THE DEGREE 33 AND BERNOULLI’S LEMNISCATA

 1.- INTRODUCTION

In point 2.1, chapter 5 of "The Symbol’s Method" (Deduction of the phase space geometry), it has been commented that:

• Multiplying any number by (-1) is equivalent to provoke a 180º turn of a vector which has that number as its module.

It has also been demonstrated that if, instead of provoking a turn of 180º, what we want is that vector turns any number of degrees, expressed in the sexagesimal system, the function which will express all possible turning degrees will be:   i ^{g / 90}

To calculate which is the numeric value that brings the expression at assigning to g a concrete angular value, we face the problem of having to deal with the imaginary number i and therefore calculus is impossible with the common algebraic methods, which is easy to show with the following example:

Let’s suppose we want to calculate the value of the expression  i ^{30 / 90}   and we substitute i by its numeric value which as we know is the square root of minus one. What we would obtain is:

( (-1) ^½ ) ^30/90 = ( (-1) ^0,5) ^0,333

That is, we should power one third the square root of minus one, which is impossible to do either with a calculating machine or with any of the methods habitually used by the known arithmetic calculus.

If those calculus could be solved we could know the size of the vector’s module in any of the turning angles, that is how to vary the associated intensity of the unitary vector as it turns.
 

2.- THE COMPLEX NUMBERS

The most common methods to deal with the imaginary numbers, consist to tackle them thru complex numbers.

A complex number is a number which has a real part and a pure imaginary part. The system of coordinates from which the complex numbers come from is a more genuinely mathematic system commonly known as a Cartesian system of coordinates. Indeed:

The system of Cartesian axis is a technique attributed to René Decartes which consists in two straight lines of real numbers at 90º of each other and with their intersection in a point called zero, calling the X axis the horizontal line and Y axis the vertical line situated at 90º of the former. But this does not emerge from mathematics but rather it is only a technique (a trick), useful and ingenious, but only a technique anyway.

The system of coordinates used by complex numbers are much different, because it emerges in a natural fashion from mathematics. This is so as it takes axis X, which represents the real straight line, and from it obtains the Y axis by a turn of 90º, mathematically prompted, that is multiplying the horizontal axis by square rot of minus one (-1) that, as it is known, is equivalent to an angular turn 90 degrees:

X = Real line Y = X . (-1) ^½

With it, the disposition of the system consists in a real axis X and an imaginary axis Y, which is the real X, but multiplied by the square root of -1.

The vector expression which will allow to represent any vector in this complex space is:  V = A + i B

As any vector turning in this space is always the vector addition of the "real" component A supported on the X axis and the "imaginary" component B supported on the Y axis.

From this graphic representation it immediately falls that the angle of turned by the vector and its module will be respectively:

• Turned angle = arch tangent ( B / A )    Vector module = ( A² + B² ) ^½

Under these conditions and if we consider the vector as unitary, the expression of vector module will be: 

Module = ( cosine angle ² + sine angle ² ) ^½

Algebraic expression which if can be directly calculated, even with a scientific calculator, for any turning angle we will always obtain 1, whatever the turning angle is, because the addition of the squares of sine and cosine of any angle is always 1.

In conclusion:

The system of representation of complex numbers does not serve for the sought purpose which is, let’s recall, to obtain the numeric expression of the algebraic function "square root of minus one power turning degrees divided by 90" and the why of this impossibility is that notation V = A + i.B is not an algebraic function, but the algebraic representation of a vectorial addition.

In other words, in the complex plane and facing again the impossibility to tackle the square root of minus one directly, we do the trick to change the algebraic to vectorial focus, thereby the representation that offers is only a vectorial notation referred to axis X (real) e Y (imaginary) of the turning status of vectors from a vectorial germ of the unitary module which constantly remains equal to one.

However, what we want is precisely to tackle that germ, though unitary in its origin (angle of turn is 0º), cannot be the same in other turning status and the solution of this problem is in Bernoulli’s Lemniscata.
 

