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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.
My apologies. I'll back up, if you're still curious.
Think of the function sin(x).
We can take a number, like x=π/3, and plug it into the function, and we get another number, in this case sin(π/3)=√3/2. (here π/3 is in radians, which when we start doing calculus turns out to be more natural than 60°) We can imagine doing that with every real number, and plotting every (x,y) on a plane, and we get a "sine wave" picture like this. That "plane" gets called ℝ×ℝ, or ℝ², because it's defined with 2 real number (ℝ) lines that form a cross intersecting at one point. It's a great picture! I can think about the function inputs as being the length of lines in one direction, outputs as the lengths of lines in another, derivatives as slopes of angled lines, etc.
But ... how about sin(i), where i=√-1? On the one hand, who cares, because it seems like √-1 shouldn't exist: there's no real number whose square is negative, and even when we found such numbers to be useful intermediate results in algebra problems we still decided to call them "imaginary" as opposed to the newly-named "real" numbers; you'd still expect to have a real number in the end. On the other hand, we soon found "complex numbers" (ℂ, all the numbers x+yi you can make by adding a real number x to an imaginary number yi) to also be useful in engineering problems (they represent oscillation a way similar to how positive numbers can represent growth and negative ones decay), and then we found them to be useful in physics problems (where a "quantum wave function" takes complex values), and at some point it's hard to ignore something as not "real" when it's at the foundation of our understanding of reality.
We can plot a collection of complex numbers on the "complex plane": for every complex number z=x+yi you just plot it as (x,y). One complex number can be described with two reals.
But how do we plot a function that takes complex number inputs and gives complex number outputs? We would need to plot it in ℂ×ℂ, two complex planes that form a cross intersecting at one point. "But two planes meet in a line, not a point", you might object, and that's true, in 3D. ℂ×ℂ only fits in 4D. If I wanted to clearly plot part of a real function y=f(x), I can plot each point as (x,y) in a square, but if I want to clearly plot part of a complex function f(x+yi)=u+vi, I need to be able to plot each point as (x,y,u,v) in a hypercube. I don't have any hypercubes lying around! I can't even visualize a hypercube.
So, we plot garbage like this instead. The xy plane there is the complex plane of inputs x+yi, and for each output u+vi=sin(x+yi), the height z of the red surface is u and the height z of the green surface is v. We plot (x,y,u) and (x,y,w) in the same cube and try to picture the true (x,y,u,w) from the result. Those two 2D surfaces twisting through 3D space are really two aspects of a single 2D surface twisting through 4D space. They're easier to understand if you use that web page to rotate them back and forth and turn them translucent, but still I can't picture the single surface in 4D that they represent. If I could actually visualize 4D then the plot of that single surface would fit in my head as naturally as that first "sine wave" plot did.
I think here it depends on what you mean by "in a 4D space".
If my movements were naturally restricted to a 3D manifold (a "surface" is just a 2D manifold) curving through 4D space then you're probably exactly right. Let's back up to 2D. Imagine as an analog a 2D version of me, living on the surface of a globe. Open my eyes, and if light also follows the globe surface then in any unobstructed direction I look I see the back of my own head one globe-circumference away, but if I'm small enough compared to the globe then it feels almost like I'm in good old flat 2D space. Even if the globe is made of taffy and some 3D monster stretches spikes out of it, mushes parts of it together elsewhere to make a torus or worse, whatever. I can still move around any weird surface I'm stuck to so long as it's smooth enough, to any part of it I want to go to so long as it's it's connected. When I'm on the globe, or on any points of "positive curvature" on a more complicated surface, I might feel a little weird (there's more "room" inside a shape than you would expect from its boundary, so it might be like my skin is getting compressed or my innards stretched). Or, on points of "negative curvature" on a more complicated shape, I might feel like my skin was getting stretched or my innards compressed. But either way, if I was small enough compared to the curvature then I'd still be just a slightly squished-around version of me.
Your "bag of holding" example actually is a 3D manifold - locally I can move parts of my body in no more or fewer than the usual 3 dimensions: up/down, left/right, or forward/backward. But those things are only consistent locally - if I stick my arm 10 inches forward into the bag and then reach 10 inches up, it won't be in the same place as if I reach my other arm 10 inches up (outside the bag) and then 10 inches forward. This 3D manifold has geometry that can't exist in 3D space, but only embedded in a space with at least one more dimension.
But with the same one extra dimension, if my movements were unrestricted? Local senses like touch would get weird too. Imagine that 2D me, previously stuck to the globe like a flat sticker (though free to move parallel to the globe surface), suddenly peeled away into the air. I can still wiggle around in my accustomed two directions, but my orientation with respect to that third direction is at the whim of the breeze. On a globe I might be able to look or propel myself north/south vs east/west, but 2D me has no muscles that can turn his limbs up/down. Even if someone took pity and stuck me back on the globe so I could move around its surface again, if they stuck me on backwards then I'd be backwards for the rest of time; clockwise would seem to be counter-clockwise and vice-versa. 3D me in a true 4D space would be in the same boat; my arm has no way to reach hyperup/hyperdown.
I think I understood the latter half of that and it was a fun ride, thanks for writing it.
Thanks! If you haven't already read Flatland, you might enjoy it. It lacks some of the mathematical sophistication (when it was written, general curved manifolds were still a cutting-edge idea) and brevity (though it is only 100 pages, and a fast read) of my ripoff here, but it does retain some attributes I had to drop like "social satire" and "literary quality".
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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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