# Life in Hyperbolic Space

## The dangers of life in a negatively curved space

Steve Trettel

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While the Euclidean world does not appear qualitatively different to an ant or a giant (Euclidean geometry admits similarities so the same set of geometric facts are true at any length scale) this is not true for geometries of constant nonzero curvature. Here we will discuss the qualitative properties of hyperbolic space at varying levels of curvature (instead of considering the lives of ants and giants we will consider the lives of humans as the curvature ramps up).

For now I have left out the computations supporting the purported qualitative behaviors, but hope to someday write another post including the details of those!

# Life in Flat Space

Let’s begin our journey by recalling what it feels like to live in Euclidean space. Of course, we are all quite well aquainted with this, but for the sake of specificity bear with me while I list a couple things about our world I find notable. (And to set the tone for this article!).
I like reading, so one thing that stands out to me is the existence of books - really, just the ability to so convienently store lots of sheets of paper by stacking them on top of one another. Of course this doesn’t just apply to books, but stacks of brinks, wooden planks, and the ability of my laptop lid to close nicely are good properties of our world to keep in mind. Speaking of reading, before books there were scrolls, thanks to paper rolling itself up nicely into a neat little tube. Even today in the age of books we reap the benefits of this property of Euclidean paper through the use of paper towels and toilet paper! Broadening our view from books, paper towels and other household objects, let’s look a little more big-picture. For instance,its bright out during the day. We are far from the sun but not so far that its light peters out by time it reaches us, allowing eyeballs to be a useful means of gathering information about the outside world. Airplanes are a time - efficient way to get around, and oftentimes digging an underground tunnel for transportation (say from NY to LA) is too much of a hassle to be worth it. And this is a bit anachronistic for us here on Earth, but one way we could have realized that our planet was round would have been to blast off into space and film the ascent!

Some other things that I’d like to recall about our world are dynamic in nature. When you spin around in circles your arms fly out from your sides and you feel a pull outwards, but when you are riding on a train you feel nothing. People float in spaceships, but not in airplanes. Speed limits on roads are imposed for safety, but speeding won’t kill you unless you crash.

# Being a Little More Precise

So, like I said, the tone of this article is going to be pretty informal. However I feel the need to make a quick interruption here and phrase things a little more mathematically.  Sheets of paper look and feel flat to us, meaning that they are locally Euclidean.  The fact that we can stack them to make a book means that in Euclidean geometry, equidistant surfaces from a Euclidean surface are also Euclidean. The fact that we can roll paper up onto a tube means (roughly) that equidistant surfaces from a Euclidean geodesic are locally Euclidean.

The fact that its bright out follows from the drop off of light intensity with the square of distance: which is really just encoding that the surface area of spheres grows quadratically with radius. Airplanes are time efficient because the increase in distance of travel by moving higher above the earth’s surface is more than compensated by the increase in speed (the growth rate of arcs of circles is linear in radius). And its easy to tell the Earth isn’t flat from a spaceship because an (infinite) Euclidean plane appears to take up half your field of view from any height, but a sphere takes up less and less of your view the farther you travel away.

Feeling an outward pull when you spin corresponds to the fact that generic orbits of points under rotational isometries of Euclidean space are circles, which are not geodesics. Not being able to feel constant velocity motion is a consequence of the fact that under a one parameter family of translations all points in $\mathbb{E}^3$ move along geodesics (likewise the floating on spaceships).

## What Changes in Hyperbolic Space?

Ok, so how do these properties hold up in Hyperbolic space? By the existence of this article, it’d be a fair bet to guess “not well.” ;And its true! The equidistant surfaces from hyperbolic planes are not hyperbolic planes (at least not of the same curvature), nor do the equidistant surfaces from a geodesic have constant curvature $-1$. The surface area of spheres grows exponentially with radius, as does the circumference of circles! Infinite hyperbolic planes take up increasingly smaller areas of your field of view as you depart from them, at a rate similar to that of a sphere.

Elliptic isometries still move generic points along non-geodesic orbits, but in a surprising change up the same is true for hyperbolic translations, and parabolics. You can feel all types of hyperbolic motion! So these are the changes we will have to deal with when investigating daily life in hyperbolic space.  However, because hyperbolic geometry is not scale-invariant, these differences will make themselves apparent to different degrees at different values of curvature - so there is not one story, but many!