3.- BERNOULLI’S LEMNISCATA

Lemniscata is presently considered as a particular case of the oval studied by Cassini in 1680 (Jean Dominique Cassini 1625-1712), but are Jackob Bernoulli (1655-1705) and Johann Bernoulli (1667-1748) who independently discovered and gave light , when they tried to solve a problem posed by Leibnitz (Gottfried Wilhelm Leibnitz 1646-1716) creator, along with Newton ( ) of the infinitesimal calculus, who challenged the scientific community of his époque to find the equation of the para centric isochronal. The found solution was the Lemniscata which, in honour of Bernoulli brothers, passed to be know Bernoulli’s Lemniscata..

In Lemniscata, when we come from an initial vector of length 1, the area bounded within the quadrant formed by X = real axis and Y = imaginary axis, is exactly a fourth (0,25 units of the area).

Presently, and since John Wallis (1616-1703) used it for the first time like symbol in its work  "Arithmetica Infinitorum" and probably for centuries of centuries ahead, Lemniscata of Bernoulli is the Symbol of the infinite (the famous laying eight), whereby it can "symbolically" be said that the numeric solution of the angular germ, function of i is enclosed in the symbol of the infinite.

Applying the equations of the para-centric isochronal, the following numeric values of the function "square root of minus one power turning degrees divided by 90" are obtained:

Beyond 45º modules of vector V are imaginary, and beyond 135º are real again. Indeed:

And it is from the previous para-centric isochronal numeric tabulations, the limits of the faint of order in the field of associated angular factors or angular space of phases contained in the Edgeworth’s Box are inferred , as we explain in the next section.
 

4.- GRADE 33 AS LIMIT OF ORDER

In "The Symbol’s Method" point "1.1.5.- Historical note and fractal account" and point "3.2.- Construction of a field of associated angular factors", we see that the probability of finding an elongation of the unitary size was 0,636 and that turning function more genuinely mathematical, that is expressed in radians, also contained factor 0,636 in its exponent, and from this it can be inferred that:

• The most probable elongations to be found in a graph will be of the range of 63,6 % of the pattern of unitary measurement, because the mathematical average expectancy is equivalent to multiply the pattern length by its probability and therefore: 1 x 0,636 = 0,636 thereby lengths more likely to be observed in set backs of any given elongation (which acts as a unitary pattern elongation) will be in the range of 63,6 % of it.

And this is what observations have massively found in the historical graphs analysed by the followers of Elliott’s Theory of Waves who, despite to keep on with the 0,618 or golden section number sponsored by Elliott, now admit that the limits of set backs are within a band between 61 and 66 % (see:  http: /www.basefinanciera.com/finanzas/publico/aula/at9_1.htm - spanish language -)
With this approach, the mean point of this band will be situated at:

61 + 66                             
                              --------- = 63,5 % = 0,635                                 
2                              

This confirm that the mean point of the band is much closer to the number set by the Fractal Theory than the number endorsed by Elliott’s Theory as 0,635 only deviates a mere one thousandth from 0,636.

On the other hand we know that chaos is an intermediate state between random and order, as displayed below:

Random is the contrary of order and therefore it is "symbolically" located at 180º of the latter. When random is introduced in an ordered system, the position slides slowly from left to right towards, and when the mix order-random is perfect, the system is situated at the central point or perfect chaos.

If, lately, the quota of random in the system keeps on growing in such a way that surpasses order, it will be positioned on the right hand part of the line, to end up at pure random, when order has completely disappeared and the system is 100% random.

Control of order over random patterns is kept until the most probable value of the fractal limit, number 0,636 as real size of the module of germ function of i and, as seen before, this strictly occurs at the 33,04551436 sexagesimal degrees although, security wise, is interesting to set this limit at the 33º as a maximum. Before these 33 º patterns, though chaotic are mastered by order, but from this point on the patterns of formation turn to be unrecognisable and hence "chaotic" in practical effects.

In the light of all these previous points we are in a position to draw on the Edgeworth’s Box the so called "square of chaos", within which the probable behaviour of charts (share quotations or anything else) faints and it becomes practically impossible to make predictions and with it we put the end to this survey.

THE DEGREE 33

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