4D rivers

[PWrong]

I think we're probably in need of a topic that's actually related to the fourth dimension. Not that the other topics aren't interesting, but we ought to have at least one related thread. So how about 4D rivers?

I once read something about how einstein discovered that the ratio of the total length of a river to the distance from one end to the other was approximately pi.

The idea is that erosion makes the river more bendy, but eventually the river doubles back and cuts itself off, so it creates an Oxbow lake and straightens up again. On flat ground, the length is about 3 times greater than the actual displacement, as the crow flies.

Does anyone know how to calculate this?

If this turns out too difficult, here's a more general question. Alkaline's introduction says that 4D rivers would corkscrew. Why is that? Wouldn't they just meander chaotically but generally travel in a straight line?

[quickfur]

Does anyone know how to calculate this?

Nope, wish I did.

If this turns out too difficult, here's a more general question. Alkaline's introduction says that 4D rivers would corkscrew. Why is that? Wouldn't they just meander chaotically but generally travel in a straight line?

A corkscrew eventually does go (roughly) forwards. I assume that by corkscrew, alkaline is saying that 4D rivers have 3 degrees of freedom in which to turn, so they'd flow in some 3D corkscrew-like shape.

What I'm still trying to figure out, though, is how alkaline concluded that 4D lakes can be easily circumvented. Assuming that the surface of a 4D lake would fill a 3D region, I don't see how walking around it is any different from the 3D world where you walk around the lake, and can't walk across it (unless you swim). A 4D analog of Earth would have a 3D surface, and the lake blocks a 3D chunk of this surface, just as our 3D earth has a 2D surface, and lakes block a 2D region of this surface. It appears to be about the same thing, unless I'm missing something obvious. Maybe 4D gravity constricts the lake more, so that it doesn't cover a full 3D area?

[Keiji]

A corkscrew eventually does go (roughly) forwards. I assume that by corkscrew, alkaline is saying that 4D rivers have 3 degrees of freedom in which to turn, so they'd flow in some 3D corkscrew-like shape.

Nope. 3 directions of movement. So only 2 degrees of freedom, because all rivers go down.

What I'm still trying to figure out, though, is how alkaline concluded that 4D lakes can be easily circumvented. Assuming that the surface of a 4D lake would fill a 3D region, I don't see how walking around it is any different from the 3D world where you walk around the lake, and can't walk across it (unless you swim). A 4D analog of Earth would have a 3D surface, and the lake blocks a 3D chunk of this surface, just as our 3D earth has a 2D surface, and lakes block a 2D region of this surface. It appears to be about the same thing, unless I'm missing something obvious. Maybe 4D gravity constricts the lake more, so that it doesn't cover a full 3D area?

A cubindrical body of water, provided it was thin enough and on the ground, could simply be walked around, just like we can "cross" a river by going all the way up to the place where the river starts and walking around that.

Similarly, a boat can avoid a cubindrical land bridge by just going around it. So bridges are almost completely unnecessary in the 4th dimension.

What you DO need a bridge for however, is when you have a larger body of water that would be impractical just to walk around.

A 4D analog of the surface of Earth would have continents "floating" in a huge 3D body of water. So ships' paths wouldn't need to cross each other.

[PWrong]

A cubindrical body of water, provided it was thin enough and on the ground, could simply be walked around, just like we can "cross" a river by going all the way up to the place where the river starts and walking around that.

We only really need to consider the surface/surcell of the water. The depth of the water is irrelevent. In other words, bodies of water in 4D should be defined by 3D shapes, rather than things like "cubindrical". Also, I think it's better to define a body of water by it's size in each direction, rather than its shape.

So, lakes in 4D would come in three types. You get long, thin lakes, like rivers, that would apparently corkscrew. You could easily just walk around it, as shown in the picture. These would form in the same way as rivers. They start on a mountain, then flow downwards.

You also get lakes that are large in only two directions. They might be long and wide, but tarrow. This would divide the land in two. You might be able to walk around, but it would take a long time. So you do need a bridge for these. I'm not sure how these would form though, so they might be uncommon in 4D.

Finally, you get lakes that fill a 3D space and are large in three directions. These would form the same way as a lake in 3D. Rain falls on the ground and fills up depressions in the ground. You can't easily build a bridge over these.