# Is there a good geometric analogy for a 'trap' dimension that you can stumble into easily but is very hard to exit?

I have a magical children’s fantasy land, which is bordered by mountains on its northern side. I want to make it so that anyone travelling too far north is likely to enter one of many trap pocket dimensions which are easy to walk into but extremely difficult to exit; once you’re inside, the outside edges wrap around and you find yourself back where you started if you keep travelling. This is likely to be fatal, unless the traveller happens to be in a pocket with a stream and a food supply and shelter and enough supplies to survive.

Now, it’s a magical kingdom, and things don’t have to obey the laws of physics, or be 100% coherent. But I do want it to be conceptually coherent enough that I can give narrative explanations for plot events and give magical locations rules that are applied relatively consistently, locally if not globally. You could say (correctly) that I have local ad hoc physical laws specific to individual locations.

So, what I’m wondering is, is there a geometry or geometric analogy that can explain/model the rules of a location where you can enter in on foot but not enter out (except at one point, if travelling in one specific direction or with a particular velocity/acceleration), while being able to continue to travel in one direction for a long time or indefinitely?

EDIT: The first comment here is quite possibly a sound suggestion: A fish trap (I had the inside of a plastic bottle in mind, which I think is the same). It's easy enough to analogise the situation as, "You feel like you're walking straight ahead, but you're actually walking around the inside of the walls". The problem for me with this suggestion is that so far I don't know what using the exit looks like in-universe, as opposed to walking across the exit without actually going through it.

• Fish trap or fyke? Commented Aug 14, 2022 at 6:59
• @Trioxidane You're analogising from a geometry with one less dimension, right? This is where I get stuck. How does a fish trap work regarding being able to travel indefinitely in one direction? Is it a case of you feel like you're moving in a straight line but you're actually curving around the walls? And in universe, how does the mouth manifest? How does one exit through it?
– user86462
Commented Aug 14, 2022 at 7:04
• Honestly speaking, I would've made their entrapment illusory instead of spatial. Blindfold a man and tell him to walk in a straight line across a field. Odds are he's just going to keep walking in circles, wandering when the heck he's going to reach the end of the field. If left to wander eternally he would eventually reach the end and be allowed to take the blindfold off, but until then his sense of direction is completely off and will result in his remaining in the field. Basically, have these areas subtly alter the perceptions of those within them to make them think they're making progress Commented Aug 14, 2022 at 7:08
• when in actual fact they're not making progress at all and just keep coming back to the same spot. This'll remove the need to explain why the pocket dimensions don't fill up endlessly with mass going in and never coming out. Commented Aug 14, 2022 at 7:09
• This is pretty relevant: preview.redd.it/… Commented Aug 16, 2022 at 18:12

A children's book needs an illustrator, and that's not me. But let's try:

At the lower right (east), we have one far upper fringe of "The Normal World". It is already decidedly abnormal, because it's gotten separate from other regions east or west that go to their own, similar, traps. The person following the blue trail at bottom moves off the east edge to the west, keeps going ... eventually (question mark) notices he's going due east. The system tried to save him! Yet he persisted, going once again true north. Eventually he reaches the "hole" of the fish trap - a place where there is only a short distance east to west before you're back at yourself. Don't shoot your bow and arrow at things in the bushes or you might hit yourself! Eventually though, he's out to the point where the right side connects to the left side of the pocket world. He keeps going for some time, then turns back again south at the exclamation point -- which is justified when he reaches it again!

The dotted line at the south of the pocket universe connects to the dotted line at the north. To borrow Sean O'Connor's comment -- the area on the left is the pocket, the area on the right is the distorted near pocket area, the traveller starts by the question mark and is being wrapped east to west each time he touches a green border line.

