This is a diversion from the usual education related posts to bring to light some concepts I personally find interesting in math. These are also not commonly understood by many people, so I hope my readers enjoy it. The first thing I want to talk about is the concept of Dimensions. It is common in science fiction, or even fantasy and new age babble to speak of dimensions as other “planes of existence”, or to describe something as coming from “another dimension”. Unfortunately, these types of descriptions we see in fiction are often ambiguous, or meaningless especially when we are referring to mathematical dimensions. In math, it all begins with a point:
This point, has no length, no width, no height. In the simplest terms possible it is an object of zero dimensions. Next, when we look at two points:
We see that we can have these two zero dimensional points define for us a one dimensional object; the line. From this point its gets a little tricky, because in mathematics, to fully define two dimensions, we might take two lines and cross them in a perpendicular fashion, like so:
These two lines define two dimensions in what is called a Cartesian Plane, named after the Famous Rene Descartes, who initially came up with the idea. (I have a fun anecdote about how Descartes developed this, but I will post that another day). These two lines are what we call orthogonal to each other (these lines intersect each other at exactly 90 degrees). Orthogonality is how dimensions are described in mathematics. In order to define two dimensions, usually you need to show two orthogonal lines. Naturally this can be extended to the third dimension.
At this point, you might be asking yourself, can this keep going ad infinitum? Mathematically, yes it can and it is something that can be shown with an extension of the Pythagorean theorem. However, when we are talking about lines and orthogonality, we are really interested in seeing how many lines can we intersect with each other, where each line is orthogonal to all of the others? Look again at the 3 dimensional orthogonal lines. Where would you put a fourth orthogonal line?
This is incredibly hard to imagine spatially, simply because it is so foreign to our understanding of reality in 3D, that our brains really cannot visualize it. In math and on paper, the calculations make sense, but the intuition does not. It’s a fun crossroads in mathematics where what you would originally think is right actually becomes wrong, or weird. However, there is a way to visualize four dimensions simultaneously, but in order to do this we must visualize the fourth dimension as time.
See, you can be in a particular place on Earth, and we could give you latitude, longitude and elevation coordinates so that someone could find you. However, we could also say that you had these coordinates at approximately 6:30 PM, and that they changed slightly by 6:31 PM. I like to visualize this as an array of sheets where there’s a two dimensional me on the first piece of paper and another on the next, and so on in a nice straight line. Doesn’t this almost remind you of… a flip book?
That’s kinda what a flip book is; you are looking at two dimensions and a time dimension, where we our positions change relative to time, and it demonstrates movement. Your movement in space is an example of 4 dimensions.
Why is this important? Because it is the foundation of one of the most groundbreaking physics theories ever discovered; The Theory of General Relativity. In General Relativity, Einstein linked the concepts of space and time into a four-dimensional construct called (imaginatively) “spacetime”. The mathematics behind it is quite dizzying, so I will relieve my readers from it for the time being. In four dimensional space time, Einstein is able to define gravity as its “curvature”. Check out this video clip from my hero Carl Sagan, he describes more eloquently than anyone else can.
Once you put it all together, you see it clearly; In order to invent the universe, you first need tiny little points.