Page x - Prologue

      continuum manifold. The relationships of this manifold obey the same general laws as any multiple extended manifolds with the same number of magnitudes. The specific manifestations of these relationships can only be determined by experience. Any problem is basically a “geometric problem”, or better said a “hyper-geometric problem” in its specific multiple extended manifold. This was the approach taken by Archytas, Leibnitz, Gauss and Riemann, to name just the most remarkable men of science that come to my mind now.

2.      Start from a good set of experimental data. I learned this from Kepler. It was his luck (and ours) that he got all the astronomical recordings of Tache Branch to work with, recordings taken in uncounted nights over many years. As a general rule start from this data and go back to it to verify any assumption you have made and also to look for new ideas about futher investigations and new directions of attack. Don’t forget that it is in this data that the pattern you are looking for is hidden. For us, the humans, the numbers are always available to us. Our mind is able to count, or inversing “the counting is a characteristic of the human mind”. Again, I will argue, any pattern that could possible exist can also be found as a pattern of numbers. From this perspective it is so easy to understand Pythagoras’ and his insight that everything that exists is a number. It is our advantage that today, using computers in the right way, we have the tools to test and verify our assumptions faster than in any time in human history. But it is also true that it is so easy to be lost in details, to lose focus, to concentrate on the wrong approach and most of the time to not be able to see the big picture. This requires a good understanding of the computer as a tool. It shall never be forgotten that the study of natural phenomena is the main purpose of the investigation.

3.      Use the complex domain. Don’t forget that in the last 150 years all new true discoveries came when using the complex domain in solving problems. Again, this can be explained by at least two reasons. First, all the simple patterns, the one that can be seen and understood using simple mathematics have already been discovered. There are tens of centuries of recorded history. There is no reason to believe that people who lived thousand or many hundreds of years ago were not as “smart”, as “evolved”, as “curious” or as dedicated to their work as we are. This is why all the simple problems and patterns have already been discovered. Second, the complex domain has proved to be the place from where all problems can be seen in a totally new light and from where apparently strange and unexplained features can be easily explained. The complex domain, its closure and its geometrical representation using surfaces, opens the door to the true and complete understanding of the Universe. That is why I believe a better name for the “complex domain” would be the “complete domain”.

4. Use transcendental functions and inversion. It was Leibnitz who said that the discovery of the exponential function (logarithm), the main actor of the first class of transcendental functions, has ended the use of algebraic functions in the study of natural phenomena. The only way to explain the rich manifestation of the physical