Imagine that you’re in a very dark tunnel, and there’s a tiny light at the end of it. This light is so small that it looks like a single point, a dot. You move closer towards it, and it doesn’t change. You move further away from it, but it’s still the same distance from you. It’s just a small, brilliantly lit dot of white light:
Forward or back (1 D)
Suddenly, you get a brilliant idea. Instead of moving forward and backward, you decide to move right, perpendicular to the dot. As you move to the right, you realize that this dot is actually a line, and you can begin to see the length of it. You walk around the dot 90 degrees so you’re staring straight at a bright, white line now. If you move forward or backward, it’s still a line. If you move to the right or left, you’ll eventually return to seeing it only as a dot, or 180 degrees around to see it as a line from the other side. Effectively, you’ve moved from a one dimensional view of it (as a dot), to a two dimensional view of it (a line with length) simply by moving around it.
Side to Side (2D)
You get another brilliant idea! Instead of moving forward or backward (1 dimension), left or right (2 dimensions), you decide to move up or down (3 dimensions). Magically, you’re able to walk upwards, as if the space around you was an invisible step ladder. As you venture up, you realize that this is no line at all, but a square with width that extends downwards. Astounded, you realize that this little dot in one dimension (forward or backward) appeared to just be a dot, but once you moved along a second dimension (side to side) you realize that it’s actually a line, and your perspective had to change to see it. Then, by moving up or down in a third dimension, you realize that this is no line at all, but a square with two sides!
Up or down (3D)
Now, this human being has ventured forward and backward, side to side, and up and down. However, we know that a cube (six squares put together with 6 sides, 8 vertices, and 12 edges) is entirely possible. However, for this three dimensional human, he’s used up all the directions he can travel in. For him in this world, where a dot is one dimension (no length), a line is two dimensions (length), and a square is three dimensions (length+width), he can only ever experience the sight of a square with width and height. He can never experience depth, although we know it’s perfectly possible for a cube to exist.
For this man in this world, a cube is a four dimensional object. Think about this: which direction could he take now to view such an object? He can’t move forward or backward (this would just move him to the other side of the square), side to side (this would just move him to either end of the square), or up or down (this would just move him to either face of the square).
He would have to travel along a new direction: in or out (depth). I use the dot as an example for one dimension in this world to illustrate the fact that although we know an object with depth, such as the cube, can exist, this man is stuck in three dimensions only viewing a square. In reality, a dot represents a “zeroth” dimension (0D), a line represents the first dimension (length, 1D), a square represents the second dimension (length + width, 2D), and a cube represents the third dimension (length + width + height, 3D).
So now, understand for us, that we are trapped viewing the cube in our three dimensions. An object of four dimensions does indeed exist, but we can never see it, nor can we draw it or properly visualize it. To view such an object, like the man in our fake world viewing a cube, we would have to travel along a new direction that’s not left or right (length/width), up or down (height), or forward or backward (depth). Isn’t that mind boggling?
We couldn’t even draw it because whatever we’re using to draw with is a product of the three dimensional world. The paper you write on, the graphite in your pencil, the pencil itself, the air around you as you write, these are all products of a three dimensional world. How can graphite trace a fourth dimensional object when the material itself only has length, width, height, and depth?
There is, however, one way. If you practice mathematics for long enough, and understand everything there is to know about geometry and the equations that give rise to shapes, and how these shapes appear due to different perspectives, you can in effect actually visualize a fourth dimensional object through the mathematics itself. This wouldn’t be your brain simulating four dimensions through three dimensions, it would be actual fourth dimensional thought. As far as we know, the brain isn’t limited to just processing and graphically displaying things for you in only three dimensions. You can imagine a point, a line, a square, and a cube in your brain, and the only reason you can’t imagine a fourth dimensional object is because you’ve never seen one.
If you knew the mathematics, however, it does seem as if you could. Your brain would calculate all the points, the faces, the vertices, and you would be able to possibly even investigate it. I can’t even imagine what amount of mental energy would be required to achieve this, but it must be magnificent.
I don’t know whether our computers could display such an object. Theoretically, they could definitely compute one, but since the display itself is made out of three dimensional pixels, it does seem impossible to graphically produce a fourth dimensional object on a three dimensional screen. As far as we know, however, your brain is limitless in terms of how many dimensions it could theoretically imagine. Your consciousness is not a construct of three dimensional pixels. In fact, I’d be willing to bet there’s a connection between your brain’s ability to imagine four dimensions and what consciousness is actually “made of”.
This is a three dimensional simulation of a four dimensional object.
PS Limited perception of 4D space by 3D creatures
Imagine a giant sphere in 3D space.
Any flat (2D) creature on the surface of the sphere would think they live on infinite plane (with no limiting borders), however for 3D creature it would be clear that the sphere has finite surface (yet there is no edge or border to it).
This analogy can be useful when trying to explain the Universe being infinite to humans, yet finite for creatures who can perceive more than 3 dimensions…
If a fourth dimension existed what would a four-dimensional hypersphere look like to us as it dropped through our “flat” three dimensions?