I think quintic functions could by understood as a rotational fractal formed by four entangled (related, intersected, interacting) functions. In this sense the quintic function or equation appears as a complex function formed by intersection of different parts of the functions related by the same kind of symmetry although different signs.

I bolded in red what would be one of the quintic equations that there are in this picture.

In a first attempt, it could be expressed in this way:

In (Z, -+Z1) Z is the Zero point at the center of the circumference, and -+Z1 is the first interval Li measured from the Zero point Z in the -+Z coordinate (the bottom right Z coordinate) in the picture below.

By crossing or intersecting the lines of the different functions in their intersection points we would be performing a subtraction (I think the result should represent a trivial zero in the Riemann Z function).

The Li (in red colour) segment is measured from the center of the circumference (the Zero point) until the center of symmetry of the square 0,50 (Inside of the central square 2).

Lr (in blue colour) is measured from Z until the center of symmetry of the square 0,25 (inside of the central square 1)

Li’ (in gold colour) is measured from Z until the center of symmetry of the square 0,75.

The bolded function in the picture above follows the Li intervals that carry the Li symmetry. The Li intervals converge at the power of 5 with the intervals of the Lr (in blue) and Li’ (in gold) functions.

Note that for projecting Li through X and Y without modifying the length of the interval – so without changing the Z symmetry that they carry or represent – it’s necessary to rotate the square 2 and so the square of 0,50), which actually implies to transform its center of the XY symmetry.

In this sense, it seems that the above picture can not be an static diagram, it must be a dynamic picture in the sense that the central circle would be a periodically variable circle that rotates on it self while expanding; Actually because the different related functions follow different directions, there should be represented different intersected circles that rotate on it selves with opposite directions having a mirror symmetry.

**Objections**

There are some points that are not clear to me yet:

In this diagram what square areas will be Li^2, Li^3, Li^4 and Li^5?

The Li segment is the root of the square 0,25. So its value is 0,5.

0,5^2 is 0,25; 0,5^3 is 0,125; 0,5^4 is 0625; 0,5^5 is 0,0321. It has non sense because the squares that we form with the Li segment should be larger each time that we increase the Li interval in the different coordinates.

It seems that we should use the division: 0,5/0,5 is 1; 0,5/0,5/0,5 is 2; 0,5 divided three times is 4; 0,5 divided five times is 8.

We should be able to form those square areas taking as sides the Li intervals (measured from the central Zero) in the different coordinates.

Then it appears a problem at Li^3. The segment formed in Li^3 f(–Z3) is not the root square of 2 as it should be. The root square of 2 seems to be the segment Lr f(–Z 4), which is the fourth blue point near to the third red one in the –Z coordinate.

Lr and Li are going to converge at Li^5 and Lr^7, one solution to be able to arrive to Li^5 could be to transform at this point the Li symmetry into the Lr one, and then to transform Lr into Li at Li^4. That implies to use the Lr function (and so the four Lr functions) as a bridge for linking Li^2 and Li^4.

Another question that appears to me would be, does represent the intersection between the functions a subtraction or a division. My initial idea was that they were divisions. In that case, the result of each intersecting point will not be zero but 1.

(Note that Lr4 is = the root square of 2, which is the value that corresponds to the square of Li^3.)

(On the other hand, tracing a segment from the red point Li^3 f(–z3) until the red point Li^4 f(-X4) we get the root square of 2 without having to change the Li symmetry.)

But even changing the symmetry at Li3 by Lr4, when it comes to Li^5 the same problem arises. The segment from Zero until the fifth Li red point at -+Z coordinate is not the root square of 8. To get the root square of 8 it would be necessary to measure from Zero until the eight Lr blue point at -+Z

Using Lr as a bridge function in Li^3 and Li^5 it would appear

This crucial point is not yet clear to me, because I tend to think that the change of symmetry only could come naturally in the convergent Z points.

Note that this problem is the same that I already commented in a previous post about calculating the area of the circumference. In the circumference there are present two kind of symmetries: the Li symmetry that rules the squares of 0,50 inside the square 2 (the square inside of the circle) and the Lr symmetry that rules the squares of 0,25, 1 and 4 (the square outside of the circle).

The square 2 touches the circle in the points on the four Z coordinates (++Z, -+Z, –Z, +-Z); while the square 4 touches the circle at the XY coordinates (X+, Y-, X-, Y+).

If only the Li symmetry was present in the circumference of radius 1, For measuring its area it would be enough with displacing (projecting) the square on the radius through the Z diagonal following the Li interval until set the bottom left corner on the red point of Z that intersects with the circle (Li^2 f(++Z2).

But those three squares of area 1 are equal to three. To get the 0,14 (etc in the case of Pi) it will be necessary to extend the last square 1 until the point that represents the Lr symmetry, Lr^3 f(++Z3). That extension appears coloured in blue in the picture below.

The difference between those kind of symmetries (Li and Lr) explains that Pi presents infinite decimals. The infinite divisions are the only consequence of comparing things that are always disproportionate. But infinite divisions will be not possible if we perform the division at the point where the disproportionate symmetries converge, at Li^5 and Lr7.

I think that is a fundamental clue to investigate on mathematical oncology; there is a hypothesis that states cancer is the consequence of multiple mutations, and more specifically precisely, that it appears at the sixth mutation. I think the scaling fields of these diagrams and the fields that can be formed taking them as base, can be interpreted in terms of stem cells and cells differentiation.

**End of Objections**

To be better understood, maybe you could think about the constant intervals rational and irrational related to the different kind of symmetries as “gauges” in the sense of Weyl’s gauges as “different distance scales” (I just took that expression from the book “Not even Wrong” by Peter Woit – Vintage Books – I’m currently reading). Also in that sense, the changes of a symmetry into another I suggested above are local transformations of symmetry.

You could find interesting the chapter about “Gauge Symmetry and Gauge Theories” of the mentioned book (pages 67 – 74). I’ve scanned these so clear pages:

I add below some other pictures about those intervals:

Yo can also read the previous posts where I explained the idea of two different kind of symmetries, rational and irrational.

On the other hand, I tried to represent the periodical expansion (until the convergent Z points) and contractions in the picture below, but it is yet incomplete because it does not show the opposite rotational movements and the intersected fields.

In this sense I’m speaking as always about intersecting fields expanding and contracting periodically. In previous posts you can read about the atomic and solar system model I developed on the bade of intersected fields that vary periodically. You can take a look at the pictures here:

Our mathematics can not be static if we try to explain Nature with them.

On the other hand, I think it is very interesting to remark the points of convergence of because I think they would represent the critical zeros on the Riemann Z function. In this sense, the square area formed with the side measured from the Z convergent point until another Z convergent point should be a non-prime number.

Felices Pascuas.