1. Idea

1. How I arrived at the idea.

Reality exists hence we say it is true. But what is really true besides that more than anything else which we can really trust, it is mathematical facts. So, to my mind I connect both since both seem to be a statement of truth. So I took a guess that reality is something akin to a circle (truth). The relations between the points give you a mathematical structure whereby you get PI which defines the structure of the circle.

So I was thinking the relation(s) between what entity(s) could give a rise to a universe (truth). To come up with a structure with some entities, the easiest way was to see if I could draw two entities and define some kind a rule for their interaction. At that time I was familiar with fractals and vaguely heard of Conway’s idea, but I did not know about either Dr. Tegmark or Wolfram’s- New Kind Of Science-. I said let me see maybe I will be smarter than Conaway and get some really fancy rule between some triangle or circles or lines or whatever. But as soon as I put a blank sheet in front of me, for a short while I thought to myself this sounds very enigmatic, first by what criteria I am going to choose my entity, and which characteristic of that entity I was going to interrelate them and what expression. Choosing by trial and error was not very natural.

 My intuition was telling me I needed something more natural. Being an engineer and a programmer we learn to be efficient in our designs. So I opted first for the simplest configuration and that was point and to start simple and not to draw points all over the paper, I restricted myself to a line. Now, if I iterate on an artificial formula I will just get fractals which has already been tried which gives you beautiful suggestive pictures but that's all. Also the different formulas I could use were most unnatural. So I thought the only way out is to throw random numbers on the line and see what happens. Off course, after a bit more than few seconds it was obvious I am going to get a uniformly distributed points on the line, I don't have to tell you that I was sad at that point( although I should have been happy as hell, you will see why). How I was to get out of this conundrum, other than mangling that paper, throwing it in the garbage can and go to a party.

The only other thing to do was to throw random lines that did not exceed an original line of length L. One more choice was necessary is to choose where those lines started, the obvious choice was random position on that line L.

After that thought analysis I went to my pc and downloaded a simple BASIC program and started coding the idea thinking I was going to get some fractal like universe or something useless. Creating the random length lines and their random positions was straightforward, but now I had to decide on what logic/constraint to use to eliminate the lines which were going outside of the original line. I tried few of them with not too complicated expressions and the output, the random positions, looked jittery but when approximated looked like some kind of a trigonometric function. So I superimposed a sine, cosine, sin^2, cos^2.

Using the simplest expression for the constraint I was in a shock, it matched perfectly sin^2, and that was the solution for Schrödinger's equation for a particle in a 1D box, with the probability of the particle position directly (no complex wave). Just from that I knew at that time that I was onto something big. Reality was nothing but random numbers (representing lines and their positions) just like what I have suspected.

The next natural thing to do is to generalize to 2D and 3D, I was in a shock again, it was so simple, just repeated the code for 1D and labeled appropriately, and plotted, a perfect probability wave for first energy level in 2D and 3D. The amplitude came out also perfectly once I normalized the probability positions to the number of throws, 2/L, L being the original line length. That was natural because probabilities had to add up to 1.

The next question that presented itself was how to calculate the energy for the particle in a box and what does it take to get the higher energy levels. It turned out, just like it should, that both questions are linked. The only obvious choice for calculating the energy was to somehow add up the lengths of the random lines. It did take some testing to see the correct expressions, because, you could take several expressions like...

but only one matched the behavior that we get from solving Schrödinger eq. to get the energy level and that is adding up all the line lengths for each point. Calculating energy that way (after normalizing) it was quadrupling when the distance of confinement was halved in a perfect match to the higher energy levels.

For a moment at that time the situation looked hopeless, how can I get any interaction going, even simpler, how to add some potential? I was having some doubts about the model, on one hand, it was reproducing sh. eq. on the other I noticed early on that I could reproduce any function in math by randomly throwing lines and applying certain constraints more complicated than the one which reproduced sin^2. I asked myself could it be that what I have discovered was just a math trick.

That situation did not last long because this model is amazing in more than one way. In a way creating these function was a blessing, because now I have a choice, very few actually, to create an interaction between the particle in a box and a potential like function ,say 1/r or exp(x) or a step function, for example. The reason why I said amazing, is that the only choice I had for interaction is to set a relation between the random lines for the different entities, mostly, if two lines crossed or not. Then I only keep the probability position for the lines that did not cross. Long and behold, you get probability waves similar to solving Schrödinger equation with a potential. The model was pushing me to do the only available choice which led to the right results.

And that led me to the next step, setting up a particle in a potential well. I could use a step function or just another particle in a smaller range within the original particle space. And the result was perfect. I got the exponential decay for the parts which were overlapping and the tunneling with continuity automatically satisfied. There was no turning back at that point.

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