Working in higher dimensions
Imagine if you will, only being able to move along a straight line. You’re now in one dimensional space. But wait! What if we are allowed, quite literally, to take a left. You are now allowed to move along a square space, this is two dimensional space. We can do better, though. You suddenly can move up and down, traveling in an area that’s the shape of a cube! You’re now in 3D space. Then suddenly you disappear from view, but where did you go? Welcome to the fourth dimension, you can’t see it, you can’t imagine it, but we can do math here and above!
We often think about time as the “fourth dimension,” but time is demonstrably different than a space dimension. We can easily travel in three dimensional space, but have no way to travel through the time dimension, so really we live in a four dimensional world, but only three of those dimensions are spatial, sorry the fourth dimension just doesn’t count like that. So we live in a three dimensional world, but that doesn’t mean we’re limited to three.
The year is 1827 when August Ferdinand Mobius (yes, the Mobius strip guy) realized that Joseph-Louis Lagrange, who in 1788 wrote that mechanics can be viewed as operating in a four-dimensional space (three dimensions of space and one of time, sound familiar?), was limiting himself. Mobius realized that a fourth dimension could be used to rotate a three-dimensional cube onto its mirror-imagine. If that sounds confusing it’s because you can’t think in more than three dimensions.
It wasn’t until William Rowan Hamilton came along that the math was formalized in 1843 and it wasn’t without its detractors, famously people argued that it couldn’t work because they tried to draw out four dimensions and it’s impossible. However, with the math proven, the thought of higher dimensions took off. In non-math purposes, the fourth dimension let fiction writers explore the implications of finding yourself suddenly able to access a fourth spatial dimension, or worse, the forth dimension having access to you!
A couple of my favorite stories about this are “And He Built a Crooked House,” about an architect who builds a house in the shape of an unfolded four dimensional cube, only for an earthquake to cause it to fold in the forth dimension and his adventures inside the newly folded four dimensional home. The second would be “It was the Monster from the Fourth Dimension,” which as the title gives away is about a four dimensional creature and a farmer who has to deal with the weirdness. It looks like a “floating piece of raw meat” and the creature randomly changes shape, size, and even disappears and reappears.
In the monster from the forth dimension case, the monster only looks like a confusing floating meat chunk because it’s a fully formed four dimensional being. The truth is that even if we were suddenly transported to a four (or higher) dimensional universe, we would only see three dimensions. The reasons for this are somewhat complicated, but in the end we’re three dimensional beings, trapped in a three dimensional viewpoint, never to see the beauty of an actual n-dimensional space* even if we could somehow go to one (*where n > 3).
I like the first book because we can actually unfold a four dimensional object into three dimensions, or a more interesting method is to “shine” a light through a four dimensional object to see its three dimensional shadow (like how we can shine a light on a three dimensional object and see a two dimensional view of it). In fact, the internet is littered with animated tesseract images (a four dimensional cube) that show what a three dimensional “shadow” projection of a four dimensional cube would look like as the four dimensional cube gets rotated.
Needless to say, when you break out of three dimensional space into higher dimensions you start to run into weird stuff. In a more practical sense, we can (and often do!) work in n-dimensional space. In my research I often find myself working in four dimensional spaces, sometimes the fourth dimension is just the dimension that I store data collected over time (so a mathematically spatial dimension repurposed as a time dimension).
Working in higher dimensions can be tricky because I can rationalize what I think will happen and typically it works, but you’re always left wondering if that will be the case. More often than not I have to think about the three dimensional analog of an n-dimensional space. I do this by imagining cubes. A cube is three dimensions and if I start stacking two, three, four, etc. cubes I imagine that as the fourth dimension, even though technically I’m still only in three.
This helps me visualize what’s happening when I want to, for example, take the mean across the fourth dimension. I’m basically smashing the different blocks together to return a single three dimensional block. This isn’t how the computer sees it, but it’s the only way I can conceptualize what’s going on. We’re stuck in three dimensions so working in more than three takes some work and really you end up just thinking of ways to imagine the space in three dimensions.
In EEG we often use different dimensions as a “feature.” A feature is just something about the signal we’re working with that help us identify something of interest. Say I wanted to list all the “features” of an apple that let me know it’s really an apple, I may list size, color, weight, shape, all different features. But you can see why using features may be problematic, apples come in different shapes, sizes, colors, etc. So we can get away with using a few features if the things we’re looking for are different enough, apples vs. dogs for example, or we may need a lot of features if we are looking at different types of apples.
The point is that we can end up in very high dimensional spaces pretty quickly. I went from working in four dimensions to five just the other day and it really hurts my brain trying to imagine a three dimensional analog for a fifth dimension, I can no longer add cubes, maybe I need to expand my cubes into a “height” dimension and not just a “depth” dimension, but no matter how many dimensions we work in, the math is always the same.
If I take the mean across the fourth dimension in a four dimensional matrix, I end up with a three dimensional matrix (my smashed blocks example again). Imagining blocks helps because I can take the mean in any dimension, one through four and I will be able to visualize what the new matrix would look like based on the dimension I am taking the mean across.
Since we can’t think in higher dimensions I try not to work above four or five if I can help it. It just makes checking my work that much harder. I can do the math by hand to verify that the output is expected, but it still feels a little unnatural, which I guess is the point since all we know in this world is three. We can easily go from one to two or two to three, but anything higher is just not something our brains ever needed to work in. That’s part of the reason it’s literally impossible to think in a dimensional space that’s higher than our own.
Still days like yesterday, when I found myself forced to work in high dimensional space, make for fun thought exercises. It FEELS impossible that you or I could get dropped into a higher dimensional world and not be able to see or access those dimensions, but it’s true. Who knows, maybe we are existing in a high dimensional universe and through biology have been locked into a three dimensional configuration. If string theory turns out to be correct, that may just be the case (the math has only been found to work in ten dimensions)
The universe is a weird place and the fact that we can play around with the math of higher dimensions just makes things even weirder. I do it all the time for my research and the work I do, but it still feels like some sort of forbidden knowledge. I fully expect one day I will perform an evil math operation in an high-dimensional space and summon the eldritch terrors.
Just fair warning, if anyone was going to have an unfortunate metaphysical math accident it would be me.