Geography on a 4D World

As noted in my last post, planets in a 4-dimensional universe would have 3-dimensional surfaces. What does that mean for geography?

First off, random landscapes in higher-dimensional spaces are less likely to have local minima and maxima. That's why gradient descent optimization works--if your problem space has enough dimensions, you can just start anywhere you like, head downhill from there, and be pretty sure you'll converge on the optimal solution--the global minimum of the landscape--without getting stuck in any local valleys first. 3D space isn't super high dimensional, but it is higher than the 2D surface of our world, which means fewer local minima and maxima. Fewer lakes, and fewer mountain peaks. And at a large scale, more likelihood of a single fully-connected global ocean (which Earth already has anyway) and a single fully-connected supercontinent (which Earth has had periodically). A 4D world with an Earthlike distribution of land and water is thus less likely to have any Australias or South Americas--large places where life can evolve in divergent ways from the rest of the world.

Rivers are still one-dimensional. No matter how high the dimensionality of space, "downhill" is still a vector! But how large and complex will river systems be? In a 2D space, random lines are guaranteed to intersect, and mergers intersections of rivers to form larger rivers with tributary systems are therefore common. Random lines in 3D space, however, will not intersect--and with more space to move around in, rivers on a 4D world will not merge quite as easily as they do on Earth. That doesn't mean they won't merge at all, though! For one thing, river courses aren't random, and rivers that begin near each other are likely to have downhill vectors that also point towards the same place. Additionally, 3 surface dimensions are not enough to avoid knots! In fact, 3 is the only number of dimensions in which one-dimensional curves can form knots and braids. (Braided rivers on 4D worlds could actually be literally braided!) And as plain-crossing rivers migrate over time, they become highly likely to intersect, for the same reasons that cords always get tangled in your pocket. However, being one-dimensional, rivers do not form natural borders on 4D worlds the way they do on Earth. Terrestrial creatures can always just walk around them, as easily as you can walk around a lamppost.

Mountains, however, are a different matter! Hot-spot volcanic mountain chains will still be one-dimensional, but they don't really form borders on Earth, either (although they will form rare local maxima in the terrain). Mountain chains produced by plate collision, however, can form borders! On Earth, plate boundaries are one-dimensional, and so mountain ranges seem analogous to rivers in forming natural one-dimensional borders--but while rivers are one-dimensional in any universe, plate boundaries are not! Tectonic plate on a 4D world are 3D structures, with 2D boundaries, and mountain ranges created by plate collisions will thus also be spread over a 2D area which can bound a 3D region. So, mountain ranges form natural barriers on 4D worlds just like they do on Earth.

A 4D world would also not necessarily have distinct climate zones by latitude--not unless it had only a single component of rotation. That is possible, but in general any object in four dimensions can rotate in two independent planes simultaneously. Each rotation induces a circular pole, which is coincident with the equator of the complementary rotation. While these two great circles are objectively deducible, though, they are not perceptually salient, and have little or no climatological significance. Essentially, there are no fixed point on the surface of a 4D world--everything moves under rotation somehow. This makes celestial navigation... not straightforward.