## Hyperbolic World

Higher-dimensional geometry (previously "Polyshapes").

### Hyperbolic World

For some time now, I'm pondering about a story set up in hyperbolic world. Let me share a few things I came up with and a few I am not sure about:

Absolute unit -- I was thinking of using 10 km as the absolute unit. It's large enough to (hopefully) not make the curvature fatal for human-sized organisms. This would make curvature more of a landscape feature -- it might not be very noticeable in a small town, for example.

Heaven and Hell -- the world is an infinite H2 plane embedded in H3 space, but with normal E1 time (since hyperbolic time axis would cause rapid expansion of space and make infinitely large solid structure impossible). It has a unique "main height level" and is bordered by two equidistant surfaces: "Heaven" at the height of 100 km (10 absolute units) and "Hell" 100 km below. Heaven is a luminescent surface that sends energy to the world, while Hell is a black-hole-like surface that absorbs all energy. Energy that is absorbed by Hell is redistributed and re-radiated from Heaven. While places where Hell is actually exposed are rare, the total amount of energy absorbed by it is infinite, so it can support infinite radiation of Heaven (it takes some SMART redistribution).

Gravity -- here is where I start speculating. I want the world to have standard gravity on the main level. Let's not get into details of what exactly is generating it, it will be 1 standard g on the plane, by fiat. What, then, is gravity in other heights?

In E3, gravity diminishes with square of distance. Strength of gravity of a point source in a given distance corresponds to a surface of a sphere at this distance -- twice the distance means four times bigger surface and so four times smaller gravity.

In the hyperbolic world, the strength of gravity should correspond to the ratio between the area of equidistant surface and the area of the plane, which can be compared by taking a bunch of lines perpendicular to a plane and comparing shapes they cut in the plane and in equidistant surface.

First we need to know what's the length of equidistant curve. If I have two points on a line with distance a, construct perpendicular lines which intersect an equidistant at distance b, then the length of the arc of the equidistant between the two points should be a*cosh(b) -- but I didn't actually calculate this from first principles, only from analogy with spherical geometry, so I don't know if it holds.

Now, for ares on equidistant surfaces, such surface can have an enlarged image of anything on the plane, without distortion. This means that if the length conversion factor is cosh(b), then area conversion factor will be square of that, cosh(b)^2, which can be rewritten as (cosh(2b)+1)/2

This means that the gravity at height h would be smaller by factor of cosh(h)^2. This would hold for both positive and negative height.

The gravity would reduce to 90% at height of 3.3 km, 75% at 5.5 km, 50% at 8.8 km, 10% at 18.1 km and at the lmit distance of 100 km, it would have strength of mere 8.2 x 10^-7 %, making things at these areas effectively weightless.

What I can't completely figure out is how air and water pressure would work. Air pressure can be computed by a differential equation (air density/pressure at given height is dependent on total weight of all air directly above it). The layers of air in hyperbolic world above a given area would be larger the higher they are, but the gravity at such height would also reduce by the same amount, so maybe the air pressure would actually work the same as on Earth? I am not sure...

As for water pressure in oceans, that works out very differently because water is noncompressible and always has the same density. If we assume normal 1 atm on the surface, then pressure in the depth d would be computed by integration like this:

In Euclidean space, regardless of our depth, each layer of water at depth x < d has the same area and same strength of gravity. So at depth d we integrate constant (ro x g) (density of water times gravity) from 0 to d, getting the standard formula of d x ro x g -- plus a constant for atmospheric pressure.
1. Each layer of water in depth x < d has SMALLER area than the area in depth d. The ratio is cosh(x)^2/cosh(d)^2.
2. Weight of each of these layers is diminished by gravity reduction. Gravity at depth x is 1/cosh(x)^2 of the surface gravity, making the weight of each layer of water correspond to 1/cosh(d)^2 -- in other words, the higher layers of water (closer to surface) will be heavier, but they will also have smaller area, making the total weight same for all of them.

So the weight is not just ro x g -- it's ro x g / cosh(d)^2. The deeper you are, the smaller the weight of each individual layer of water above you. So full pressure is d x ro x g / cosh(d)^2 + 1 (for atmospheric pressure on top), i.e. smaller than pressure at the same depth on Earth (or in Euclidean space) by a factor of cosh(d)^2.

