When you think of dimensions what comes to mind?
When I hear the word dimension I immediately think of the measures of length, width and height. For example, when I want to describe the size of a shoebox (i.e., to know its dimensions), I say it’s 30 cm (12 in) in length, 20 cm (8 in) in width and 15 cm (6 in) in height.
Length, width and height are especially meaningful because we can describe the size of pretty much every object simply by defining these dimensions. And using the size of an object is equivalent to say how much space that object takes up in our world.
If you understand that idea, then you are in a very good position to understand the quirks of Flatland.
But let’s start from the very beginning. You know our universe is three-dimensional. But, can we imagine what a lower dimensional object might look like?
The only object that has 0 dimensions is a point. But as in all things “mathematical”, we need to be very precise how we define a point.
Whenever people think of a point, they will imagine something like this • . But that isn’t 0-dimensional. It’s actually two-dimensional, because I can tell that it has a bit of length and width, say 1 mm x 1 mm (see figure on the left above).
A 0-dimensional point is an infinitesimal point. It has zero length, zero width and zero height, and, thus, cannot be measured with existing tools.
You might reasonably argue that a 0-dimensional point then simply doesn’t exist. And, yes, 0-dimensional points do not exist in reality (at least, not in our reality) – they are a purely theoretical construct.
Bear in mind, however, that some scientists have suggested that point-like particles such as quarks and electrons may be zero-dimensional (see the above figure on the right), but that view is not unanimously accepted.
OK, now imagine you had a bunch of those infinitesimal 0-dimensional points and connected them. What would you get?
A line!
Again, not a line like this — (that’s a two-dimensional line, because it has length and width, see the above figure on the left), but a line with only length as a dimension.
To state the obvious, a one-dimensional line has only one dimension, which is its length, and it has 0 height and 0 width.
How would a one-dimensional line look like in our world? Well, can you picture a line without width and height? Likely not.
So, as with 0-dimensional points, one-dimensional lines do not exist in our universe. Some physicists theorise that the strings postulated by string theory may be one-dimensional, although there is no scientific evidence to back it up.
Now what about two dimensions? In most cases, people will tell you that a square drawn on a sheet of paper is two-dimensional because it has only length and width.
However, that is technically incorrect.
The square you draw on a piece of paper isn’t two dimensional because it too has a bit of height. No matter how flattened your drawing of a square is, if you had a potent microscope and zoomed in on your drawing, you could see that it had a tiny bit of height as well (see figure above).
Graphene, a material made of pure carbon, is also usually described as a truly 2-D object. However, even though graphene is the thinnest possible object since it is exactly one atom thick, it is still three-dimensional, because atoms are themselves three-dimensional.
Nevertheless, scientists are happy to ignore its thickness, since, on the one hand, it is negligible, and on the other, it is likely the closest thing to a 2-D object we will possibly ever get.
It is more accurate to think of a square drawn on the computer screen as a truly two-dimensional square (a shadow has also two dimensions). But again, we need to be careful with the kind of labels we give to these things. A square on the PC monitor (just like a shadow) isn’t really an object – it doesn’t interact with our world, so it does not take up any space.
And that is a really important point here. Could a 2-D square ever exist in our 3-D world?
Let’s imagine we could take the square drawn on the PC monitor and somehow represent it on our 3-D space. The square wouldn’t have height, so if you tried to measure its height with the most potent of microscopes, it would appear as though it disappeared. But then, you would look at it from above and it would suddenly appear (see figure above).
So weird!
Because this sounds like an impossibility and defies the laws of physics, truly 2-D objects are unlikely to exist in our 3-D space.
So what kind of objects have three dimensions? This should be easy-peasy: everything!
That’s right! Every physical object that exists in our universe has three dimensions. That’s because atoms, the stuff of all matter, are themselves three-dimensional. And what makes an object three-dimensional? Well, the ability to describe its size using the three dimensions of length, width and height. And if an object has non-zero values in all of these dimensions, then it takes some amount of space in our world.
We could extend the argument that adding another dimension to those three would allow us to represent an object in 4-D. What would that object look like? Hah! That is a topic for a whole different article coming up very soon!
Great! Having established that objects with less than 3 dimensions cannot possibly exist in our world, let’s now ask the reverse question. Could a 3-D object exist in a 2-D world?
Now, based on what I said above that all atoms are three-dimensional, it would be impossible to reduce atoms to two dimensions without collapsing its atomic structure. No biological and chemical interactions as we know them would be possible, and no matter would be formed, so it’s clear that we could not exist in a 2-D world.
Still, for the sake of fun, let’s imagine you could get inside the computer screen, how would 2-D beings actually see a 3-D object?
For this exercise, I’d like you to think of the original 1980 game, Pacman (see figure above).
As we have established above, the images on a computer monitor could be thought of two-dimensional, so we could consider Pacman and the fiendish ghosts as two-dimensional objects.
Why? Well, would you be able to pull the original Pacman off the screen and into our 3-D world?
No, not really.
Don’t fool yourself thinking that if you removed Pacman from the screen, he would somehow acquire the shape of a sphere in our world. Again, 2-D Pacman would have no thickness so it wouldn’t be able to exist in our 3-D world, since we could not represent Pacman’s height.
But what if a human were to cross Pacman’s land?
If you look at the figure above, where a 3-D woman is interacting with Pacman’s 2-D world while falling down, you will have an idea. Pacman would essentially only see a cross section of the woman, the section that is interacting with his 2-D world.
But, since Pacman has no concept of height, he is unable to differentiate the different cross sections of the woman. To him, every cross section is the same, a bunch of lines.
But remember, even though I represented the cross section of the woman entering Pacman’s world as a black line (which is a two-dimensional line because it has length and a bit of width), Pacman would actually only see a one-dimensional line (i.e., 0 width and 0 height, only length).
Alas, I cannot depict a one-dimensional line on the screen, as it would be invisible to our eyes… but not to Pacman’s eyes.
And that brings me to a crucial point.
Our vision is possible because light reflected off objects enters our 3-dimensional eyes and converges on the retina in each eye. The retina contains photoreceptors (rods and cones) which convert the incoming light into electrical signals that are then sent to the occipital cortex. The occipital cortex processes this information, allowing us to “see” in 3 dimensions.
Colloquially, we tend to speak of our vision as 3-D. Even textbooks will tell you that our brains convert the electrical impulses from both of our retinas into a stereoscopic 3-D image.
However, that is technically incorrect. If our brains were truly capable of generating a pure 3-D image, we would be able to see the back of the objects that are usually occluded from our sight.
It is perhaps more accurate to say that we possess 2.5-D vision. That is, we see two-dimensional images with the addition of the illusion of depth.
Right. The problem is that Pacman has no 3-D eye or 3-D brain. Both his eye and his brain must be two-dimensional.
If incoming light from objects form 2-D images using our 3-D eyes and brain, we can extrapolate that 2-D objects would be projected as a 1-D object on Pacman’s retina, and so Pacman would see in 1-D (or maybe in 0.5-D; check Do we really see in 3-D? Not really, more 2.5-D.).
And what is a 1-D object? A line, of course.
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