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Having some thoughts about 4 dimensional spaces.
I've heard it said before that humans can't conceive of or perceive 4d spaces.
I was thinking that this isn't a mental limitation it's a perceptive and specifically a visual limitation.
Vision is basically a 2d sense, so it is limited in that it can only accurately perceive 2d environments (like a map). Our brains are able to do some juggling and work this 2d sense into perceiving our 3d environment. And it helps to have more than one eye to convert the 2d senses into a 3d understanding.
Some of our other senses are what I might consider 1d. Hearing and smell are just intensity detectors. They aren't really for navigating our 3d environment so we don't often think about how limiting they are in that way. Hearing doesn't feel 1d because we have ears that alter the sound and allow our brain to figure out directionality.
Here is the fascinating thing: we do in fact have a sense that it is 3d. Our sense of touch or our basic bodily self awareness.
If you try to imagine 4d spaces visually they make no sense, but if you instead imagine being able to contort your body inside a 4d space it can seem a little weird but not 'my brain is totally broken' levels of weird.
Simple exercise:
Imagine a bag of holding from dungeons and dragons. It's a small 6 inch round bag. But if you reach inside there is about a circular yard of space. This is a 4d space. Visually it's confusing as all hell, especially if you imagine the outer material of the bag being see-through. But imagining reaching in their with one arm while your other hand holds the bag is not all that confusing.
So by string together multiple 3d senses, via our sense of bodily space and touch, we can perceive 4d environments. We don't really have 4d environments so our brain doesn't have any built in hardware to make this easier.
5 spacial dimensions is where things might actually go off the rails. I have no idea how to even describe such a space. Certainty nothing as simple or widespread as a bag of holding. Curious if anyone else can think of 5 dimensional spaces used in fiction?
Inability to perceive 4D spaces just kills me. It turns out that "imaginary" numbers are actually at the root of reality, and most functions we're interested in are rooted in analytic multivalued functions, visualizable in ℂ×ℂ. That's a 2-complex-dimension space, so it's 4-(real)-dimensional, so we're screwed. Best you can usually do is to switch back and forth between plots of output magnitude and phase (or between real and imaginary components of output), or plot magnitude as height along with phase as color. Fortunately we don't have to be able to visualize something to describe and compute with it, but I feel like it could have helped a lot.
The "bag of holding" trick is clever, it gets you the topology of a 3D manifold that can't be embedded in less than ℝ⁴, but to me it "feels" like a very fixed geometry - two parallel 3D spaces, with the "hole" of the bag's opening connecting them.
TVTropes is shockingly empty of 5-D spaces in fiction. There's a Greg Egan book that takes that seriously, there's a Douglas Adams joke, there's a corny Superman villain, and it's sparse and downhill from there.
I don't think I understood any of this.
Its the simplest 4D space I could think of, but I think our touch perception would still work just fine on the most complex 4D space available. In an unobstructed 4D space your 3D senses would continue to work just fine, just as in an unobstructed 3D space your 2D sense of vision works just fine. Its when the space is obstructed that the lower dimensional perception becomes difficult or confusing. A wall obstructs 2D vision. But if there are no obstructions there isn't much to perceive either. In 3D outer space you can turn any direction and see infinite nothing (except the stars). In 4D outer space you'd be able to turn more as you twist into that 4th spacial dimension but you'd feel nothing different. You could do some visually weird things like phase your hands through your own body. But the actual sensation and mental model of you doing that wouldn't feel weird. You can sort of do it right now if you have a big enough beer gut, just press your hand into some soft tissue and move it out of the way, your hand is now where your body normally is. The only difference is that in 4D you wouldn't have the dual feedback of the skin pressing against each other.
I think the best spacial sense would work something like knowing the fluid shape of the area around you. Going back to fantasy, imagine a slime monster. A gelatinous round ball that can only feel its "skin". For a slime navigating any dimensional space is all the same. If you magically found yourself in a 4D space you might be best off acting a bit like a slime by closing your eyes and feeling your way around. Your eyes will lie, your touch won't.
Stuff with boundary conditions is modeled by differential equations. Differential equations have wave-based solutions. Waves must be represented with 2 numbers, and one way to do this is to separate the “real” and “imaginary” components. If we treat these like (x,y) pairs, we can graph them on a plane just like any other pair of numbers. The set of all these complex numbers is denoted ℂ.
Royst is talking about a different (but similar?) class of equations which have solutions that require 4 numbers. To graph them, we’d need two simultaneous planes: ℂ×ℂ. So we’re out of luck unless we want to get cute with color.
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