This blog is about the fundamentals. It concerns the ultimate cause of all effects. One might say that I’m all about what’s essential. Such an enterprise tends to become rather one-pointed. It converges into nothing really. Most people would find that rather boring. The multiple effects are more entertaining for sure.
Today we will have a quick look at an essential singularity. It is a beast of mathematics which was unleashed by the french genius Emile Picard
Great Picard’s Theorem: If an analytic function f has an essential singularity at a point w, then on any punctured neighborhood of w, f(z) takes on all possible complex values, with at most a single exception, infinitely often.
Since I suck at math I must translate into layman lingo: If you have a mathematical function that describes complexity (a lot of stuff going on), and that function has an essential singularity in it, then anything that happens at that singularity can become any event in the whole picture except for one single event. Whatever happens out of the essential singularity can be anything. And it will be anything, or rather everything, infinitely many times. There is only one thing it can never be.
The one thing it cannot be is probably all possibilities at once. That is, one operation can be one value/out put, any output actually, but one operation can never be >1 output. Everything seems possible, but not at once. We must allow for some evolution, right?
Plot of the function exp(1/z), centered on the essential singularity at z=0. The hue represents the complex argument, the luminance represents the absolute value. This plot shows how approaching the essential singularity from different directions yields different behaviors (as opposed to a pole, which, approached from any direction, would be uniformly white). (from wikipedia)
Note that a “pole” is probably related to defined positive and/or negative values. But in this case we are looking at what I suspect is a monopole. A monopole is…well, different. If approaching the singularity/monopole from different directions show complexity, it is reasonable that the same holds in the opposite direction. Out from the singularity emerges complexity. But you can never enter it. That’s the bummer with an ultimate cause. Everything comes out of it, but nothing comes back.
The black and white in its sub-zero center, that is the “punctured neighborhood” from Picard’s Theorem, is the simultaneous/instant contraction and extension of the spinning viscoelastic contraction which is the monopole I imagine. A true monopole has 2 ends, a dipole has 4 ends. The electromagnetic field in the Beautiful picture is only possible to generate in a mathematical simulation of a monopole. In physical reality, monopoles always come in pairs of 2 making them a dipole function. A solitaire monopole has no em-field around it and can therefore not be detected as it is. But it can reside inside a Black Hole I think. A monopole should hide in the center of a Black Hole.
The hardest part in this is not the complex math. The challange is to imagine the black and white singularity being one single unit of polarity. it is not a sequence which flips from one to the other extreme. There’s nothing linear in this, so no matter how fast we imagine a black/white switch to be, it won’t end up an essential singularity. Not even the speed of light is enough. To generate all of universal complexity, the monopole singularity must do its thing at once.
Can you imagine something that does opposite actions at once? I guess not. To paint such a picture, you must abandon all linear thinking. You must abandon scales that have two directions. In fact, you must leave all of space behind you. You see, at the very point of everything, there is a point. A point has zero directions in space. The point of the monopole is that it spins. It spins as a “pole” but it doesn’t look like a pole. There is no axis of rotation inside the zero point. There is just rotation. So how do we make it go separate ways in relation to itself? Here goes:
Make it elastic so it can stretch and relax.
Make it contractive so it doesn’t break from the rotation.
Don’t be surprised if there is extension at the equator of the point just as its “poles” are both going towards the center of the spinning point. There you have it! One expanding circle/torus around a pole with two contracting ends. That’s about as “all at once” I can get it.
Today was a round about the essential singularity as a mathematical image of a physical monopole. To me it is one and the same, the infamous Singularity suggested by Einstein in his General Relativity. There will be more on the funny little monopole in coming posts. I will soon try to enlighten everything in a slightly unconventional way.