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So, my world is a sphere (roughly) within hyperbolic space. Since the surface areas of spheres in hyperbolic space grows exponentially with radius, it has the same surface area of the Earth (indeed, we can assume that the surface is the exact same as the Earth (continents, countries, and all) for the purposes of this question), but only has a radius of 4 km (2.5 mi).

This means that you could, for example, start in the US, dig a 8 km tunnel, and end up in China. Indeed, you can reach any point from any point with a 8 km tunnel (ignoring the heights of mountains and stuff).

In the past, the hyperbolic nature of the universe has not affected the world, since humanity was stuck to the surface of the sphere (spheres in euclidean and hyperbolic geometry are the same), but towards the beginning of the twenty first century, the inhabitants started to be able to build tunnels through the world.

This of course has awesome economic potential, but many militaries are worried. How do you stop enemies from digging under you country? How do you stop enemies from digging into your base, or into your country, to either deploy troops, plant bombs, or even just dig a giant pit under you, your buildings, and your land?

Remember, they don't need to be next to you. Enemies anywhere in the world with the digging capability can easily get to you from below. Indeed, even if they are next to you, going through the center of the world is probably quicker. Not only that, but by going below, they can hit you anywhere, not just the border.

Note:

  • It may be hard to visualize how a sphere with the surface area of earth but only a 4 km radius would even be possible. To do this, it might help to visualize it within the Poincaré ball model, centered on the world. Note how circles grow exponentially with respect to radius in this model, instead of quadratically. For example, look at the heptagon tiling. A large circle of heptagons has a much smaller diameter than you would expect (note that all the heptagons are the same size).

  • Once countries realized the economic, and more importantly military implications of the situation, they invested very heavily in digging and tunneling technology (similar to how on Earth, we invested heavily in nuclear technology). As such, tunneling technology is particular will be much more advanced than in our world. In particular, I imagine they might have developed "worm tanks", i.e. vehicles/drones that move through the ground without needing a tunnel.

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    $\begingroup$ Well then, seems to me that would be a great way to build fast travel options. China to U.S. in a few minutes through a tunnel that looks long, but isn't. I'd suggest they probably already built the tunnels during a time of friendship, and just need to collapse them if things turn to war. $\endgroup$ – Tim Feb 16 '18 at 3:37
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    $\begingroup$ Only one end needs to be collapsed. Besides, digging in 8km tunnel is work of several months. You'd know they're coming, have time to prepare a counter strike. $\endgroup$ – Tim Feb 16 '18 at 3:43
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    $\begingroup$ Minecraft bedrock $\endgroup$ – Amoeba Feb 16 '18 at 4:03
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    $\begingroup$ You need to think how gravity, pressure and temperature scale. I do not think digging the tunnel would be possible. $\endgroup$ – Ville Niemi Feb 16 '18 at 4:39
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    $\begingroup$ How can they dig so deep without having access to more than 200 km3 fossil fuels globally? $\endgroup$ – Real Subtle Feb 16 '18 at 8:46

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Israel has developed several ways to find tunnels made by Hamas. There most advanced one relies on a series of sensors and computer systems that detect slight vibrations in the ground and triangulate them. But simpler models have worked relatively well for decades.

In WW1 where mines were common, soldiers developed a whole host of means to detect mines before they were blown. These involved detecting the vibrations in the earth to digging their own tunnels and listening to the tunnel walls with stethoscopes. When a tunnel was detected they were broken into and blown up.

Considering how this world works, it would be reasonable for countries to develop a large warren of tunnels all over their country and set up listening posts, manned at first than automated as the technology developed, to ensure no one was trying to tunnel into their country.

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    $\begingroup$ You might also look at oil exploration, which probes depths measured in km. $\endgroup$ – user25818 Feb 16 '18 at 17:34
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How do you stop enemies from digging under you country?

enter image description here

You dig under your country first, small tunnels with seismic sensors (which isn't going to be so easy, since you're receiving a planetary surface's worth of seismic waves from above). Then deploy mines, and an underground force to intercept enemy excavations.

With current technology it is relatively easy to drill a hole one kilometer deep. So drill lots of such holes, several at the same time to prevent seismically locating them. Then deploy sensors in every hole.