## Curvature

The worlds we will investigate come in two main flavors, determined by their curvature. When we say that hyperbolic space has curvature -1 we are implicitly choosing a unit to measure distance in, and we will call that distance the “characteristic length” of our world.  Low curvature worlds are those for which the characteristic length is on astronomical scales with respect to earth - in these worlds we would get along just fine day to day, but our knowledge of the greater universe would change drastically. The second class of worlds are those for which the characteristic length is on human scales

• its in these worlds that our daily life becomes radically different.

# Life in Low Curvature

As we tour the different levels of curvature, we will mark each world by its characteristic length, that is, the distance unit along which the curvature is measured to be $-1$. This gives us a useful measuring stick, helping us keep in mind approximately where the effects of curvature become important.

## Galaxy

The difference between the world we live in and a hyperbolic world with curvature on the scale of 100,000 lightyears would be all but imperceptible as we went about our daily activities.  In fact, no differences would have likely been noticed throughout our history up until relatively modern times, except by those astronomers with particularly good eyesight.  While all the stars in the sky would still twinkle with their usual brightness, the band of stars making up the milky way would shine approximately 17 percent dimmer towards the horizon, where the galactic core lies.  This small deviation from its Euclidean glow would be almost imperceptible : our eyes being masters at adjusting for varying light levels and all. The only clear difference would lie a little below the constellation of Cassiopeia on an inky black night - for where the dim smudge which is the Andromeda galaxy usually lies would be noting but a patch of dark sky.

In fact, due to the exponential divergence of geodesics here at the galactic scale, Andromeda would appear smaller, fainter, and dimmer than a galaxy in our Euclidean world lying at a distance of $10^16$ lightyears - over a million times farther than the cosmic horizon itself. And that’s the closest galaxy to our own - meaning that in this hyperbolic world no matter how carefully we scoured the skies, no matter how sensitive of a telescope we built, we would never come to know that space is teeming with other island universes just like our own.

## Lightyear

If we increase the magnitude of curvature by a factor of 100,000 so that the characteristic length in our world is on the order of one lightyear, there are still no noticeable changes to life on earth (for instance, the sun appears to be within 1 part in a billion of its current size in the sky, and thus certainly everything on earth appears essentially identical). Again we need to move to nighttime where objects can be seen at appreciable distances before we notice the difference.

The closest star, Alpha Centauri, lies at a distance of 4 lightyears from earth. However due to the divergence of geodesics in hyperbolic space it appears in the sky with the angular size and intensity as though it were 46 lightyears away. Certainly this would still be visible (as in the real world we can see stars which are up to 16,000 lightyears away with the naked eye), but it would appear approximately 100 times dimmer than we are accustomed to here.

Currently the brightest start in the sky is Sirius, which outshines Alpha Centauri not due to its proximity (its twice as far away) but by its relative closeness coupled with its size.  However, in a hyperbolic world with lightyear-scale curvature, Sirius would shine with the brightness of an equivalent Euclidean star at a distance 2745 lightyears away - and while still remaining visible would be significantly outshone by Alpha Centauri.

What other stars (if any) would be visible? A little computation shows that stars at a distance of 10.3 lightyears appear to be 16,000 lightyears away, which is approximately the visibility threshold for the human eye on a black night.
The stargazing opportunities in this world would be bleak - the only stars visible at all would be Alpha Centauri and Sirius, one star in each hemisphere.

However, the curvature is still so large with respect to the size of our solar system that everything locally remains the same: the planets orbit the sun at the same distance and shine in our sky with the same brightness.  Five planets a moon and a star.  Thats the night sky in a world with curvature 1 lightyear.

Of course, with a telescope a very few extra stars may become visible: Vega which lies at 25 lightyears from Earth appears to be at a distance of $10$ billion lightyears - however this is at our current threshold for being able to perceive galaxies from Earth, not lone stars. So, even armed with the Hubble our night sky would be dark indeed.

