Moiré interferences in the map of orbits of the Mandelbrot Set

•From old Zeno's paradox of Achilles and the Turtle to modern Superstring Theory or Planck's Quantum Theory, the debate over the Continuous and the Discrete has always been a disconcerting subject that has been kept alive by the Physics and Philosophy fields. At the same time, mathematics has been able to create a perfect abstraction to dominate the Continuous in the uncountable set of real numbers. The uncountable nature of real numbers is able to create mathematic wonders such as the Mandelbrot Set. Real numbers allow for expressing any magnitude with an infinite precision; however, when we use them to model the behavior of a reality considered as continuous, we then must operate on discrete samples, that is to say, we must demarcate with a finite precision.•We cannot manage the Continuous, value by value: there are always infinite real numbers between two given real numbers. Then, what would happen if we studied from a discrete point of view a reality that we actually consider as continuous but it turns out that such reality is discrete instead of continuous? And, if we confront these two discrete layers (one from mathematical sampling and the other from reality) and both have similar orders of magnitude, or if we sample spaces by taking sampling intervals of the Planck length (about 1.63E-35m) or even if we sample time by taking sample intervals of the Planck time (about 5.39E-44 seconds), what would happen then?•The result would be that we would find Moiré interference patterns and our perception of reality would be screened by such interferences. This is just a theory, but let's see how far it takes us.

This article shows the presence of Moiré Interference patterns in the map of periods of the Mandelbrot Set. It describes the requirements for their appearance and shows that such interferences are highly sensitive to the original conditions that define their calculation. The specific case herein studied shows that the Moiré interference patterns appearing in a picture of a section of the map of orbits are unpredictable, even if we obtain different maps from very similar original conditions. It begins with a brief description of the Mandelbrot Set and some of the characteristics of its orbits, the Moiré Patterns, as well as a concise introduction to a description of the Discrete Wavelet Transform. In order to develop the proposed specific case, a Multi-resolution Analysis method based on the Discrete Wavelet Transform has been used. It is significant that Moiré Interference Patterns always appear when the order of magnitudes reaches a certain limit where, what is considered as hypothetically continuous, behaves as a discrete pattern. The patterns as shown by the Wavelet analysis change drastically at the slightest modification in the original calculation conditions and it does not seem possible to predict their shape beforehand. This article ends with some conclusions and suggestions.