But for fun, let's add a mismatch: the north edge of the wrap-around is higher. That means that "!" might actually be on a stream, down which you might take a raft until you reach your starting point again. The constant downward flow of water might be an energy source for heating the pocket dimension if the Sun never makes it there. All material eroded away simply comes back again above, so the stream never digs a canyon downward. Which is convenient for the story, since I imagine that downward must pinch inward, with less and less distance across from east to west, until there is no where left to dig down at all - your pick axe would hit itself, striking from another angle! The gravity is frozen in the shape of space, emerging from the bottom point without needing to have an actual source.

The sky, however, is open - there is some "out" from this pocket dimension by going upward. Whatever you like could be up and out there. The sky is dark, and the darkness allows the landscape to radiate the heat produced by perpetual motion. Anti-solar cell photosynthesis (not making those up!) would allow plants to survive, if not necessarily thrive, under that dim sky.

• It took me a couple of goes to digest this. Your illustration talent is similar to mine! But the more I read it, the more I liked it. The vertical offset at the wrapping edge is inspired. Thanks!
– user86462
Commented Aug 16, 2022 at 7:04
• For anyone else who is confused, the area on the left is the pocket, the area on the right is the distorted near pocket area (another good idea, as it gives a warning, and it's fun in itself), the traveller starts by the question mark and is being wrapped east to west each time he touches a green border line.)
– user86462
Commented Aug 16, 2022 at 7:16
• @SeanOConnor - thanks! Left, right ... sigh. Fixed now. Commented Aug 16, 2022 at 15:55
• I imagine you'd have some weird-AF visuals at the portal since the same rules would apply to light rays Commented Aug 16, 2022 at 20:17
• @user253751 If you look in any direction you see lots of trees and bushes. In that sense it's fairly normal. But you're right that at the question mark you might see a Sun to the southwest and due south at the same time. However, it's also possible that the "fish trap" at right closes over on top, and doesn't have a direct view of the Sun at all - looking up you might be seeing the ground you're standing on, if you can get a clear view, but it might be quite dark with only a few distant images of your torches far off in the sky. Commented Aug 16, 2022 at 22:31

## The Hyperbolic Plane/Space

Hyperbolic geometry grows exponentially with respect to radius. This means that it's very easy to get lost. Anyone who has played HyperRogue is familiar with this: the final boss is

the challenge of back tracing a mere 100 steps accurately. It's only really feasible if the player has a way to create a trail, since there are trillions of destinations you can get to in 100 steps

so in a sense, the world you want already exists; it's HyperRogue.

• You beat me by 39 seconds :) I think escaping the Haunted Woods is a closer match to what's being asked for is than getting from the key back to the Orb of Yendor, though. Commented Aug 14, 2022 at 19:53
• I'm interested. How many play hours would I need to sink into it to get that far, though?
– user86462
Commented Aug 15, 2022 at 6:40
• @SeanOConnor the game does take a long while, but even after two hours or so you'll already have an impression of the hyperbolic space's quirks. — BTW there's also another game that does 3D hyperbolic geometry. (I haven't played much of that one; it actually had a bit to much blabla in it for my taste, distracting from the hyperbolic geometry.) Commented Aug 15, 2022 at 12:16
• @leftaroundabout the effect of hyperbolica is a bit different: the geometry is the same, but it has a lot of navigational cues. It's still a bit disorienting, but not nigh impossible like in HyperRogue. Commented Aug 15, 2022 at 13:51
• @SeanOConnor although getting to the final challenge of the game is difficult, there's an easy way to simulate it: modify the terrain (this is easy in Land of Eternal motion and Alchemist Lab), walk 100 steps away (through a land where you won't leave a trail), and then try to get back. Commented Aug 15, 2022 at 14:58

Spheres.

A sphere is a 3 dimensonal object. We can walk on a sphere and we will eventually wind up where we started. That is how our Earth is.

In your world, other 3d spaces intersect with ours through the 4th dimension. These spaces contain spheres and humans can move around on these spheres. When 3d spaces intersect / overlap one can move from one to the other.