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    $\begingroup$ "Going nearer to the center of a planet in hyperbolic space would mean that you become smaller." Uhm, I don't think this is true. Moving around shouldn't change your size. In certain models, it might appear that your size is changing though (due to the model), but that's because the model distorts distances. (Also, in the Poincare ball model, you appear to get bigger as you approach the center, not smaller.) $\endgroup$ – PyRulez Feb 16 '18 at 6:28
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    $\begingroup$ Also, I guess this kind of turns war into a game of 3D dig dug. $\endgroup$ – PyRulez Feb 16 '18 at 6:28
  • $\begingroup$ Yes, that it would. As for the size change, as I see it it either happens or not happens. If it does not happen, though, I suspect that there wouldn't be enough gravitational force to make the planet stick together. $\endgroup$ – LSerni Feb 16 '18 at 6:34
  • $\begingroup$ If your worried the planet won't have enough volume, you can see math.stackexchange.com/q/1445278/49592 to calculate the volume of a sphere. Intuitively, though, its better to imagine the surface of the planet being "smaller", not the center. $\endgroup$ – PyRulez Feb 16 '18 at 6:40
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    $\begingroup$ Relevant: Tunnel Warfare is messy, see Battle of Messines (1917). $\endgroup$ – KlaymenDK Feb 16 '18 at 11:30
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Build up

You would want to build up anyway since there is so much more room up there. But building up also allows you putting things like heavily reinforced concrete in the enemy's path.

Of course, even the heaviest concrete can be breached, but it will take time and be very noticeable.

One might think that building a concrete floor under your entire country is not feasible, but in a hyperbolic universe it turns out you can.

That is, if you aren't too hung up on the idea of defending the original surface area. Instead you put down a thick and strong concrete slab on a few square kilometers. Then you build a few hundred meters up from that and suddenly you have room for everything and everyone you want.

Let the enemy have the rest of your old country, you don't need it anymore.

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    $\begingroup$ Except that you need resources and you can't pull them out of thin air (it's the same problem that "floating island" nations face); you'd need a novel economic model to sustain your country. $\endgroup$ – LSerni Feb 16 '18 at 14:04
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    $\begingroup$ @LSerni So you do it in reverse: go down and put that concrete a km below your country so it requires less surface area. $\endgroup$ – Loduwijk Feb 16 '18 at 17:06
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A 8km deep tunnel would collapse as the weight and pressure of the surrounding rock makes the rock behave more like a soft pliable plastic than a solid. Heat is the other big issue at about 4km down it is too hot for humans. In hyperbolic space the pressure may actually increase faster than on earth, more mass above a given area, so it will get softer and hotter faster than on earth.

the other issue os of course it takes a long long time to dig that deep, the deepest* a human has ever gone on earth is about 3km. On top of that it is really really obvious so its not a surprise attack. Note this is digging into rock not cheating by finding a deep water trench.

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    $\begingroup$ Note the volume is much (4 orders of magnitude) smaller than Earth, so the number of radioactive heating is smaller by 4 orders of magnitude, and even the surface to radiate away is proportionally bigger. And of course, gravity has to work very differently (no shell theorem, for starters). $\endgroup$ – Radovan Garabík Feb 16 '18 at 9:48
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    $\begingroup$ @RadovanGarabík But, intuitively to me anyway, each unit of dirt is supporting the weight of more dirt above it than it does in our universe. So the pressures and temperatures might climb at a much higher rate. If this were the case, I do not claim to know at what rate. $\endgroup$ – Loduwijk Feb 16 '18 at 17:04
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    $\begingroup$ @RadovanGarabík by that logic it has to be even hotter and softer as you dig down, as each unit of rock is supporting the weight of many many more units that in would in normal euclidean space. you also need a hotter core to get similar surface temperatures. $\endgroup$ – John Feb 16 '18 at 17:33
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Water would be another way, like @Dan Clarke suggested Israel isn't alone in suffering from Hamas tunneling, Egypt is another country with the exact same problem (and the exact same cause to said problem) and their solution to it was to dig a moat and fill it with water around the Egypt\Gaza border and connect said tunnel to the ocean, any tunneling attempt will simply be flooded once reaching that tunnel.

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From a story point of view, this could lead to an interesting mutation of submarine warfare. Each country builds a fleet of subterranean transports. They have digging apparatus in front and they pack the dug material behind them, so they can freely patrol under their own countries without literally undermining their own land and buildings.

They have seismic sensors to detect enemy craft or sappers. Drama could be created in a believable way by limiting the technology such that the sensors can only detect another craft if the sensing craft is not moving (since the action of tunneling creates too much "noise". That would mean that once an enemy has been detected, the captains of such craft would have to predict the motion of the enemy and decide when to start moving to intercept knowing they will go blind once they move, and they will be detectable. Then they could stop periodically to listen and "hide". Some kind of slow, tunneling torpedo would be interesting. It could be detected seismically like the craft, and it could attempt to home in on a tunneling signature, and may or may not lose lock when a target stops moving.