## Solar System

Moving along to a world a little over 2,000 times as curved, we may now expect to notice some changes in our solar system. First off, the night sky is completely star-less: even Alpha Centauri is as dim as a star would be at $10^{415}$LY in our world, forever out of reach. In fact, the visibility range is a mere 535AU: meaning that if a star were to pass within this distance of Earth it would lie just at the limit of perceptibility with the naked eye.  (Note that this is 17 times the distance to Pluto, but still well within our solar system’s collection of comets).

Within the solar system we would begin to see changes as well. The size of the sun (and hence the planets) remains nearly unchanged; but how would they be laid out?  As this is a fictional exercise it’s up to us, and I’m sure its clear one way this could go: if we left the planets at their actual separations and continued ramping up the curvature, they would quickly recede from view. Then we’d loose the sun, and the Earth world forever be in darkness. But a planet in complete darkness is not a good place to learn hyperbolic geometry: let’s instead say the planets formed in hyperbolic space near where the gravity from the sun is felt the same as in the real world.  This will have the effect of contracting the solar system as the curvature grows, keeping earth bathed in light.

Under this plan then, where does the Earth lie? Rather un-excitingly it still orbits at $99.98$ percent of its current distance from the sun, meaning we would notice nothing qualitatively different.  However, due to the fact that spheres grow exponentially in size in hyperbolic space, the further out a planet is the further in it needs to move to keep the gravity the same: this amounts to Neptune orbiting at 88 percent its current distance from the sun.  This makes it slightly closer to earth and so you may think it’d make it slightly larger in the night sky, but the divergence of hyperbolic geodesics neatly cancels this making it appear approximately the same; only visible with a telescope never with the naked eye.

Structures on the order of galaxies would arrange themselves radically differently if we forced them also to follow our convention above, contracting inwards to mere percentages of their current sizes. Not that we’d know - the outer solar system is the farthest we could ever hope to see.

## Earth’s Orbit

Time for another drastic increase in curvature: what happens if the world was so curved its characteristic length is approximately radius of Earth’s orbit? This scale is still large enough to leave life on Earth relatively unchanged, as our orbit is quite large with respect to the size of planetary bodies even the sun remains  99.99998 percent its current radius.

Not that we’d ever know here on Earth, but the effects of curvature have begun to show themselves on the larger stars in the universe: Betelgeuse (the largest star currently visible in the sky, 1000x the size of the sun) would now have a radius only 55 percent of its current one to have the same volume.

Turning our eye to the positioning in our solar system; Earth now lies at $88$ percent its current distance from the sun, mars at 79 percent its current orbit, Jupiter at 45 and Neptune at 13- meaning Neptune lies only 4AU from the sun (this is inside the current orbit of Jupiter).  This surprising distance - dependent contraction shows the characteristic non-scale invariant properties of hyperbolic space - Neptune’s orbit is altered by nearly a factor of 10 but Mercury’s is effectively unchanged.

This condensed solar system means that the planets all orbit closer to the sun, and thus pass closer to earth than we are accustomed to. But what does this do to their sizes in the nighttime sky? Can we tell the difference between this universe and the last one purely from Earth-based observations?  If you’re an avid watcher of the nighttime sky you’ve probably noticed that the planets not only change location in the sky over the course of a year, but also vary in brightness depending on our distance from them. In the spring of 2016 Mars was at its brightest in our sky, but in this hyperbolic world Mars would shine 2.45 times as bright at closest approach. Due to the fact that all orbits are contracted in this fictional world Mars’ furthest point from Earth here is actually closer than its furthest point in the real world - however this is more than made up for by the exponential divergence of light rays in hyperbolic space - Mars at its most distant would appear less than 0.4 times its current dimmest.  This variation - brighter at closest approach and dimmer at farthest - only continues to grow as we move to more and more distant planets: Jupiter goes from $4\times$ its current brightness to $0.2\times$ its current dimmest - and Neptune from $5.4\times$ to $0.18\times$ at the extremes.

While our day to day life has not been much altered yet, this level of curvature would be a huge boon to planetary scientists: it would only take years instead of decades to get to Neptune, and the communication - delay with our robots on mars would be less!

