I get 1, 2, and 3 dimensions and their visual representations. For instance, 3-dimensional space has width, length, and depth but how about a 6-dimensional space, how can we grasp it in n intuitive manner free of mathematical analysis?
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The best way for me personally to conceptualize more than 3 dimensions is to use lists of discrete numbers or objects. So, the first dimension is just a list of numbers. Maybe 1, 2, 3, 4, 5. This looks like a few dots on a page in a line. However, if we turn each of these numbers into a list, or a folder, where there is another list of numbers inside or behind it, then we would have two dimensions.
So let’s say we started out with the numbers 1, 2, 3, 4, 5 in the first dimension.
To add a second dimension, now we have folders labeled, “1,” “2,” “3,” “4,” and, “5,” that each have pieces of paper inside labeled, “1,” “2,” “3,” “4,” and, “5.” So we could refer to something like, “1,5” and know that means folder 1, page 5. We just need to decide on a standard for what the first number refers to, and what the second number refers to. Traditionally, we think of these as X and Y coordinates on a coordinate plane. The first number tells us how far left or right we are. The second number tells us how far up and down we are.
But this is somewhat arbitrary. The “first” dimension doesn’t have to have anything to do with left and right. The “second” dimension doesn’t have to have anything to do with up and down. We traditionally use those because we often use dimensions to talk about where something is located in space, or how it is moving through space and agreeing on what each dimension “means” helps us communicate with each other more effectively.
However, you and I could be oriented differently. If I am across from you at a table and you tell me something like, “the 3rd one from the left,” I might have to clarify whether you mean your left or my left. There is nothing inherently sacred about left and right, up and down, etc.
In fact, dimensions don’t even have to refer to space at all (although I suspect that is what you are trying to visualize and wrap your head around). Going back to my folders analogy we can create as many dimensions as we want by turning each new dimension, which is just a list of discrete numbers, into another folder.
If I tell you the piece of information I am referring to is on page 3 of folder 1 in drawer 4 of filing cabinet 2 in building 1, that’s 5 dimensions right there. Building, filing cabinet, drawer, folder, page.
If the folder analogy isn’t intuitive or is too far removed from imagining spatial dimensions, let me try one more.
A line is one dimension. A square is two dimensions. A cube is 3 dimensions. And we can split each of them up if we want. We can have a line broken up into 3 smaller lines. We can have a square broken up into 9 smaller squares. We can have a cube broken up into 27 smaller cubes.
If we label those discrete sections we just cut up, then we can use, “coordinates,” to refer to specific ones. The sections of a 1-dimensional line could just be 1, 2, or 3. But how to refer to a specific square? We’d have to specify which of the 3 horizontal positions it is in and which of the 3 vertical positions it is in. So we could have (1,1) but also (3,2) and it sounds like you intuitively understand that. The same for which specific cube we want to talk about, but now we need 3 coordinates. A 4th dimension means we would need a 4th coordinate because each of those cubes has 3 more possible options. But where can we go physically that hasn’t already been covered by the first 3 dimensions? One option that people seem to understand is time. You could make what time you are referring to the 4th dimension. Are you talking about the past, the present, or the future for that specific cube? You could refer to each as 1 (past), 2 (present), or 3 (future). But it doesn’t have to be time, that’s just one of the few things we can imagine moving through that makes sense and is somewhat similar to the spatial dimensions.
Maybe instead of making time the 4th dimension we add 3 smaller cubes inside each of our 27 cubes from the 3rd dimension. To differentiate each of the 3 new cubes inside every one of our original 27, we can refer to whichever cube is closest to the center of our larger 3D cube as, “1,” and the new cube farthest from the center of our larger 3D cube as, “3.” We do the same for every one of the now 72 new cubes (3 each inside of our original 27). We haven’t used a 4th dimension of space, but we have added a 4th dimension of information. If you want to refer to the exact position of one of the 72 smaller cubes you would need 4 pieces of information. Which horizontal section it is in, which vertical, how far back the cube is, and then how close to the center.
I’m sure that’s a little unsatisfying because we didn’t actually introduce a novel 4th spatial dimension. That’s why I prefer to picture dimensions as folders nested inside of folders (or inside drawers, inside cabinets, inside rooms, inside buildings). It’s easy to imagine having to go to the Nth level of a filing system to find a specific document, but hard to imagine finding a certain physical point of an Nth level object.
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