The trick is that the intersection points are in motion through the 4th dimension. The place where they overlapped and you came thru is not there any more. The overlap might not exist, or might be somewhere different in the respective planes. Or might be with yet another 3d plane. I like the idea of the character unwittingly moving on to successively smaller spheres with the same terrain. Finally she sees someone in the distance, facing away. She shouts but the person does not turn around. She runs after the person but the person runs away, then stops when she stops. She is looking at her own back.

For the childrens book, the analogy is walking off the train platform onto the train. The 3d spaces of the station and the train temporarily interect. When you are on the train and turn around the door you came through is closed. It will open again but it now connects somewhere else because the train moved. You can wait on the train until you come back around to the station they got on.

Your characters will need something to orient themselves. Maybe a map like for a subway system? I could see that being an excellent cover art. Or something like a 4d compass that will show the direction of large 3d masses nearby through the 4th dimension. That will lead them to where the planes might overlap enough for them to move through to somewhere different.

I am reminded of the map from Time Bandits.

https://geektyrant.com/news/2010/3/1/must-have-time-bandits-time-portal-map.html

• The problem is that the damage done by a moving portal (on the ground) would invariably leave a trail, unless it was something like sand maybe. Commented Aug 14, 2022 at 23:13
• @PyRulez - that is a slick idea. Maybe the damage occurs at predictable intervals / spaces and by examining gouges in the forest and the forest regenerating you can predict where and roughly when the next one might come. Commented Aug 14, 2022 at 23:57
• You might find that the intersection points are actually quite predictable and tend to produce circular regions of terrain which are different when merged with another location. Fairy-Rings come to mind as a good example in myth. Commented Aug 15, 2022 at 10:40

Your question is a little confusing because you seem to be asking for 2 different things:

• You want a dimension that is easy to enter but hard to exit
• you want a dimension that wraps around itself so that you end up at your starting point after walking

You also mention that you want the solution to be geometrical in nature, but I'm not sure which of those two aspects has to show up geometrically. What confuses me is that, strictly speaking, those two things are mutually exclusive. You enter from point A. If you walk 100m and walk back where started, you are back at the entrance. You can exit. So you'd need to have a "trapping corridor" after your entrance (that doesn't loop you back to the entrance), and only then, have the loop effect. I will treat of those two things separately

I) For the one-way ticket thing

You can not achieve that purely geometrically. As far as geometry is concerned, if you can draw a path from A to B, it is also de facto a path from B to A. To make it harder to walk one way than another, you will have to introduce other factors than pure geometry (forces, limitations of the human brain, etc..). Here are some of the most straight-forward propositions:

• a slope/drop:

As we said, introduce forces, cause you need to introduce something more than geometry. Gravity is a good one. Everybody loves gravity. It's just so adorable. Place a steep slope right after the entrance. Like a slide. Boom, the guy is trapped. If you wanna be extra safe, just make it a drop instead. I felt this level of complexity definitely needed an illustration:

aka an n/p junction. It speeds you up one way but pushes you back if you try to go the other way. Put that right after your entrance

• A maze:

you enter, you walk around a bit along a path, and you don't notice that there are some other paths coming from another direction that seem to connect to yours. When you turn back, you notice, and you forgot where you came from. If you like to get fancy with geometry terms, feel free to make it fractal to have plenty of branching

• a door:

Open it then close it. Done. More generally, any moving part in your geometry can achieve the same goal (a bridge that collapses prevents you to go back, a straight line that wraps around itself locks you in a loop, etc...)

As you see, those few solutions to trap you into the dimension are not very complicated and I'm sure you could come up with a dozen more. For the other aspect, it's not much harder.

II) for the self-wrapping stuff

• I introduce to you, earth:

Or any old sphere. Or any old variant in lower dimension like a loop. Or a cylinder. Or really any hypersurface where you can draw a self-intersecting line. If you walk in a straight line on earth and you happen to be able to walk on water (it's a good move to start a successful cult, look into it), you end up where you started

• Alcohol:

Give me enough of that and I promise straight lines will become a foreign notion to me. More generally, have something that trumps the senses of your victim to make them run circles

• Have the exit moving:

You can have just a regular plain old euclidian geometry, but if the only guide is that one shining light that is the exit, and this light happens to be moving in a circle, then you're just gonna run circles after it.