# High Curvature

Now we will ramp up the curvature somewhat drastically, and consider life in worlds where the non-euclidean nature begins to interfere with everyday life

In a world where the curvature is on the order of the Earth’s radius, the sun will experience radical changes. Its radius has dropped down to 7.5 times that of the Earth while keeping the same volume, but its surface area has grown to nearly 73 times that of its Euclidean counterpart. Earth’s orbit has continued to radically shrink to 0.00055AU, or approximately 7 current earths lined up end to end, or 1.7 times the hyperbolic radius of the sun! The amount of the sun that’s visible has continued to drop as well; the area now visible at any given time is approximately the area you can see on the earth from space. This means that entire sunspots can take up our field of view of the sun as they pass across the disk.

The size of the Earth has finally begun to change: the same volume of rock now fits into a sphere of 94 percent its Euclidean radius and the surface area has increased to 118 percent. This relatively small change means that geometric figures drawn on the earth, from triangles on a sheet of paper in a classroom to a road map for a city, deviate little from their Euclidean counterparts.

So, throughout our day-to-day lives would we remain oblivious to the non-Euclidean nature of our world?  Probably not.  While the same volume of rock has been crammed into a sphere of slightly smaller radius (without changing the density!) - gravity in this world would decrease not by the square of distance, but with the inverse surface area of a hyperbolic sphere. Gravity would drop off exponentially! (Note obviously I make no claims about how or what “gravity” would be in a hyperbolic world, like brightness and orbital distance this is just a story telling device to explain the peculiarities of the hyperbolic metric). Thus despite the decrease in radius the gravitational pull is actually less here, at $8.28m/s^2$, or 84 percent its Euclidean value!

As visitors from Euclidean space, we would also notice a slight difference in how far we can see along the sphere: while the surface area has increased our field of view has actually decreased. This phenomenon is similar to the small portion of the sun we can see - spheres in a space of ambient negative curvature appear ‘more curved’, almost ‘pointy’ in comparison to our Euclidean experience. Walking along the beach you could see 2.57 miles out to see versus the 3ish visible here, and from cruising altitude in an aircraft you would only be able to see 201 miles, instead of 235.

The size of the Sun, Earth and Moon (which is 99 percent its current radius but orbits at 8 percent its Euclidean distance) work together to produce some interesting changes in our sky.  Because we would orbit the Sun only 0.7 solar radii away from its surface (and a solar radius in this universe is only 8 earth radii) the the sun is measurably closer at noon than at sunrise or sunset. Further, because the angular size of an object decreases exponentially with distance, this should correspond to the sun changing size throughout the day! And indeed it does: the sun at noon is the same size as the sun here in our world (by design; remember we moved earth to keep it equally bright) but the sun at sunrise is only 15 percent its usual size, giving a 650 percent change in size over the course of the day. The moon’s likewise is 0.58 times its current size at moonrise but 3.866 times larger at its highest point in the sky.

Now that we have a fairly good feel for what the world looks like at this level of curvature, let’s acquaint ourselves with what its like to move around. As we’ve seen, geodesics tend to spread out from each other, and as matter wants to travel along geodesics, it wants to spread out. Thus, moving bodies feel a force: there is no relativity here you can tell if you are translating forwards in empty space if your arms try to move out from your sides (just like you can tell if you are spinning in our world)

This is the first time in our journey towards ever more curved spaces that this property begins to matter: at larger curvatures even giant starts felt these effects only negligibly. However now the sun, moving forward at only 1m/s, feels a force pulling its two sides outwards by $1m/s^2$. And this force increases with the square of the velocity: indeed, were the sun to move at the speed it currently does around the galactic core it would be ripped apart as the outer ring would be pulled off at 7billion $m/s^2$, or 4 million times the surface gravity.

We however, would never notice here on earth. Even the fastest spacecraft to date would have only experienced a force of 0.0001$m/s^2$ during its fastest moment, and no human, astronaut or other has traveled fast enough to have noticed any sort of change.  However, this distinction spells doom for humanity’s future interstellar travel aspirations (not that we’d have any here - remember we don’t know there are any other stars out there!). Trying to get a ship 10m in radius to go 1/10th the speed of light would lead to forces of 23g pushing against the outer walls as objects in the interior tried to follow the diverging paths of geodesics only to be stopped by the hard walls of the ship.  So maybe we could send robots out in the blackness, but our feeble bodies could never take the forces associated with high speed (but constant speed) travel in this hyperbolic world.