This has a surprising consequences: at certain point, as you dive to the hyperbolic ocean, the pressure will reach a local maximum and then it will start to decrease. In this world with absolute length of 10 km, this will happen at the depth of roughly 7700 m, where hyperbolic pressure will be around 448 atm (compared to 771 atm on Earth). Not sure what will happen below this limit -- will the air bubbles and such move down?
Marek14
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### Re: Hyperbolic World

I usually work with something like {5,3,4}, and {5,4}, since the edge of these are of the radius of space. It's a little longer, but it gives you a grid you can work to.

You are right about the curvature not being noticable at the scale of a small town. A 50 km walk, with right-angle turns at each 10 km would bring you back to where you started. But at the scale of dekametres, you would be pretty much euclidean.

Heaven and Hell

If you replace the heaven bit with something that simply reflects the rising radiation as light etc, then you open the skys up to allow other bodies. The energy would come from underneath, and radiate upwards and mostly fall like rain does in the water-cycle. That which does disappate would render the object visible to passers by.

You could then have an adjacent planet that would only be visible from five or six pentagons. So it would like be travelling to Brno to get a trip to the moon, because ye would not see it in Prague.

Gravity

You are right that an equidistant surface is an exact replica of the ground, but distances are increased. It is actually the cosh(h/x) function that lengths expand by. Gravity would decrease by cosh²(h/x). For a height of 10 km, i should not expect this to be much different to 1.618, and being close to exponential at this level, i suppose at ground level, the gravity goes at an increase of 1/2000 per height (ie g + hg/2000).

The relation of g*area is preserved over any given ground area.

Water and air pressure would not be changed. You do have extra mass and less gravity, but the net relation is given by g*area, but this is constant, so 50 feet of head is 50 feet of head in both schemes.

If you go down below the plane, then I imagine that gravity would reverse, or disappate something. In any case, the plane and what is below it is someone elses' problem.

Topology

If we suppose a topology with hills and valleys etc, we could suppose the world sits on a H2 table, and that the land and water is added to the top of this, like people build model trains etc. This would mean that something like Bohemia would be a slightly larger area than something like the Netherlands, simply because Bohemia sits higher above the base-board. It further depends on whether you want the ocean to sit on the base-board too.

This means that distances over land are longer than they are on the table, but the villages etc are still Euclidean in nature. Taking the high road from A to B would be longer than the low road by a smallish amount (6% or so).

The same logic applies to equidistant curves in hyperbolic geometry as does equidistant curves in spherical geometry, except you poke in cosh for cos and sinh for sin. So the equidistant at 10°N is an exact replica of the equator, reduced by cos 10. So the 10° from a hyperbolic plane is a replica of the equator (ie the H2 table), increased by cosh 10°. This is if you travel in the arc so formed. The actual straight line in both cases is shorter.

The chord over 15° is R cho 15° = 2 R sin 7.5° If you go north 10° then then R = R cos 10°, and the distance is 2 R cos 10° sin 7.5° = 2R sin x°. Here x would be less than 7.5, because the points are closer on the sphere. But if you up all of these to hyperbolic functions, then x > 7.5.

On the sphere, you travel less km to cover a distance, by increasing the lattitude. So a new-york to moscow flight flies north of iceland, although the rhomb or mercartor line between these towns is south of iceland. On a H-space, the opposite is true. You travel less distance by getting closer to the 0-level, so the corresponding line between NY and Moscow, would run over africa.
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### Re: Hyperbolic World

wendy wrote:I usually work with something like {5,3,4}, and {5,4}, since the edge of these are of the radius of space. It's a little longer, but it gives you a grid you can work to.

My computations show edge of 1.06, yes. I like the truncated {3,7}, which has edge length very close to 1/3 of radius.

You are right about the curvature not being noticable at the scale of a small town. A 50 km walk, with right-angle turns at each 10 km would bring you back to where you started. But at the scale of dekametres, you would be pretty much euclidean.

Heaven and Hell

If you replace the heaven bit with something that simply reflects the rising radiation as light etc, then you open the skys up to allow other bodies. The energy would come from underneath, and radiate upwards and mostly fall like rain does in the water-cycle. That which does disappate would render the object visible to passers by.

Yes, could work.

You could then have an adjacent planet that would only be visible from five or six pentagons. So it would like be travelling to Brno to get a trip to the moon, because ye would not see it in Prague.

