Chapter 9: NEW YORK
I landed in the USA, it was in October 2000. A few months earlier, my patent had been approved in Israel. I arrived on a flight from Israel, in order to meet with my American lawyer Mr. Harold Nissen and have my patent protected in the USA.
I had come at the week-end, so I found myself strolling the streets of New York, walking as I usually do for several miles, browsing in the book, computer and stereo appliance stores and then returning to the hotel where I was staying.
But one of the treasures of my whole life I found in a bookstore I had visited. It was one of the many bookstores, and one of the reasons I so loved to visit the USA. It had a huge variety of books and scientific literature. And then I pulled the blue-bound book from the shelf and the title on its front cover read: “Catastrophe Theory for Scientists and Engineers” by Robert Gilmore. Published by Dover.
I leafed through the book on mathematics and found it full of formulas, calculations and also interesting graphs. On page 478 I discovered a very interesting graph, which had a symmetrical and pleasant shape. Preceding it were several pages of equation solving, leading to the graph. The solution included high mathematics, derivatives and partial derivatives. In conclusion, the solution was displayed on the graph.
“I wonder whether I could solve this with the aid of my software”, thought I to myself. I bought the book, which had sat on the shelf. Prof. René Thom had developed the theory during the 50's of the previous century, when he won the Fields prize in 1958 for his work in differential topology (he had competed for the prize with John Nash, who later got the Nobel for Economics).
I returned to the hotel, connected the computer to the mobile printer I had especially bought for this purpose and opened the book to page 475. It was Problem No. 3 and the question was: What is the Phase Portrait of the potential? I entered the equation into the software I had programmed:
r = x^2 * y – (y^3) / 3
and I activated the software, searching for a suitable scale whereby one could see the graph. Wonder of wonders, after a minute of calculation, the picture appeared:
I made a printout. The outcome was astounding, colorful as a photograph. But upon further observation, I discovered something interesting. Additional data had accrued – many more shapes of stars appearing on the face of the picture. I took a ruler, positioned it on the picture and saw that there were stars, which had arranged themselves in a straight line. I drew straight lines and in a surprising fashion I got the solution, which had appeared, in the book. I got the lines of the solution, which determined the a, the dx/dt and the dy/dt, without the need to use derivatives, and partial derivatives, which appeared, in the book's solution.
The solution, as it appears in the book:
On transparent sheets, I drafted the lines in different color and I presented myself to my lawyer with the fresh news: I have succeeded with the aid of my algorithm to solve a difficult mathematical problem of the Catastrophe Theory.
The title request for the patent reads: Method and System for the Display of an Additional Dimension. The American patent was approved.
If we observe the picture, we can see its colorfulness. The color is the additional dimension that actually expresses the potential dealt with in the previous chapters, where each color indicates a potential value.
I programmed the software based on the algorithm in C language, and 16 colors were placed at my disposal: Black with value 0, Blue 1, Green 2, Cyan 3, Red 4, Yellow 14, White 15. And what happens to the larger numbers? Well, one divides every number by 16 and the remainder is what gives the value of the suitable color. For example: 16 will give Black, since there is no remainder when dividing 16 by 16. 17 will give a remainder of 1, that is to say Blue, and so on.
In addition we do not see the function itself, but an entire family of the function, where the difference between the function and the family, is in the constants (1, 2, 3 etc). This
affords observation of an additional dimension to the regular observation which takes place on the solitary function.
In the example I described, we observe an environment of a critical point and the flow in its area. That is to say – what the dynamic behavior is in the environment. The arrows show the direction of the flow.
By means of the algorithm, we get this by the colors according to the direction: from Black to Blue to Green until the White, or from White to Yellow…and in descending order to Black.
This was astonishing. Prof. René Thom got the Fields Prize given to mathematicians, which is parallel to the Nobel Prize. And here I had succeeded with my algorithm, to achieve a solution in a simple, colorful and delightful way.
Back home. The view of the buildings seems as a chaotic graph with some extreme points. All covered with shiny lighting points of the buildings. Most of the airplane passengers were sleeping, some were looking by the windows as the airplane left the Big Apple, excited, so am I.