Alternatively, there could be seismic outposts that detect activity and use 3D triangulation to map out exactly what is moving where, and then they could use limited radio communications to guide their craft. Ferromagnetic ore deposits and veins could prevent radio communications, or merely "wet" earth with dissolved salts. All kinds of interesting technological challenges and solutions would arise in subterranean warfare.

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In the 1980's the US released radar generated images of underground water channels of the Nile River - this 1996 article shows some of them.

The real reason for releasing the radar images in the 1980's, was to show the Soviets that the US could spot their underground missile silos, no matter how well hidden they might be, by looking for the rather large gaps in the ground. A missile silo requires a fair amount of open space underground, and there's no way to mask that without looking like you're really trying to hide something.

So an advanced society could use space borne synthetic aperture radar to spot such tunnels early in their construction, and take appropriate countermeasures. That's in addition to seismic sensors... tunneling, especially through rock, is a very noisy affair.

If that planet had a molten core, as the Earth does, tunneling straight through the middle would be impossible.

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Note: I understand the approaches mentioned in this answer may be entirely infeasible for some literary scenarios. I'm not saying this is necessary the most universally useful answer for all scenarios. However, I propose this as a valid approach that may be useful in some scenarios, and therefore be worth considering (even if not for an exact scenario that you're currently actively thinking of, perhaps useful for some other scenarios that may be usefully worthwhile to include somewhere in your world).

If you have a bunch of friendly tunnels that pre-exist, then creating a new tunnel may seem like a pointlessly expensive venture. (If someone wanted to become an enemy, they may take an entirely different approach that is less expensive.)

How do you stop enemies from digging under you country?

Take Switzerland's approach. The "Swiss army" may be most famous for something that is not a weapon of mass destruction that typically causes tremendous amounts of death, but for a useful tool. (Wikipedia's article for “Neutral Country” notes that Switzerland and Sweden avoided the conflicts of World War 1 and World War 2.)

This approach may best be explained by this Presidential quote:

"The best way to destroy your enemy is to make him your friend."
-- Abraham Lincoln

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Very interesting geometric puzzle.

If I were an army inventing armaments to deal with that, I would take a lesson from Ender Wiggin: the enemy's gate is down. I would make a lot of weapons that are designed to be utterly vicious when fired downward. It'd definitely involve exploiting the properties of the world (whatever they truly are), but I'd try to make it so that the enemy is welcome to take advantage of this really fastpath, but to do so puts them in an incredibly weak position because they're going to be in line with my most dangerous weapons.

In theory, this is really just an adaptation of standard warfare policy. You always have more rapid responses prepared for the most rapid paths an enemy can take to your position. If they take the long route, I always have time to react.

For an amusing irony, this means "guns down" is actually the aggressive position, while having guns leveled at the enemy might actually be "average" level hostilities.

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You stand on a rock.

Just as the radius of the planet is absurdly small when compared to Earth, so are its inhabitants' brains when compared to ours. For the same ratio that you have between that planet and Earth's radii, a cranium with the same surface area as a human's would have a very small brain. Anyone's enemies probably have the intellect of a worm. They may dig as much as they want but they will not figure a way to dig through rock.

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  • $\begingroup$ "a head with the same surface area as a human's would have a microscopic brain" Uhm, that's not how hyperbolic space works. At small scales (like, say, the scale of a human), its approximately euclidean. $\endgroup$ – PyRulez Feb 16 '18 at 4:57
  • $\begingroup$ Don't circles grow expinentially in that space? I expect their brain volumes to be a fraction of ours. $\endgroup$ – Renan Feb 16 '18 at 5:07
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    $\begingroup$ They grow exponentially, not shrink exponentially. At small scales, volumes are cubic. See math.stackexchange.com/q/1445278/49592 $\endgroup$ – PyRulez Feb 16 '18 at 5:10
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    $\begingroup$ @PyRulez That does not make sense. If they grow exponentially, then they must shrink exponentially. If going from a combination of radius R1 and surface area S1 out to a larger radius R2 produces the exponentially larger surface area S2, then going back down to R1 should result in the original S1. Either they grow exponentially or they do not. Is your last comment suggesting that they grow exponentially, but not at small scales? IE: from R1=1cm to R2=2cm surface does not grow exponentially, but from R1=1km to R2=2km surface does grow exponentially? $\endgroup$ – Loduwijk Feb 16 '18 at 17:09
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    $\begingroup$ Just trying to interpret the relationship between surface area or volume and brain ability, and how the two might differ in hyperbolic space, then thinking about how special relativity actually implies that actual space is non-euclidean, seems to raise some interesting questions. $\endgroup$ – Darren Ringer Feb 16 '18 at 19:41

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