I must say that I'm impressed you even KNOW of Brno Gravity

You are right that an equidistant surface is an exact replica of the ground, but distances are increased. It is actually the cosh(h/x) function that lengths expand by. Gravity would decrease by cosh²(h/x). For a height of 10 km, i should not expect this to be much different to 1.618, and being close to exponential at this level, i suppose at ground level, the gravity goes at an increase of 1/2000 per height (ie g + hg/2000).

I got rid of the "/x" by implicitly making all calculations in the natural unit system (with radius equal to 1).

The relation of g*area is preserved over any given ground area.

Water and air pressure would not be changed. You do have extra mass and less gravity, but the net relation is given by g*area, but this is constant, so 50 feet of head is 50 feet of head in both schemes.

That corresponds to my thoughts about the air, but not water. In water's case, the deeper you are, the smaller is the cross-section of the water column above you. But that is related to the fact that I wanted to have the surface of the ocean as H2, not the bottom.

If you go down below the plane, then I imagine that gravity would reverse, or disappate something. In any case, the plane and what is below it is someone elses' problem.

Topology

If we suppose a topology with hills and valleys etc, we could suppose the world sits on a H2 table, and that the land and water is added to the top of this, like people build model trains etc. This would mean that something like Bohemia would be a slightly larger area than something like the Netherlands, simply because Bohemia sits higher above the base-board. It further depends on whether you want the ocean to sit on the base-board too.

Actually, I wanted to avoid this because then the topology of the surface would be mostly equidistantial -- still negatively curved, but a bit more complex than it could be. I wanted to make H2 the sea level, with sea extending below the plane and land and air upwards. Other advantages of this approach are that it would allow for large subterranean caves and mines, as there would be lots of space underneath even a small area on the surface. In this model, it's the Hell equidistant that's the true source of a gravity, so it still pulls in the same direction even under H2, but gets diluted as you hit larger and larger equidistants.

This means that distances over land are longer than they are on the table, but the villages etc are still Euclidean in nature. Taking the high road from A to B would be longer than the low road by a smallish amount (6% or so).

The same logic applies to equidistant curves in hyperbolic geometry as does equidistant curves in spherical geometry, except you poke in cosh for cos and sinh for sin. So the equidistant at 10°N is an exact replica of the equator, reduced by cos 10. So the 10° from a hyperbolic plane is a replica of the equator (ie the H2 table), increased by cosh 10°. This is if you travel in the arc so formed. The actual straight line in both cases is shorter.

The chord over 15° is R cho 15° = 2 R sin 7.5° If you go north 10° then then R = R cos 10°, and the distance is 2 R cos 10° sin 7.5° = 2R sin x°. Here x would be less than 7.5, because the points are closer on the sphere. But if you up all of these to hyperbolic functions, then x > 7.5.

On the sphere, you travel less km to cover a distance, by increasing the lattitude. So a new-york to moscow flight flies north of iceland, although the rhomb or mercartor line between these towns is south of iceland. On a H-space, the opposite is true. You travel less distance by getting closer to the 0-level, so the corresponding line between NY and Moscow, would run over africa.

One thing I was also thinking about is orientation. I've envisioned a compass with four separate needles pointing to four ideal points. At the place where such compass is issued (like a major city), it would work like a traditional compass, but as you move out, the angles between the needles would start to deform until, sufficiently far away, they would all basically seem to point in the same direction.

Three-needle compasses would be also possible, those would have one unique point where all three needles point 120° from each other.

The question of mountains is interesting by itself, though. Bigger space and lower gravity in high altitudes could produce some weird-looking mountains (after all, a simple cone shape with mountain rising at a given angle from a circular base, could lead to infinitely tall mountains).

I also did some calculations about parallax, supporting what I learned elsewhere (that even very distant objects would move zoom around you pretty quickly as you move).

One problem I haven't tackled yet is the question of orbiting velocity. How would a body thrown in a direction parallel to the surface move? In absence of gravity, it would move in a straight line (which would be, necessarily, divergent to the surface). Gravity would pull it towards the ground. I am unable to even DRAW this situation properly, but my reasoning is that with small velocity, the body will fall down and with big velocity, it will escape. There should be a critical velocity, where the body will exactly follow the curvature of an equidistant, and thus stay at a constant height above the surface, but this velocity is unstable -- a bit less and the body will fall, a bit more, and it will fly away.

