how can someone visualise fourth, or higher dimensions?
can our brains actually learn to comprehend, to envision dimensions beyond the perceptible three? how could you describe higher dimensional shapes in a way that would allow someone to visualise them?
Honestly, we already have trouble just imagining the capacity of 1, 2 and 3 dimensional spaces to be subdivided infinitely. We can't even address (as in give names to) the majority of non-rational numbers very well, and the majority of that line-space remains effectively unmapped mentally on an infinitesimal scale.
4th dimensionality gives each point in an infinite and infinitely divisible 3d matrix an infinite degree of variation, each point now contains a whole line. 5th dimensionality gives each point 2 degrees of that variation, each point containing a plane. And, 6th dimensionality gives each point in an infinitely dividable matrix its own whole 3D vector space. Past 6th gets even more tricky as we're forced to stack metaphors, infinite lines in infinite spaces in space for 7th and so on.
Beyond that stacking of metaphors, I don't think we can really picture it, if you mean visualizing in the more traditional sense wherein you could attempt to commit a scene to canvas or something.
There’s a book called Flatland that you’d like if this kind of thinking interests you. It’s a pretty short and easy read.
Imagine you’re trying to describe “up” to a 2D person. You would have to tell them to move in a direction orthogonal to to what they can comprehend. To them it would be more like moving “inward” or “outward”. Once they are out of their plane it would make sense.
Now imagine being a 4D person trying to do the same to us 3D types. You would need to move “outward” in a completely different direction then we’ve ever headed. It would be orthogonal to our 3 axis. Once we broke free and move in this outward direction everything would make sense.
The way that works best for me is to use time as the extra dimension. Each moment you observe is a 3d “slice” of the universe as you move forward in time at a fixed rate.
To analogize, imagine a circle that lives in an XY plane, moving around in this plane as the plane itself glides along in the Z axis. You’d see some weird snakelike structure growing off towards the sky, but the circle only ever experiences it’s movement in the XY plane. It surely remembers where it was before and has some idea where it will be in 5 minutes, but you can see every point of it’s existence in 3D space all at once. Likewise a 4d being could in principle see your entire timeline at a glance.
This has some weird Lovecraftian implications though if you imagine what that 2d circle would see if it a 3d sphere happened to cross through its plane, and then extrapolate that to the 3d world… The exercise is left to the reader 😉
I was doing something similar; for one dimension, I imagine a sequence of dots. the 2nd dimension adds a series of new lines of dots, forming a flat sheet, the the third dimension adds new rows of these sheets.
for the fourth dimension, you regress back to the first; now all dimensions before are encapsulated within each dot.
however, in the past I came across a website which allowed you to manipulate a hypercube, rotating it through our familiar dimensions as well as the third it extends into. I found myself utterly unable to predict how rotating the two dimensional image of a 3d representation of a 4d object would alter what was displayed.
ever since then I've been curious to learn to envision what such an object would "look" like. these ways of thinking about higher dimensions don't really shed any light on that.
Some people have managed to visualize the 4th dimension of space-time, though my brain finds it a very slippery concept lol, meaning I think I understand it but I can never remember why it makes sense. I believe our brains are capable of eventually visualizing/comprehending further dimensions, but I think that will take hundreds of years, assuming humanity doesn’t die of climate change or war in the meantime
I guess it depends why you want to visualize the shape, and what each point represents. If a point is a bunch of attributes expressed as a tuple of dimensions, the visualization probably doesn’t have much value.
In some cases, the only reason why an item is expressed as a point is so software can perform operations on it. You can visualize those operations as happening in a two or three dimensional space without losing meaning.
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