If you combine any of those solutions, you have a trap that also gets you back to your starting point

If you're willing to give up "once you’re inside, the outside edges wrap around and you find yourself back where you started if you keep travelling", then you can satisfy the rest of your question's requirements by having your travelers end up in a dimension with hyperbolic geometry, which is much "bigger" in some sense than Euclidean geometry is (e.g., the circumference and area of circles both grow exponentially with radius, unlike in Euclidean geometry where they're linear and quadratic, respectively). In hyperbolic geometry, it's basically hopeless to ever end up somewhere you were before unless you retrace your steps almost exactly (you'll instead just keep ending up in new places).

You can get a feel for this firsthand by downloading the game HyperRogue and venturing into the Haunted Woods. The only way out of that land is the way you went in, and once you're far enough in that you can't see the exit anymore, you'll find that it's basically impossible to return unless you somehow marked your path.

• "If you're willing to give up 'once you’re inside, the outside edges wrap around and you find yourself back where you started if you keep travelling'" It's actually not too hard to fix this: you can use a quotient of the Hyperbolic plane instead. Commented Aug 14, 2022 at 19:59
• @PyRulez That might work. I wonder how you'd strike the balance between quotienting enough that you do eventually end up wandering back where you started, without quotienting so much that it becomes easy to find the exit point. I'm thinking in particular about the Minesweeper minigame, which is quotiented so much that you can see the entire world several times over from anywhere. Commented Aug 14, 2022 at 20:14
• No such tradeoff is necessary. You had make 2n+1 exists for absurdly large n, where 2n wrap around to each other and only 1 connects to the outside world. Commented Aug 14, 2022 at 20:26

# Möbius strip

A möbius strip is an object with just a single side. However you move over the edge,you will move along the strip on both the 'back' and 'front' of the strip to end up in the same place.

An excellent example is shown here:

Möbius strip gif

By Sketchplanations

There are many more complex shapes possible for such a strip, but this is the easiest way to show it.

Now imagine there is one tiny strip leading to the Möbius dimension from your main dimension. Much like a fish trap you can have the entrance become smaller. Then they arrive unknowingly on the Möbius dimension. They can move around and will end up in the same place if they walk straight. There is a chance to find the tiny strip leading to and from it, but as the Möbius strip can be much wider it might be hard to stumble upon the small entrance/exit.

• If I understood your answer, an equally wide circle would work just as well in terms of trapping efficiency. And a simple straight line, better (since your assumption is that they always walk forward, and not just come back where they came from). It feels like you're proposing a Moebius shape cause it's cool, but it doesn't really serve the purpose that is sought after in the question Commented Aug 14, 2022 at 10:55
• "There is a chance to find the tiny strip leading to and from it" One could imagine a scenario where traversing this tiny strip destroys it, leaving you truly trapped. Commented Aug 14, 2022 at 19:03
• A 3D analogue would be a Klein bottle. Commented Aug 14, 2022 at 21:37

Anything with a preferred direction of motion will do.

For example there several plant with hair on their stems: if you slide your fingers in the good direction of the hairs, you feel like a smooth velvet, but if you do in the opposite direction you will feel a spiky resistance. Those plants use those hair to make it more difficult for bug and pests to climb up their stems.

Same can happen if you try to move in a cane groove: advancing in the direction where the canes are bent is way easier than advancing against the bent canes.

The same concept is applied in the anti-rape condom, with its jagged teeth which get stuck in place when activated. So, easy to get in, it takes a surgeon to get it out.