And talking of flying bodies, moving things in this world would (thanks to discrepancy in space and time geometry) experience a force caused by their different parts moving along differently curved paths. Basically: even though this is not an expanding universe, it will BECOME one if you're moving; you will feel the space expanding and pushing you apart, with the effect being the stronger the faster you're moving. Each body would have some limit speed above which it could no longer hold together.
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### Re: Hyperbolic World

One thing I've always wondered about a hyperbolic universe is, can "spherical" planets exist? and if they do, what would they look like, and how does the curvature of space make them different from spherical planets in Euclidean space? Also, assuming curvature is high enough to be noticeable at distances below the radius of the planet, what implications would there be as far as topology, etc., are concerned, from the POV of a surface dweller?

In other words, suppose you have a planet embedded in, say, 3D hyperbolic space with rather high curvature, where the surface of the planet is defined as the boundary of some radius r from a fixed point O (the center of the planet). How would the curvature affect the shape of the planet's surface? My naive assumption is that there ought to be a lot more surface area than the equivalent planet in flat Euclidean space. But what of the topology? Would it still be topologically a 2-sphere just like in Euclidean space? Or would the ambient hyperbolic space warp this topology into something else?
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### Re: Hyperbolic World

quickfur wrote:One thing I've always wondered about a hyperbolic universe is, can "spherical" planets exist? and if they do, what would they look like, and how does the curvature of space make them different from spherical planets in Euclidean space? Also, assuming curvature is high enough to be noticeable at distances below the radius of the planet, what implications would there be as far as topology, etc., are concerned, from the POV of a surface dweller?

In other words, suppose you have a planet embedded in, say, 3D hyperbolic space with rather high curvature, where the surface of the planet is defined as the boundary of some radius r from a fixed point O (the center of the planet). How would the curvature affect the shape of the planet's surface? My naive assumption is that there ought to be a lot more surface area than the equivalent planet in flat Euclidean space. But what of the topology? Would it still be topologically a 2-sphere just like in Euclidean space? Or would the ambient hyperbolic space warp this topology into something else?

Spherical planet in hyperbolic world would be, in a way, simultaneously smaller and larger than we're used to.

Larger in the way that a planet with a given radius would have massively bigger area and volume, but smaller in the way that planet with a given AREA would take much less space.

Let's take my world with absolute unit 10 km as an example, but first, let's talk about the topology of a sphere. Yes, the topology would still be the 2-sphere. And this is a very useful because it gives us a very easy way to compute the surface area!

In Euclidean world, circumference of a sphere is c = 2*pi*r while its surface area is S = 4*pi*r^2. We can combine these two formulas into this:

S = c^2 / pi

This is a great formula, because it's independent on anything in the surrounding space -- it will hold for ANY sphere, in any geometry. After all, if you are a 2D being living in the surface of a sphere, you can't know anything about higher-space embedding of your sphere (indeed, not even whether there IS such space or not).

For a proof, we can go one dimension lower and imagine a plane and a horosphere intersecting in a circle. Since the circle is an intersection, it follows that the circle, as a curve, has the same properties whether we draw it on a plane or on a horosphere with its Euclidean geometry. It's the same with sphere.

Now, we can use our surface/circumference relation to compute surface area of a sphere with relation to radius.

c = 2*pi*sinh(r)
S = 4*pi*sinh(r)^2

We can use the identity cosh(2x) = 1 + 2 sinh(x)^2 here:

sinh(x)^2 = (cosh(2x) - 1)/2

S = 2*pi*(cosh(2r) - 1)

(This, btw, would also be the proportional strength of gravity of a point source in H3.)
The volume formula can their be obtained by integrating this from 0 to r, obtaining a bit ugly formula

V = pi*(sinh(2r) - 2r)

At this point, we can imagine three version of Earth - a planet with Earth's radius, a planet with Earth's surface (and circumference), and a planet with Earth's volume.

Mean radius of Earth is 6371 km, i.e. 637.1 absolute units. A planet of this radius would have circumference of 1.54 x 10^278 km, which is a lot. Its surface area would be 7.50 x 10^555 km2. This is 1.47 x 10^547 times bigger than OUR Earth. A prime living space, definitely. The volume would come out to 3.75 x 10^556 km3.
(Though I should note that "km2" and "km3" are just Euclidean constructs and they do NOT come out as squares or cubes of 1 km edge in the hyperbolic space. 1 km2 is simply 1/100 of absolute area unit and 1 km3 is 1/1000 of absolute volume unit.)

