4+ spatial dimensions and how to manage them

[Khantazm]

Basically, I'm writing a story that involves beings of 5 spatial dimensions visiting Earth. And when I say "writing" I mean "putting thoughts down on paper as they come to me" or almost that, i.e. not much thought has gone into world-building. Mostly 5-dimensionality manifests in crazy transformations of architecture and vehicles, and psychodelic walking into completely different places while apparently staying still. Also, as a map gets infinite increase in stuff when it's tranformed into a globe, the 5d Earth has much more geography than our 3d Earth, but for some reason most of the extra geography is barren, except for, lately, some lichen and giant fungi (a half-formed reason for that might be that only those fungi have a mutation allowing them to travel in additional directions).

Additionally, the story is inspired by Doctor Who, so the 5d beings look completely like humans (except maybe for wings that are maybe the part of them that extends into the other dimensions, or there is more to them than is seen)

And finally, 5 dimensions could be an overkill, but 4 seemed to be too pedestrian at the start, plus if I did 4 it would be too easy to assume that I mean space and time, but they make plotting a mess, because as I understand it, at any moment my characters can end up anywhere in space (if not in time, but I didn't want to copy Doctor Who completely).

It all seems to me to be almost incoherent, but I wonder, if via a few generous cuts and edits, the world could make a bit more sense*, and if anyone here would have any suggestions as to how that might be accomplished.

*That is, put more "science" into "science fiction".

 

[Drakkith]

Imagine pushing your finger into a 2d world from 'above'. (Which can't be experienced by the 2d observers in the 2d world) From their perspective there is a small point that magically appears out of nowhere and gradually expands as you keep pushing the tip of your finger through. To me, that's what someone visiting a world with less dimensions than themselves can do. Your 5d beings can, quite literally, step 'out' of the universe through the 4th or 5th dimension just like you can pull your finger 'up' out of the 2d world. This may or may not allow them to move about quickly, it depends on scale and such.

 

[Khantazm]

Wut wut wut! How does it depend on scale and such? Please clarify! When I compare to the map-globe analogy, there would only be an increase in speed if there were a tunnel. By the way, does anyone know what shape matter tends to take under gravity in 4 (or 5) dimensions? Hypersphere?

 

[Drakkith]

Wut wut wut! How does it depend on scale and such? Please clarify!

I don't know really, I don't quite remember what I was thinking when I put that in. Now that I think about it, it doesn't make much sense, so I'd just ignore it.

 

[Curious3141]

Scale may not matter, but curvature does. If you visualise a 2-d manifold embedded in 3-space, the manifold can be completely flat, or it can be curved. The latter is far more interesting, in that it affects the appearance and speed of sections/projections onto the manifold.

Let's say your 2-d manifold is curved sharply like a U or V in the middle, between two infinitely large flat expanses on either side. You are a 3-d being playing with 2-d creatures (let's unoriginally call them "Flatlanders" (FLs) after Abbott) that are unaware of your existence. A 3-d cylinder that enters the flat part of the manifold (planar) orthogonally will appear as a circle to the FLs. If it starts moving (in 3-d) at a constant speed parallel to the 2-d manifold, it will appear as a circle moving at constant velocity.

Now when that moving cylinder hits the curved part of the 2-d manifold, interesting things will happen. It will suddenly become an ellipse (or more generally, a conic section of some sort). It will also appear to suddenly speed up from the perspective of a FL, because it is traversing more of their space in the same amount of time.

We can only imagine 3-d projections to 2-d, but you can try to extrapolate the same sort of thinking to higher dimensions. But some things about higher dimensions are nonintuitive. For example, in 4-d, two planes can actually meet at a single point (in 3-d, two non-parallel planes necessarily meet at a line). So beware of traps like this.