• Lol, now I need another question, "What real life object is a sound analogy for an anti-rape condom like dimension that could actually go in a children's book?". So what does being in an anti-rape condom dimension work when you stumble into it? If you turn 180 degrees as soon as you think you're in one, you can get out? Or if you don't travel in on an orthogonal axis or something (that could work)? I'd prefer the exit strategy to be harder than saying "bother, I need to turn around".
– user86462
Commented Aug 14, 2022 at 7:18
• @SeanOConnor "What real life object is a sound analogy for an anti-rape condom like dimension that could actually go in a children's book?" - How about a chinese finger trap with barbs inside? It's easy to put on, hard and painful to take off. Commented Aug 14, 2022 at 19:05

Your description reminds me of a wasp trap:

You are guided to enter in the middle of a large space, and once you wander away from the middle, it becomes very difficult to find your way back out.

It's like a circular railroad track, just in all compass directions

With one inbound track. As soon as the train is on the circle, it won't ever find the way out because a train is always going forward.

Yeah, it's an incomplete analogy. Trains can go backwards after all.

The full story is: There's an additional direction that humans can't perceive; let's call it "feyward". Entering a circle happens if you go "feyward", to leave them you need to go "anti-feyward". Thing is, humans simply cannot go anti-feyward, just as they can't go back in time.

• +1 for "Feyward" as a good term for an extradimensional direction. Commented Aug 15, 2022 at 10:42
• @Ruadhan The term was inspired by (or stolen from) the writings of Jack Vance, I believe. Commented Aug 17, 2022 at 8:59
• Presumably, you could go anti-feyward from the human realm as well, I think of it as kind of like the complex number plane - real numbers are at 0 on the imaginary axis - that's like our normal boring dimension, but you can go up or down on that axis to access either the positive feyward direction or the negative. Up to you what "positive" and "negative" mean in this context... Commented Aug 17, 2022 at 15:00
• @DarrelHoffman Can you explain how the complex plane is a more helpful analogy than Euclidean two-dimensional space (i.e. adding another direction, whether it's called "feyward" or otherwise)? Commented Aug 17, 2022 at 17:00
• @toolforger Just as the 2D complex plane is an extension of the 1D number line, and you can go above or below it based on the imaginary dimension, the 4D "feyspace" would be an extension of the 3D physical space. But again, it'd have to go in both directions. Let's say humans can go feyward from humanspace to get to feyspace, but maybe only the fey can go anti-feyward to get back, then the fey could also keep going anti-feyward past humanspace to get to anti-feyspace. (Perhaps the only way for a human to escape, or visit anti-feyspace, would require assistance from a friendly fey?) Commented Aug 17, 2022 at 17:33

You don't need a place's outer edges to wrap in order to get someone trapped. As long as the place is either featureless, or of its features all look the same in all directions, people can get lost im them. Happens more than often with people in deserts and forrests.

If the passage to the pocket dimension is not something you can notice with your eyes and the pocket dimension is in a forrest, even if a small one-hectare piece of woods... good luck finding your way back.

Just make it so that the pocket universe is curved, and that will mean that walking in any direction will eventually lead you back to where you started.

# Equilibrium:

This made me think of stable and unstable equilibrium. The equilibrium of your two universes is opposite. staying in the primary universe as you travel beyond the edge is like an unstable equilibrium - possible, but any variance from the perfect course means the path back becomes rapidly near impossible. But once inside the pocket universe, the equilibrium becomes stable as you attempt to leave - it keeps self-correcting to the default state (being in the universe), and getting out requires an extraordinary effort.

# Needle in a Haystack

If there are many, many interconnected pocket dimensions and they don't look very different to each other, then entering one is easy, but finding the exit is difficult since there are so many windows to other dimensions that finding the window to the correct one is difficult. I.e. it is easy to get to a different dimension, but hard to get to a specific one

An extra dimension. Humans can navigate and move in 3 dimensions. But this pocket plane is in 4 dimensions. And just like time domain, you constantly move forward in 4th dimension without any chance of going back. As a constant in 4th dimension, the door out of the plane moves away from you. In a few seconds it will be forever out of reach. Most things in this dimension moves in the 4th dimension just like you, but some things will wrap in and out just like the door. You can have mages using some magical energy to move in 4th dimension at their will.