Surface:
The surface of Earth is 510,072,000 km2, that is 5,100,720 absolute units. The radius corresponding to this is 71.5 km. This Earth would have correct circumference of 40 031 km. But its volume would be 2.55 x 10^9 km3, 425 times smaller than the volume of our Earth.

Volume:
The volume of Earth is 1.08321 x 10^12 km3, that is 1.08321 x 10^9 absolute units. The equivalent hyperbolic planet would have radius of 101.8 km. Its circumference will be 824 985 km and surface area will be 2.17 x 10^11 km2.

Now, how would those planets actually LOOK, if you were on them? Well, here's the thing -- each and every of them would be quite visibly curved, even the big one.

If you look at a sphere, any sphere, you see it under a certain angle. Construct a tangent from your eye to the surface of the sphere, then a straight line from your eye to the center: if we call the angle between them "alpha", the visual angle of the whole sphere is 2*alpha.

Now, in Euclidean geometry, for small values of alpha, you basically see half of the sphere. Moon is a good illustration -- we often say that we can see one half of Moon's surface, even though, strictly geometrically, we see less than half, by a very small amount.

But this does NOT hold in hyperbolic geometry! In hyperbolic geometry, you will never see a half of a big sphere, or anything close to that.

We are at point A, with center of sphere at C and tangential point at T. The sphere has a radius of r and we are in distance of d beyond that. Let's call the distance from us to the sphere along a tangent t. We can establish a triangle ATC with sides t, r, and d+r and the angle at T is right. The sine theorem tells us that

sin(A)/sinh(r) = sin(T)/sinh(d+r) = sin(C)/sinh(t), but T is right and so sin(T) is 1 and we see that

sin(A)/sinh(r) = 1/sinh(d+r)

so

sin(A) = sinh(r)/sinh(d+r)

We can examine what will happen when we keep our distance from the sphere (d), but its radius will grow (r), and we find that the angle A will grow, but only to a certain extent.

If we are 1 absolute unit (10 km) from the sphere, then the absolute limit of A is 21.6 degrees, corresponding to full visual angle of 43.2 degrees. It can never be more than that, regardless on the sphere's size (that's because the limit angle is valid for a HOROSPHERE, which can be consider a sphere of infinite radius). For comparison, if you are 10 km above Earth, you will see it under the angle 173.6 degrees -- it will take almost half of the view!

Even at height of 1 km above the sphere, the limit angle of viewing is still 129.6 degrees. 100 m, it's 163.8 degrees. And 10 m, it's 174.9 degrees. To achieve the same visual angle Earth has from 10 km, you only need to go 16 m above the surface!

Basically, if you go on a tower built on a surface of this hyperbolic planet -- and it doesn't even be overly HIGH tower -- you will see the curvature pretty easily, even if the planet was actually horospheric, and thus infinite in extent!

But, to be fair, this would hold even if the "planet" was actually planar -- if you rise above a plane, you will see its whole surface (unlike sphere or horosphere), but it will take smaller and smaller part of your vision.
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### Re: Hyperbolic World

Spherical planets can exist. If you set the size of curvature at around 4000 miles, the earth would be about the size of the cell of {5,3,4}. The sun would be something like five of these, but distant from the earth by about 40,000 miles. The thing is that the sun shines on half of the earth, but a sun that remote would not even be able to light up the antipedes, (eg the poles on an equinox). Thus, we have a lesser part of the earth lit by the sun then in euclidean geometry.

Have a look at this picture. It shows an x4x3o8o, but the x3o8o are indeed flat. The circle in the middle is the whole sky, the circles are completely flat x3o8o, so you can see the size of what is visible. Not a lot.

