Need for simulation

First I’d like to argue for the need for simulation.

Let us revisit the Mandelbrot set, which we used as an analogy earlier. The complex details exhibit a lot of regularity, as shown in this picture.


However all these regularities, we could call them emergent behaviour, are secondary effects. Connecting them to the primary generating formula by proof is either impossible or at least very hard. The only way of connecting the patterns to the generating function is through simulation.

In my opinion all of the current study of physics, outside of  research into fundamental cellular automata, is at this higher level, and investigates secondary effects, such as particle interactions.

Another famous function in the complex plane is the zeta function. Here is a plot:


You can see that the coloured lines all seem to converge on points that lie on a line that passes through 1/2 on the x axis. A straight line seems a much simpler shape that the Mandelbrot swirls.

Bernhard Riemann made that conjecture in 1859, the proof of which has since then advanced to one of the most intriguing open problems in mathematics. In spite of many brilliant mathematicians dedicating large parts of their careers to this, it still has not been solved. This means that the proof is either very hard or impossible. If it is impossible then proving so is also very hard or impossible. Simulation is easy though. Computers have calculated billions of points that all lie on that line. Simulation is equivalent to a proof of specific cases. We can prove that the first 10 points lie on that line, but our proof is equivalent to simulation; we just calculate the coordinates of these 10 points.

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