How to Visualize 4(+) Dimensions

Note: I am not a mathematician or a philosopher, so basically everything I write right here is coming straight out of my ass. As if I needed to explain that to you.

Ok, so everyday people ask me questions. Marcus, what is the meaning of life? Marcus, what is the most kick ass film of all time? Marcus, how did you become so sexy? Marcus, blah blah blah. But the one question I get asked the most is: how do I visualize the 4th dimension?

Now, I don't know the meaning of life. Everyone knows the most kick-ass film of all time is Evil Dead 2. As for the question of my innate sexiness - genetics and grooming, baby.

You see , most questions people ask me are easily explained with one-liners. But when people ask me about higher dimensions, I have to take my time. Now, I have a limited life span, just like every other creature on this planet. If I actually sat down with everyone who asked me about dimensions and gave them the proper explanation, the universe would have time to burn out and re-expand many trillions of times. So I'm just going to put it here, on my website so that I can just respond to all you simpletons with a one-line response.

So lets get to brass tax here. Most of us don't have a problem visualizing one, two or three dimensions. Why? because we have examples all around us. 1D - anything in a straight line. 2D - a picture on paper. 3D - the physical computer sitting in front of you. But the 4th dimension is a little stickier. They say that the 4th dimension is time. That makes sense. Physical objects are 3D, but they change position over time - the next dimension. Still, it's hard to visualize time. We live in it - we see its effects all around us, constantly, but it doesn't hit our eyes 'all at once' like the other 3 dimensions do.

So, in order to 'see' the 4th dimension, you have to kind of transpose your concepts of simpler dimensions into higher dimensions.

In the first ever programming class I took (Pascal at SBCC baby!), I learned about arrays. Arrays are a type data structure, which make life easier for programmers. Sections of memory are referred to in code by things called variables. A variable represents a piece of memory, and if you need to look at that part of memory or change it, you use its corresponding variable. Think of a variable as a non-dimensional point in space. An array is literally just a bunch of variables clustered together.

The simplest type of array is a one-dimensional array. It is a bunch of variables in sequential memory, lined up, one after another. Am I drawing a clear picture here? A 1D array is representative of a 1D line in space.

The next step up is a 2D array. Instead of being a line of memory, its a rectangular block of memory. Kind of like a chessboard. Each block on the chessboard refers to a different variable, or piece of memory. Also, your computer screen is like an array. It is made up of a 2D rectangle of pixels, each having a unique coordinate.

Then there are 3D arrays. They can basically be visualized as a Rubic's Cube of memory. Now, its possible you're thinking - isn't memory laid out in a 2D manner on the actual chip? How can you have a 3D block of memory? Don't worry about the physical layout of RAM here - the computer is just manipulating things so you can access memory in a logical way.

But the cool thing about arrays is that you don't have to stop there. They can be as high dimension as you want. Let's start with a 4D array. It's simple to access each of this array's elements in code, but how do you visualize it? Well, this is what I do, which I'm sure is not original or anything, but it works: Go back to the 3rd dimension - remember the Rubic's Cube? Now, for the 4th dimension, stretch that Rubic's Cube out into a long line of Rubic's Cubes, one after another. Each one of the Cubes in that line represents one element of the 4th dimension, and the line is what comprises that dimension. Basically what we're doing is thinking of the 3rd dimensional cube (or rectangle or whatever) as a point in space. What dimension is a point in space? No dimension. It's a line of these points that make up something in 1D. This is just a line of cubes.

For the 5th dimension, the next logical step is to take that line of cubes and stretch it into a rectangle, like the 2nd dimension.

6th dimension: a cube of cubes. Pretty cool. But all we're doing is applying our simplistic understanding of 3 dimensions and forcing it to visualize any number of higher dimensions. 7th dimension is a line of cubes of cubes. 8th is a rectangle of a cubes of cubes. 9th is a cube of cubes of cubes. Every 3rd dimension is a cube. It's all divisible by 3.

Hopefully this is helping to visualize something in the real world as 4 dimensions, 5, 6, etc. But it doesn't really cover all the bases.

See, arrays are, by their nature, discrete. Not discreet. Discrete is a synonym for distinct. There is an episode of Futurama (I believe the one with Lucy Liu robots - but I could be wrong) where Bender opens a dating service. The sign on the door says 'Discreet and Discrete'. A little CS in-joke for nerds. That show is chock full of interesting nerd information. Arrays are atomic. Not atomic like bombs, atomic in that they are discrete - distinct. Anyway, each variable in the array is an atom, and each array is just a cluster of atoms. Not a cluster of bombs. A cluster of discrete particles. Ok, I'll stop that now.

I'm not a quantum physicist, but I understand there are differing opinions on how the universe works on a subatomic level. One possibility is that space is continuous, where it is completely nondisctinct - though there are some problems with that idea. Another is that, somewhere, on a deep subatomic level, everything is discrete. When something moves, it doesn't actually make a smooth motion. It jumps. Like on a grid. Like on your computer screen and you're playing video games that look smooth, but if you pay close attention, you can see that its just distinct pixels 'jumping around'. Ok, pixels are really not jumping around, but lighting up and darkening in a controlled manner so that they appear to move, but you get the idea.

Anyway, if you believe that the universe is distinct, then its relatively simple to apply the array simile to the real world. Every motion in time can be broken down to a sequence of 3D pictures in a line (albeit a really really long and detailed line) - much like movie film - and each frame is 3D. But if everything is continuous, how can you visualize the 4th dimension? By a smear. Like a paintbrush across a fresh painting. Like those long exposure photographs where everything that moves leaves a trail of light. But the smear travels in the direction of time - in a line. Or at least, I think that is a way to visualize it. Maybe not. I'm not an expert, just an asshole.

But since I am more confident in describing a discrete universe, I'll go with that train of thought. If you want to think of something in 5 dimensions, simply take that filmstrip of 3D pictures and stretch it into a rectangle. 6th dimension, a cube and so on. What does the 243rd dimension look like? Well, 243 is 3^5, which is obviously divisible by 3. So its a cube of cubes of cubes of cubes of cubes of cubes..., etc. Simple yet complex. Useful information? Probably not. But I think its interesting. Spiral out; keep going.

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