gonegahgah wrote:[...] Just as our 3D rivers tend to follow one preferred direction for most of their lengths with occasional splits - instead of splitting off into several different north, south, east, west or between directions all the time - I tend to believe a 4D river will tend to follow one preferred direction as well.
Generally splits occur to form forks and you would get something similar with 4D rivers.
Yes, one would expect that would happen.
This topic of river formation is actually quite interesting. I experimented with automated terrain generation algorithms some time ago, where you start with an essentially random elevation map (which looks nothing
like a realistic terrain, with unlikely spikes and precipitous pits sprinkled everywhere with almost no flat surface) divided into tiles. Then you simulate rainfall by assuming X amount of rain falls on the terrain, and then trace the direction of water flow by following the path that leads to the lowest altitude from each map tile. This in itself doesn't produce anything realistic either, since the water just collects in randomly-scattered dots where the pits are. But then you add water erosion to the simulation: every time water flows from tile X to tile Y, it carries some "topsoil" along with it, so that tile X reduces its elevation and tile Y increases its elevation. You can also optionally add wind erosion to the simulation, by causing very high tiles surrounded by very low tiles to lose part of its height and distribute them into surrounding tiles (simulating rocks falling from steep cliffs into valleys below). After running this simulation for some cycles, all the strange jutting spikes and precipitous pits have evened out into a more realistic-looking terrain, with many more-or-less "smooth" areas that one can actually walk on.
Now what's most interesting is this: as you run the simulation, store a "water saturation" value in each tile, which basically represents how much rain falls on it plus how many times water flows over it. At the end of N cycles of the simulation, color the tiles whose water saturation is below a certain threshold T brown (the idea being that soil can soak up a certain amount of water), and color the tiles whose water saturation is above this threshold blue (soil can only soak up water up to a certain point, past that point it can't hold anymore and the water level rises above the soil). What's really amazing is that when you adjust the value of T appropriately, you get an amazingly realistic terrain complete with rivers that begin at mountainsides and fork every now and then until water collects at the lowest elevations as little pools. You can even get oceans if you fill up tiles below a certain elevation (the "sea level") with water.
So basically, I started out with a completely random elevation map, and ended up with beautiful mountains, valleys, and plains, complete with rivers with forks and lakes, just by simulating the action of rain and water erosion. Of course, the terrain is still not 100% realistic-looking, but it's pretty darn good for such a simplistic algorithm.
So assuming in 4D we also have rainfall and water erosion, both of which are very reasonable assumptions, we certainly expect that 4D terrain would also have mostly-linear rivers with the occasional fork, with water collecting in lakes and seas.
[...] One of the cool things with 4D that you can actually touch the inside of 3D shapes.
You can't touch the inside of hypersolids but you can touch the inside of any of their hyperfaces.
So if you have a tesseract you can touch one inside of each of the 8 cube faces from the one hyperside.
And in fact, most of the time 4D beings would touch the inside of the cube facets--just like most of the time, we don't handle a cube by its corners or its edges, but by its faces.
[...] I must concede that the chimney hyper-riverbank provides a means to encircle the river without needing to cross it.
Bingo. Now you know how to walk around a 4D river.