## California

Increasing the curvature now so the radius is approximately the size of California, the exponential dependence of volume on radius has caused the Sun and Earth to grow ever closer in size: while our star has over a million times our volume, it now fits in a sphere of only four times the radius! The mass of the sun has managed to pack itself into a sphere of radius only 7600 miles, and the Earth has shrunk too - its now only 54 percent its Euclidean radius (and orbiting at a distance of only 0.5 solar radii away from the sun!).  As we began to notice in the last worlds, the ratio of surface area to volume is quite different in the Hyperbolic and Euclidean worlds - despite its much smaller radius, the Sun here has a surface area 374 times its current size. The earth’s surface area has grown as well to 3.52 times the area we are accustomed to, meaning that meaning that if the oceans took up the same percentage of the land area they would only be 3,500 feet deep on average (compare to 12000 here).

The difference in the size of the sun throughout the day is truly impressive: when it rises it is only 0.002 times the size of the sun we see, but reaches the size we are accustomed to around noon. This is a 374 times increase in apparent size! And that difference in apparent size is caused only by our distance from the sun changing by a single earth radius - this gives very good insight into how dark hyperbolic space is. Remember we decided to place the Earth where it would receive light in our story to keep things interesting - but at high levels of curvature this is a very precise choice to make. In the real world, its bright on Earth, bright on Venus, and bright on Mars - the quadratic drop-off in intensity leaves large swaths of the solar system bathed in light.  Not so in highly curved hyperbolic space. Were the earth’s orbit to be increased in size by a single earth radius our sky would be black - the sun would fade from view.  Traveling through the solar system at even moderate speeds, explorers would find the inky blackness would suddenly give rise to a small speck of light, which rapidly grew in size and brightness - the sun - only to quickly fade from view once again after traveling a mere 2000 miles away.

Enough space - the curvature has finally ramped up to a level where the effects on Earth are pretty interesting. What’s life like down on the ground? Well our gravity has decreased to 2.7m/s which is only 28 percent what we are used to, close to the Euclidean gravitational pull of mars.  The increase in curvature has also made our Earth look ‘rounder’ decreasing the distance to the horizon.  Standing on flat ground the horizon appears 1.3 miles away (instead of 3); and from the height of a 6 story building the visibility has dropped from its actual 9.5miles to 4.1.  Air travelers would begin to loose the impressive views we are accustomed to from atop the clouds - in this world you would not be able to see 235 miles away but only a little less than 100 at cruising altitude. One consequence of these shrinking horizons is the cost of cell phones: the horizon from a tower of 200ft is only 7.6 instead of 17 miles, meaning the area of coverage is over 5 times less.  Thus; need 5 times as many cell towers, making your monthly bill that much more expensive to pay for their construction and upkeep!

As circumference increases exponentially with height; we can see that if the international space station orbited at the same height as it does in Euclidean space (249 miles) it would have an orbit 1.38 times as long as on the ground (vs in Euclidean space where the difference is 1.000004). The Earth would also look smaller; it’d take up about half the angular size that it does here (which is still a giant portion of your field of view).  Higher up satellites are affected even more greatly: at a height of 1000 miles the orbit is 3.67 times the ground distance, and at a height of an additional Earth radius the orbit is 19times as long (vs twice, as in Euclidean space).

The rotation of the Earth is starting to make a difference around the equator: points here are rotating quick enough that the centripetal force they feel trying to pull them off the planet is 0.6$m/s^2$, which is about a fifth of the local gravity. Thus, when hyperbolic physicists explain to their freshman lectures that they weigh less on the equator, this isn’t some obscure effect with no real world consequences (as it is in our world) but instead something anyone whose ever traveled is already well-acquainted with.  Because the quicker you rotate the greater the feel of an outward force, on flights from LA to London (against the Earth’s rotation) the centripetal force felt is lower and consequently more of gravity is felt, whereas from London to LA its the reverse - you feel significantly lighter! (This difference is not small: at typical cruising speeds it changes by over 3 times in magnitude depending on the direction of your flight).  This provides an interesting opportunity for high-speed aircraft makers: flying 2159 miles per hour with the earths rotation exactly cancels gravity, allowing people sustained weightlessness.

