"Energy-Geometric Code of Nature"

Such a title has the book written by the famous Polish journalist and scientist Jan Grejzdelsky. The book published in Polish and having a limited edition (1500 copies) contains a number of very deep scientific ideas. Let's consider some of them.

Cassiny's ovals

As is well known a sphere was considered in the ancient period as the "ideal" geometric form to simulate the laws of Nature. The idea about spherical character of planet's orbits brought into the creation of trigonometry and was put forward by Ptolemy as the basis of his geocentric system of the Universe. The discovery of some mistakes in the basic principle of the Solar system organization ("the cult of sphere") was the greatest shock to Kepler and led him to the ellipsoidal insight as to the character of planet's orbits. As is well known the ellipse is a geometric plane figure meeting so-called "additional" principle since the sum of the distances from some point of the ellipse to the its focuses is a constant value. It follows from the "ellipsoidal" insight that the geometry of the Solar system is the "additional" geometry based on the "addition" principle.

In his book Jan Grejzdelsky returns to the ideas of the French astronomer and mathematician Cassiny (1625 - 1712) who was Kepler's contemporary. In Cassiny's opinion, the first Kepler law is not correct. Cassiny affirmed that planets move in accordance with Cassiny's oval. The basic geometric peculiarity of Cassiny's oval consists of the following (Fig. 1).

Let's suppose that F1 and F2 are the focus points of the oval and OF1 = OF2 and F1F2 = 2b. Then a geometric definition of Cassiny's oval consists of the following: MF1 ´ MF2 = a2. This means that the product of the distances from some point M to the focuses F1 and F2 is a constant value. Then the equation of Cassiny's oval in the rectangular coordinates x and y has the following form:

(x2 + y2)2 - 2b2(x2 - y2) = a4 - b4.(1)

It is clear that Cassiny's oval is the curve of the 4-th order. In contrast to the ellipse, which does not change its form in dependence on the focus distance, the form of Cassiny's oval depends on the focus distance. If a ³ 2b Cassiny's oval is a convex curve (Fig. 1-a) similar to the ellipse. If b < a < 2b there appears a negative curvature in Cassiny's oval form (Fig. 1-b). If a = b Cassini's oval equation has the following form:

(x2 + y2)2 - 2b2(x2 - y2) = 0.(2)

It is the equation of the curve having the form of the number of 8 (Fig. 1-c) and called Bernoulli's lemniscate. Just this figure is supposed to be chosen by the ancient Greeks as symbol of infinity (¥).

At least for the case b > a the Cassiny oval falls into two separate geometric figures (Fig. 1-d).

Cassiny's oval
Figure 1. Cassiny's oval.

It was Jan Grejzdelsky who was the first after Cassiny to advance the idea that the geometry of Nature is the geometry of Cassiny's ovals and ovaloids. Moreover, the addition geometry following from the Kepler laws is replaced by the multiplication geometry (Cassiny's oval). The basic advantage of such an approach to the geometry of Nature consists of the fact that it allows to give a logical and energetic explanation of the division processes widely observed in Nature phenomena. The cause of the "Cassinyable" divisions is the change of the equilibrium conditions of the system. Geometrically this is expressed in increasing the focus distance (Fig.1-b,c,d). Upon overcoming the certain energy threshold, the rotating solid, having Cassiny's oval form in its cross-section, strives to the stability state but this process demands not only the energy change but also the form change.

Thermodynamic equilibrium

Grejzdelsky spares a special attention for Bernoulli's lemniscate (Fig.1-c) and its space form called lemniscatoide, which is the expression of the system thermodynamic equilibrium. Grejzdelsky found out the Golden Section in Bernoulli's lemniscate and advances the idea that just the Golden Section is the proportion of the thermodynamic equilibrium.

As is well known the Golden Section is presented in the form of an infinite fractional:

(3)

which contains only coefficients 1 in its representation (3).

The unique mathematical property of infinite fraction (3) consists of the fact that it is the most sluggish infinite fractional among other infinite fractionals. Grejzdelsky affirms that "this property is connected with the thermodynamic equilibrium and the given sequence presents very nice the idea of the most sluggish movement". Just the latter is suggested by Grejzdelsky as the alternative to the Newton doctrine of the "absolute rest".

Grejzdelsky demonstrates the idea of the thermodynamic equilibrium by an example of optical crystals. As is well known the ellipsoidal model permits to explain of the light rays spreading in the optical crystals. Grejzdelsky advances the hypothesis that the "golden" ellipse is the optimal model for demonstration of the thermodynamic equilibrium in the optical crystals. The "golden" ellipse is formed with the help of the two "golden" rhombi ACBD and ICJD inscribed into the ellipse (Fig. 2).

The "golden" rhombi ACBD and ICJD consist of 4 right "golden" triangles of the kind OCB or OCJ. Note that the isosceles "golden" triangles ACB and CJD are similar to the triangle forming cross-section of the Cheops Pyramid.

The 'golden' ellipse
Figure 2. The "golden" ellipse.

Let's consider the basic geometrical relations of the "golden" ellipse. Let's suppose that the focus distance of the ellipse is equal to AB = 2. In accordance with the ellipse definition there exists the following correlation:

AC + CB = AG + CB.

Besides, there exist the following relations connecting the sides of the right "golden" triangles OCB and OCJ:

OB : BC = 1 : t;OB : OC = 1 : ;
OC : CJ = 1 : t;OC : OJ = 1 : ;

It follows the next proportion from the similarity of the triangles OCB and OCJ:

CB : CJ = OB : OC = OC : OJ = 1 : ,

where t is the Golden Section.

In Grejzdelsky's opinion, the latter correlation expresses the proportion of thermodynamic equilibrium in the optical crystals and creates optimal conditions for the photon arriving to the focus with minimal energetic losses.

In the preface to Grejzdelsky 's book Marek Zysek wrote the following words:

"This work is very extraordinary. There arises the question: can it find its application? A time will show. But one fact is certain. If any critic of Grejzdelsky's work shows that he made a mistake but the person who will use his arguments, could facilitate himself the way to great discoveries".