Living in the surface-volume of a hypersphere [closed]

Context:

I'm designing a "demi-universe," accessible from Earth (via appropriately supernatural means), that has the following internal topological property: given any point in 3D space, and any 3D vector, the ray from that point along that vector will eventually reencounter itself from the opposite orientation. In less formal terms, if you go in any direction, in all three dimensions, you will eventually "loop" back to where you started:

• If you go east, you will eventually arrive at your starting point from the west.
• If you go west, you will eventually arrive at your starting point from the east.
• If you go north, you will eventually arrive at your starting point from the south.
• If you go south, you will eventually arrive at your starting point from the north.
• If you go up, you will eventually arrive at your starting point from below.
• If you go down, you will eventually arrive at your starting point from above.

Moreover, in any of these directions, or any other you could care to name, the distance required to make exactly one complete "loop" is the same (and is large by human terms but still significantly smaller than Earth's diameter). These conditions force the world to be geometrically a hypersphere, or rather the "surface" of one.

In addition, the world has a form of gravity with a strength approximately that of Earth's, but always pointing in the same direction (which thus becomes "down" by virtue of there being no other way to distinguish directions).

Assume that this world contains air with approximately the right pressure and composition for humans to breathe comfortably, and further assume that there are mechanisms that will "refresh" the air's composition as humans convert the oxygen content to carbon dioxide. Also, assume that any food and material requirements that can't be produced on-location can always be imported from Earth without too much trouble, and that any unusable waste can be dealt with similarly. Finally, assume that the world is a completely closed system aside from any deliberate transport of matter and energy to and from Earth (and the above-mentioned air refreshment mechanisms).

Question 1:

What kinds of weather patterns can one expect from a world like this?

Question 2:

What sorts of societies would form from long-term human colonies within this world?

• I'm picturing the land as taking one of two forms: either islands floating in space, or a solid surface, in which case the underside of the surface is always above you, so there is no sky (essentially, a massive cave w/o cave walls). I get the sense you intended the later. If so, the weather is going to be dependent on a number of factors: how much space is there between the top of the surface, and it's underside above it? Is the solid part uniform, or does it have mountains, oceans, and the like? Is the underside flat or varied? Is the amount of heat/light received over the surface uniform? – LindaJeanne Aug 21 '15 at 10:46
• I think we need clarification on land, energy source and water source (are there oceans?) Before this can be answered. – Tim B Aug 21 '15 at 13:00
• This must be the surface of a hypersphere? Other hypersurfaces satisfy the start-in-one-place-and-return-to-it requirement. – HDE 226868 Aug 21 '15 at 16:19
• What fraction of the sphere should be filled with landmass? (If we look at earth as model, at which latitude should be the surface of land?) – celtschk Aug 21 '15 at 19:07
• About analogy to "Always pointing south": note that there is north pole and south pole. Analogically on your hypersphere there would be the highest point and the lowest point. In the bottom there would be ground, and above it the sky. The surface would be a sphere, like the surface of Earth, only that there is not cosmic space, but above the highest point you go down again. – BartekChom Aug 22 '15 at 9:37

The weather will be quite boring. As you say

"assume that the world is a completely closed system aside from any deliberate transport of matter and energy to and from Earth"

That means the system will be nearly at its maximum entropy. Everything is the same temperature. Hopefully that temperature is hospitable to life. The only gradients that exist are there because of this portal from Earth, and that is most likely a very small source/sink in a very large world.

While it might be novel to visit this place, it wouldn't be a very interesting place to live. There is nothing that I know of which would be easier to do in such a place (except not get lost if you can keep going in the same direction). All the energy that people need to grow food would need to be brought by the humans. The societies would be similar to those that might develop deep underground, but there couldn't be very much branching from Earth culture because constant deliveries of supplies and energy would need to be made from Earth. Isolation would mean death.

the distance required to make exactly one complete "loop" is the same (and is large by human terms but still significantly smaller than Earth's diameter).

The Earth's diameter is about 13,000 km. I think a hypersphere circumference (length of the "loop") of 1200 km (about 9% of the earth's diameter) can be considered "significantly smaller", so I'll take that value. Thus, since the circumference is a great circle, we can calculate the radius of the hypersphere as $R = 1200\,\mathrm{km}/(2\pi) \approx 200\,\mathrm{km}$. This is certainly large enough that for human-sized objects, the curvature effects are negligible (that is, the space feels flat; being in the hypersphere doesn't feel fundamentally different than being in Earth's space). The total volume of the hypersphere is $2\pi^2 R^3 \approx 16\cdot 10^{7}\,\mathrm{km}^3$, According to Wikipedia, this is about the volume of Lake Baldegg, Switzerland.

Since you assume a gravitational field pointing towards the "south pole" of the hypersphere, all the landmass will be collected in a sphere around that point.

As of your comment, the landmass should fill 20% of the hypersphere. According to my calculation (see section "Calculations" below) this gives a landmass with surface area of about $11 R^2 \approx 440\,000\,\mathrm{km}^2$. This is according to Wikipedia between the size of Japan and the size of Spain.

Next, let's look at the floor. Since the rays, following great circles of the hypersphere, apparently are bent away from the land surface, the surface looks more curved than it is, as if the sphere you're standing on were smaller than it is. Behind the horizon begins the back of your head, and when you follow upwards, you'll see the rest of your body from behind, until you reach the floor you're standing on. That will then reach up to the "sky", as if in a gigantic cave. Note that all that is still turned around by 180 degrees.

Apart from these optical effects, I don't expect anything particularly interesting in the hypersphere. As the others already wrote, there will be no weather to speak of. Any humans living there would not form a colony, but more the equivalent of a moon base, where everything important has to be brought from Earth.

Calculations:

If $w$ is the 4D coordinate along the "axis" of the hyperball, the volume of the landmass (which is spherical in the hyperspherical geometry) is \begin{align}V &= \int_{-1}^{w_0}4\pi\rho^2\,\mathrm dw\\ &= \int_{-1}^{w_0}4\pi(1-w^2)\,\mathrm dw\\ &= \left.4\pi\left(w-\frac13w^3\right)\right|_{-1}^{w_0}\\ &= 4\pi\left(w_0-\frac13 w_0^3 + \frac23\right) \stackrel!= \frac15\cdot2\pi^2 \end{align} So we end up with the equation $$w_0 - \frac13 w_0^3 + \frac23 - \frac\pi{10} = 0$$ Solving this numerically with Mathematica gives as only viable solution $w_0 = -0.369295$.

Question 1: No weather

You have even heating everywhere so beyond occasional breezes there is nothing to generate atmospheric currents. You have no sources of atmospheric water so no evaporation to form clouds and no precipitation.

Question 2: None, they are all dead.

With no water cycle there can be no life here.

Assuming this was somehow fixed (for example water magically appearing in the sky) then it would have very little affect on society. The world you describe is large enough that for most people the details become irrelevant. The lack of compasses might make navigation harder. The lack of seasons would mean there wasn't much need to develop food storage techniques and shelters could be simple.

A good starting point would be looking at our own tropical societies, places that live with very little seasonal variation.