What Are the Spatial Dimensions?

Don't forget about motion!

We live in a three dimensional space. The theory of relativity added another one. However, various miss-conceptions persist regarding the meaning of spatial dimensions. Surprisingly or not, the most common explanation of what spatial dimensions are is at least miss-leading, if not flat wrong.

Why is the computer screen bi-dimensional, while the world is three-dimensional (or forth-dimensional)?

The common explanation is that you need two numbers to express the position of some pixel on the computer screen (one number is not enough, while more than two are superfluous), while you need three numbers to express a position is space.

Truth is you certainly don't need two numbers to express the position of a pixel on a computer screen. A computer screen has a limited number of pixels, you could simply label each one of them with a different number. However, you won't find any computer programmer in the game industry abolishing the two-dimensional system. But why don't they?

Interestingly, even if the computer screen would have had an infinite number of pixels infinitely small and it would have been infinitely wide, you could still have used only one single number to label all the pixels. This had been demonstrated in the late 19th century by the famous mathematician Georg Cantor. (He was as surprised as you are now, if not even more, that he managed to prove such a thing!)

Intuitively, you can imagine covering an entire infinite bi-dimensional plane with the help of only one number by using a spiral.

What is true for the bi-dimensional plane is true for any kind of space no matter how many dimensions it has. The three-dimensional space could also be covered with the help of a single number.

Thus, in what sense are the computer screen bi-dimensional and the world three-dimensional?

The key point is to think dynamically and not about static things. You have to think about the motion of objects in space (on the computer screen or in the real world).

Suppose one would use a spiral for covering a 2D space. Then, when an object would move in a straight line it would pass through the spiral - from the point of view of the numbers used to express the positions, the object would seem to jump from number to number, skipping a whole bunch of intermediary steps. However, when one uses a bi-dimensional system the motion over a small distance is reflected by small changes in each of the coordinates. A continuous motion is described by a continuous change in the values of the coordinates.

Thus, the fundamental reason why space is three-dimensional (or forth-dimensional as the theory of relativity asserts) has to do with motion and not just with the static description of positions. It is the only way a continuous motion can be represented mathematically by a continuous change of coordinates. If one would use a single number or more than three (or four) one would necessarily represent some continuous motions as discontinuous in the mathematical realm.

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