However with the exception of some pretty cool airplane rides and the strangeness of the sun appearing abruptly out of the blackness around noon, life at human scales still would not have changed all that much locally. If you stayed in your room you’d not even notice the difference.  So let’s push onwards towards higher curvatures!

## Campus

What if the characteristic length was on the order of the size of a college campus (say, a mile) What havoc has curvature of this magnitude wreaked on our little planet? Well first off, its gotten a lot smaller!  The entire mass of our world now fits in a sphere of radius 13 miles!  The surface area has continued to balloon out to 2639 times its current value - so the ‘oceans’ here are 4.5feet deep (if they covered the same percentage of land). But while our planet’s livable area has grown, the increasing curvature has given us an ever more local view of it - even from the vantage of a small mountain peak (3000ft) you could only see 0.88 miles, and you would really feel like you were on top of the world - the horizon would appear 62 degrees below the horizontal (instead of just slightly below half, like here). Of course, this makes things a great struggle for the cell phone companies as they now need 4100 towers to cover the same area of land that one suffices for in Euclidean space. As you’d probably expect, the view from an airplane isn’t that great, you can only see about 3 square miles of ground and you have to look nearly straight down to do so; the earth has shrunk to the point that the horizon is only 1 degree above straight down from 5mi up! Just as the sun abruptly appeared out of nowhere in the sky at our previous curvature magnitudes, so does the Earth here even on short flights - you don’t have to travel far for earth to disappear into the darkness, just slightly above cloud level suffices.

Not that you’d ever fly that high. For one, it’s terribly inefficient. As the earth’s radius is only 13 times the hyperbolic unit, moving upwards one or two or seven miles greatly increases the length of your journey. At just one mile up your trip is 2.7 times as long, and at 5 miles it is 148. That means if you were traveling at 500mph in an airplane and your friend was going at 3.36mph (ie, walking) on the ground, you’d cover the same amount of ground in the same amount of time. So, no airplanes and no cellphones.  Bummer.

While it would be foolish to ride an airplane for travel, you might want to take one up for the cool views. This might be a bit uncomfortable as well - even if the Earth were not spinning (so the airplane was the only cause of motion) flying at 500mph at a height of 5miles would cause all things inside the aircraft to feel a force of 70,000g squishing them to the ceiling, ripping the plane apart. So to live, go lower and slower. It turns out the best combination for sightseeing is to take a plane up to a height of 2 miles and fly at 125mph; here the outward force would be -1g and so you would be pressed onto the ceiling of the plane with a force equivalent to what you currently feel from your chair.  Comfortably looking up from your seat and through the glass floor of the plane at the tiny earth below (above?) now thats a view I’d pay for!

So, if planes are out of the question for rapid transit, what are we to do? Remember the surface area of the world has grown by thousands of times - theres tons and tons of places to visit how are we to get there?  Since going upwards was the problem (exponential increase in circumference and all) what is ground transport like?  Could we survive on high-speed trains or the like? For medium length journeys this turns out to not be that bad of an idea, down at ground level traveling at car-speeds is no problem, traveling at 500mph results in 0.3g of centripetal force and going supersonic results in a pull of 1.2g (it quickly ramps up from there). Thus we could build ourselves a hyper loop type system; high speed trains moving at approximately the speed of sound - with the only difference being that we’d be sitting on the roof again! However the outward force increases exponentially with speed, and so anything faster is pretty much out of the question - and with 2000 times the surface area to cover, the fastest possible circumnavigation of the globe would be 77 days, so the best case scenario appears to be travel times would look more like they did in the 18th century and the world would forever feel unimaginably big.

Luckily this turns out not to be the case - we have just been thinking about things wrong, using our Euclidean intuition.  Airtravel and ground travel are not the best ways to get around a hyperbolic Earth - tunneling is!  Remember, the radius of the planet is a mere 13 miles!  So while Australia might be over half a million miles away across the surface, its only 26 miles straight down! Thus, if it could be accomplished, digging geodesic paths through the earth would drastically change things. For instance, two points which are 100 miles distance on the surface could be connected via a subterranean path only 9.8 miles long, and points ten times further apart still by a tunnel of length less than 15 miles! These tunnels do go rather deep underground (3.9 and 6.2mi respectively) but I’m sure we’d figure it out. This insane length saving continues, with a path 10,000mi on the surface being only 18mi long underground, and a path directly through the planet is 26mi but would take 638,973mi up on the surface.

