color. If not, the sky is just black. If the exterior universe has a finite or non-looping lifespan, the sky is just black.

  If the escaping stuff gets dispersed, then perhaps incoming stuff does too. Except then the planet would never get sunlight (the light would be distributed evenly, and each lap would get an infinitesimal amount of light).

  So maybe everything entering the atmosphere gets duplicated across every lap. This allows starlight and sunlight (and radio transmissions from the moon) to look the same from any given lap. It also means entering the atmosphere (or wherever the division between normal space and infinite space is) duplicates you infinitely. And further means you can never escape, unless you somehow synchronize all infinity of your copies to move out at the exact same time in the exact same way.

• C. Additional comments. I can't think of any reasonable way to connect an infinite space to a finite space without either destroying everything that tries to escape or infinitely saturating the finite space surrounding the infinite world.

  Also, an infinite world means infinite races have developed an infinite variety of technology, and there would be infinite civilizations capable of space travel (if such a thing were possible at all), as you suggest. But there's really no way to be right at the cusp where the very first civilization travels to space, because in the very next moment of time, infinitely more follow.

  The only thing I can sort of think of is if there are an infinite number of finite spaces outside the planet, and entering/leaving the planet basically puts you at a random lap/space relative to where you started. If it were truly random, things would be very chaotic.

  But maybe it's a normal distribution, so you generally end up finitely close to where you started. People leaving the planet would tend to be separated, though a few might end up in the same space. Perhaps enclosed objects, like spaceships, end up in one space (otherwise people would get separated on a molecule-by-molecule basis and die pretty quickly).

  Then when you re-enter the planetspace, you end up within a few laps of where you started. Mostly. By re-entering near the poles, you could quickly determine which lap you were on because there would be labeled (probably numbered) signs along the way. If you're from lap 30, and you land on lap 68, you know to head west (or east, depending on convention) 38 laps, then fly south (or north, depending on which pole you entered) to get home.

  Note that the sky would be very fuzzy, since you'd see light from a number of nearby spaces quite readily. Depending on the standard deviation of the normal curve, you might see one sky mainly, with others being almost invisible, or you might see dozens of skies at once.

  This would allow space travel, and solve the problem of infinite space-faring races (there would be an infinite number, but they would practically never converge to the same space or lap). But it requires infinite spaces which isn't what the question asks.

Answer 2

To simplify things I will remove one dimension from my explanation. Imagine a 2-dimensional universe: it's just an horizontal plane, with planets as circles. YOUR planet is actually a sphere, with its centre lying in the plane of the universe.

To build the planet imagine to take a wire. You start from the north pole, make a first loop and then you start a second loop to the right of the first one. Once you reach the south pole you cross the first loop, thus when you reach the north pole again you are on the left of the first loop. There you go on the right again, and again on the south pole you go on the left. The crossing does not overlap the existing loop, but compenetrates it. You will eventually close the sphere, but since each loop has 0 width it will require an infinite number of loops to do that.

From outside you can only see the intersection between the sphere and the universe: it's a circle, thus a perfectly legal planet, but it's made of 1 point for each half-loop. From the planet the sky you can see is just a line (the intersection between your loop's plane and the universe), but the closer you are to the pole the less you can see: at the poles you cannot see the sky at all. Only from where your loop intersects the universe you can leave your planet, 'cause from everywhere else you would go outside of the universe.

If you apply a similar concept to a 4-sphere planet in a 3D universe you can get something similar to what you want, although I think that the equator is the intersection between a loop and the universe, and you should walk through the pole to advance... But my knowledge of geometry is limited to 3D Euclidean spaces, I must leave to you how to figure out the mathematical details!