I doubt if the sort of compasses that Marek is suggesting would work. If you suppose something like x14o7o, which contains 24 heptagons of x3o7o, and 14 triangles of x3o14o, now draw your map in this, and move an edge of x3o14o, and only two triangles of this are shared in the different views. The four right angles of the compass would be entirely in two triangles, unless you move close to one of the axies.
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### Re: Hyperbolic World

wendy wrote:I doubt if the sort of compasses that Marek is suggesting would work. If you suppose something like x14o7o, which contains 24 heptagons of x3o7o, and 14 triangles of x3o14o, now draw your map in this, and move an edge of x3o14o, and only two triangles of this are shared in the different views. The four right angles of the compass would be entirely in two triangles, unless you move close to one of the axies.

Well, the compass would only have four right angles at one spot -- they would deform anywhere else.
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### Re: Hyperbolic World

Yesterday, I found a really weird hyperbolic fact.

Imagine you have an isosceles triangle with base b and sides s. The angle at the top is alpha. The question is, if we go along the sides, up and down, how big a detour do we take compared to just going along the base?

Cosine theorem states that sinh(s)^2*cos(alpha) = cosh(s)^2 - cosh(b). I plotted this and found that both paths (b and 2*s) become asymptotically equal; 2*s is, of course, always larger because of triangle inequality, but in the limit, there is no substantial difference between the two. And this is regardless on the angle alpha -- small angles will keep the difference bigger, but it doesn't ultimately matter.

This means that while in Euclidean space straight line is the best path between two points because significant deviations from it make the path longer, it's not so clear-cut in hyperbolic space. On a long journey, you can take even pretty extreme detours from the straight path without affecting the total travel time in a significant way.
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### Re: Hyperbolic World

... at least if you there then don't take a rest for BBQ, go shopping, go sightseeing, enter a pub, or the like! --- rk
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### Re: Hyperbolic World

Klitzing wrote:... at least if you there then don't take a rest for BBQ, go shopping, go sightseeing, enter a pub, or the like! --- rk

Sightseeing is an interesting question by itself. In our world, the network of roads that connect various places is hierarchical: you have important roads connecting important places, then side roads that connect less and less important places to the network. The result is that it's generally easy to reach an important or popular place, but it's comparatively harder to get to a place that is lesser known to obscure (New York is just a plane trip or two from your nearest major city, while it's pretty hard to figure your way to, say, Kurokawa Onsen [it's a small hot-springs village in Kyushu we accidentally found mentioned in some tourist materials when we were planning our trip through Japan, and we were served an incredible traditional dinner there]).

In hyperbolic world, this behaviour would be turned up to 11. Imagine a circular empire of radius of 100 km, with the capital in the center.

The area of such empire (with our assumed 10-km absolute unit) is around 7 million km2. If the whole area is more or less temperate, how many people could live there?

About 5000 m2 of farm soil is necessary to sustain a human being (probably depends on whether there are seasonal changes, but let's assume that). USA has around 1 600 000 km2 of arable land, roughly 1/6 of its total area; 1/6 of 7 million is, say, 1 150 000 km2, and since each km2 can sustain 200 people, this would translate into around 230 000 000 people living in the empire.

I'm not sure how they would be distributed, but we can probably expect smaller number of larger cities and larger number of towns and villages.

It would be very easy to travel from anywhere to the capital in the center, but traveling back out would require very precise navigation and you could easily miss your village if the signage was bad and you didn't remember each turn of the road. The circumference of the empire would be almost 700 000 km long, after all. (This also means that over a half of the empire's area (4 400 000 km2) would be "squeezed" in a 10-km wide strip at its edge.)

A hypothetical "megaempire" of radius 1000 km, meanwhile, would have area of 8.4 x 10^45 km2 with 1.4 x 10^45 km2 of arable land and 2.8 x 10^47 inhabitants. Its circumference would reach 8.4 x 10^44 km and 10-km strip at the edge would, once again, contain over a half of its total area: 5.3 x 10^45 km2.

Imagine sightseeing there!
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### Re: Hyperbolic World

What I like about hyperbolic geometry is that any polytope or tiling is possible if its size matches the curveture of the space. Size has something absolute in that kind of space. Convex polytopes can have infinate size and you can go in a circle and never return to your staring point.

I think hyperbolic space is interesting for visualizations or artwork related to association, possibilities, dreaming and the development of consciousness, as its ever branching structure matches these things much better than euclidian space (though not perfectly).

Spaces with positive curveture (or toratopic spaces) on the other hand are nice to depict opposites, closed cycles of time, relations or causation, predictable developments or how things interrelate and connect together.
What is deep in our world is superficial in higher dimensions.
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