What does the ground look like when you walk around?  At this level of curvature the ‘pointiness’ of hyperbolic spheres becomes apparent even at day-to-day scales: the surface of the ground will not appear flat like we are accustomed to, but will appear positively curved around you, as though you are standing on the top of a small hill (that moves with you as you walk around) This apparent sloping away of the ground is gradual over short distances (for instance, over the span of 100 ft the ground appears only to dip 0.92 feet below the horizontal), but this effect grows quickly with distance (as we’ve likely come to expect!) - standing on one side of a football field the opposing team’s goalposts appear to be 170ft below your feet!

This property leads us to an interesting question; how do buildings work at this level of curvature? In Euclidean space, we can build our floors flat because that shape is both approximately an equipotential for gravity, and totally geodesic. In hyperbolic space, these two notions diverge. In building a building, you want the furniture not to all roll to the same side, so you want the floor to be perpendicular to gravity.  This means the floors should be sections of concentric spheres.  And the walls? You want them to be straight up and down (so the building isn’t top heavy or anything) which means you want them to run parallel to gravity.  This leads to the interesting property that the higher up floors have more room than the lower floors: the math building would have barely any perceptible change, but a skyscraper of height 600feet would have a top floor with area 1.35 times its bottom. The freedom tower in New York would have a top floor 2.7 times as spacious as its base.

## Human

Now for fun, lets be a little wild. We’re going to set the radius of hyperbolic curvature to be 1m, so everything is happening on a human scale. The radius of the earth is now only 75feet!  Keeping the volume the same has let the surface area run wild however and we end up with over 4million times more land area.  Oceans of the same volume would barley be enough to wet the ground over a similar proportion. What would a world look like if we opted to fix the surface area instead? Turns out the world would have one ten-millionth the volume and a radius of 53feet, meaning that for the sake of the things we will be considering; you can think about either case!

From sitting height you can see at most 0.9meters away from you before the world is behind the horizon, and when standing (or being any higher up actually) you can only see up to 1 meter away. The horizon is at much less than 1degree away from the vertical, meaning that the entire world appears down in a little tiny ball by your feet.  Standing in a cubicle, the corner where the wall meets the floor would be out of view, and just like the stars faded from view before, nearly everything does here. You need a head-lamp to see pretty much anything; as light from a light source appears 60ft away from only 10ft, and all but invisible from 15ft.

Buildings are out of the question: if you want to keep the floor level to the ground the ceiling of a single story has over a million times the radius of the floor (and only a 6ft ceiling at that! you’d have to crouch) and over 10 to the million times the area.

Ok, well if we can’t build buildings, would universities just conduct all their courses outside? Turns out that’d be pretty hard as well. If we wanted our books to have flat pages (ie locally hyperbolic planes) they wouldn’t close, if you try to stack a bunch of hyperbolic planes on top of one another they crinkle up.  Even though reasonably sized books would approximately close they wouldn’t stack well on a bookshelf for the same reason. Scrolls are also out of the question; hyperbolic paper doesn’t roll up onto an equidistant tube (these are euclidean). Desks would be trouble as well; if you want the top to be hyperbolically flat it would curve down in the middle right to the floor, and so all objects would roll to the middle. If you made it curved you couldn’t set paper on it without crinkling.

Ok, so its dark out and we can’t do math even outside with our headlamps. Can we at least move around?  Turns out even that is complicated: the speed at which geodesics diverge means that a spaceship traveling at the “slow” speed of 17,000 mph will experience 10million g’s (6ft from central axis) blowing it apart. In fact, sitting in the passenger’s seat of a car would cause 131g at 60mph. Thus, all cars would be long and skinny so you could sit on the translation axis. But even this would lead to 12g on the shoulders at 60mph, pulling you apart. You could stand sideways and experience only 6.2g, but that’s not very helpful. Plus, your car axels would have long ago been blown apart. Bikes are the option! You can bike at 15mph while experiencing only 0.7g (for a net 1.4g pulling you apart) uncomfortable, but probably livable.