The system of the Symbol works with adjusting curves. Concretely, it has available more than half a million functions, 516.961 exactly. We will study the problem of adjusting curves and then will see where we get the exact number of 516.961 functions.

The function which adjusts to a straight line is: Y = A.X + B

The function which adjusts to a parabola is: Y = A.X² + B.X + C

And in complex graphs the function is: Y = A.X⁵ + B.X⁴ + C.X³ + D.X² + E.X+ F

We see that the first expression contains all possible straight lines, and to adjust it to a graph which represents a straight line to give the correct values to parameters A (pendent of the straight line) and B (height at the origin) is only needed. Yet, if the graph has not the shape of an straight line but of a parabola, however much we endeavour to find the parameters A & B we will never get the adjustment, as we will have to use the second adjusting function.

The formula which the system of the Symbol employs to obtain the necessary set of functions to be able to adjust any potential graph it may come across, is based on the so called "Lie’s algebra" which allows a function to be rotated and extracted from it all the contiguous functions and concretely, and in technical jargon, allows to "the germen of a function to be rotated and extract from it, in smooth transition, all the contiguous functions, until arriving at the function opposed to the primitive function."

Imagine that we take a column and in its upper part we place at the most pushing up function possible, for instance a strongly pushing up parabola, and then we place below and orderly (one below the other) all the contiguous functions obtained by Lie’s algebra, until we arrive at a function which is contrary to the first, that is a parabola strongly pulling down, and we will have finished the process.

At the end of this process, we will have in the upper part of the column an ADJUSTING FUNCTION strongly pushing up, with which we could represent graphs in free rise, and in the lower part of the column we will have another adjusting function strongly pulling down, with which we could represent free fall graphs. In the middle of these two functions an ordered series of functions, where will be included those to allow the adjustments to straight lines and complex curves.

So far we will have completed all referred to FUNCTIONS, but parameters are still missing.

To take on the problem of parameters, let us imagine again the column explained before, now full of formulae (functions) ordered in descending sense, from the most pushing up (free rise) to the most pulling down (free fall) and let us continue imagining that we draw horizontal lines above each formula, and these lines start before the column (right part) and prolong to the left side of the column and beyond.

Figure 12

Under these conditions we take each function and begin to modify their PARAMETERS, in such a way that from the centre (of the column) to the left side the parameters are increasingly pushing up FOR EACH FUNCTION and from the centre to the right side the parameters are increasingly pulling downs FOR EACH FUNCTION. When we finish this process will have a gridlock of parameterised functions ordered in such a way that each function will be contiguous and ordered in relation with the others and in this gridlock :

• The more upper and left the parameterised function is situated in the gridlock , the more pushing up the graph which adjusts to this function will be.

• The more down and right the parameterised function is situated in the gridlock, the more pulling down the graph which adjusts to this function will be.

If we now allocate all the parameterised functions obtained, which will be CONCRETE MATHEMATICAL FORMULAS, in a computer and then enter data conforming whatever curve or graph, the computer will seek which concrete mathematical formula among these adjusts better to this graph. Once we get this we may know where in the total mosaic of formulae which the System handles this graph is located. This the basis of how the IT System of Symbol works.

The aforementioned gridlock of formulae is scientifically named "space of phases". This name comes from Chaos Theory, where space of phases is how a diagram is named, which represents all the possible status a phenomenon can take. Each of these possible status is being represented by a point of the diagram.

In view of this, if a mathematical formulae gridlock is adequately and extensively constructed, any phenomenon expressed in a form of a graph, curve, or chart will be able to be represented as a position (a point) on this gridlock, its geometry will be equivalent to the geometry of the space of phases and in this case to "the space of phases of the graphs".

If we wish the space of phases geometry of graphs be entirely formal, it should be completely deducted from mathematics without any aprioristic consideration and, therefore, to do so we can not start from a system of Cartesian axis, because a system like this does not come from (inferred or deducted) mathematics.

The Cartesian system of abscises (x axis) and ordinates (y axis) is a technique, it is an invention of Mnr. Descartes, which represented a big step forward in the representation of functions, the graphic resolution of equations and interpolations of functions, but what we pretend here it is not to adopt a technique but rather to obtain a diagram which can be directly inferred from mathematics.

Figure 13

We observe that:

• Multiplying any number by (-1) is equivalent to provoke a turn of 180º of vectors which have this number as a module.

And this will produce without any need to specify an axis of reference, because the position of reference is the initial position where the vector is located before its turn , whatever this initial position is. Therefore, there will not be any privileged or fixed axis.

Finally, if we want to obtain an easily viewed unitary multiplying factor to turn the vector any amount of degrees expressed in the sexagesimal system, we should find a function (a formula), in which the sexagesimal degrees are the independent variable, in such a way that entering the degrees we obtain the multiplying factor for the turn.

Which can be checked easily as in the following three cases:

The previous formula "i power degrees divided by ninety", is expressed in only a single dimension. Indeed, despite the viewing of turns may be illusory of being on a surface, in reality we are in an unidimensional straight line, as only on a straight line we obtain real numbers (1 and –1) from that expression. Yet, when the turn brings us to the surface, the imaginary number i always appears, as it can not be in other way because this plane does not exist, is imaginary.

• The formula "i power degrees divided by ninety" expresses in one dimension all the turning angles of a unitary vector.

To find the equivalent function of the previous one in two dimensions, that is, to find the expression of "all the turning angles of two vectors in a surface", we can square the previous unidimensional formula, the same way as we can square the length of a segment to get the surface framed by this segment, and then the following happens:

=

In the light of the above we already know what the true geometrical structure of the gridlock of functions (commented in section 2) is. In other words, we have deducted the geometry laying under the "space of phases of the graphs".