Abstract: Projecting the square 1 through the diagonal of its hypotenuse we can build a new prime square 1 with an irrational symmetry. Combining the rational and irrational symmetries we can get new prime squares which roots will be irrational. The zero points displaced in this way through the infinite diagonal should be coincident with the Riemann function zeros. But it seems it also could be necessary to consider other different lines related to the mirror symmetry of the Riemann’s critical line.
Draw a circumference of radius 1 in a space R2. The center of the circumference is our zero point of symmetry.
Build the square area 1 taking as side the radius 1. Now we have our referential square 1 for measuring areas. Trace the two diagonals inside. Their point of intersection is our primary central point of symmetry, the center of the rational square 1. We also can draw the 4 squares of 0,25 and trace their central points. Those points are proportionated with respect to our rational primary proportion, de primary zero (the center of the circumference in the above picture).
Displace the square 1 through the right diagonal, setting its bottom left corner on the central point of symmetry of the bottom left square of 0,25. The distance between the primary zero point of symmetry and the central point of the 0,25 square is the distance we are going to use for displacing the square 1 through the irrational diagonal.
The symmetry of the displaced square 1 is irrational with respect the primary square 1 because it has its center of symmetry displaced with respect the central point of symmetry of the rational square 1. It follows and represents a different inner proportionality. Now we have then a rational primary square 1 and an irrational, displaced or projected square 1.
Trace the right diagonal inside the displaced square 1 (in red colour in the attached picture) and take the segment placed from the primary zero point of symmetry until the upper right corner of the displaced square 1.
With that segment, build a new square setting its bottom left corner on the primary zero point of rational symmetry. The area of the square that we just created is a prime number, and its root square is an irrational one. Why? The area is prime because we have used a new referential symmetry – the irrational symmetry of the displaced square 1 – to build it, and its square is irrational because it is formed by mixing two different symmetries related to the prime rational and the prime irrational squares. When we compare two disproportionate measures of reference we always get disproportionate results.
So, the side of this prime square area represents the golden ratio of the hypotenuse of the irrational square 1 + the distance between the irrational and the rational zeros.
We can repeat the same procedure infinite times, always displacing the last displaced square 1 through the diagonal in the taken proportion. We can also create new square areas displacing through the diagonal the new created prime squares taking as reference the new created points of symmetry (in red colour in the above picture).
It’s easier to see if we draw the central point of symmetry of each new square 1 or whatever drawing their diagonals inside and the 4 central points of symmetry of their 4 squares of 0,25.
It’s also easer to see that the displaced squares have a new and irrational (I mean, disproportionate with respect to our referential and rational points of symmetry of our primary square 1) by observing that the displaced square 1 has the same central point of symmetry of the square area 2, which root square is irrational.
I think the blue and red points on the diagonal represent a displaced zero that would coincide with the zeros of the Riemann Z function. At the eight zero the rational and the irrational points (look at the zero points as the carrier of a kind symmetry) are coincident. In musical terms that represent a first octave with their tones and semitones, being formed the Do note with two different notes, the rational and the irrational ones.
In this sense, the Riemann hypothesis seems to be right because of those of zero points are on the same right line.
But we also can draw at the same time other zero points on the line that comes from the another diagonal of our rational square 1. It would imply to work with a second circumference intersected with the first one, and so with a second rational zero that will be the center of a mirror symmetry. Although in this case it also appears a new center of symmetry in the center of that intersection. In this case we can construct new complex areas that will combine positive and negative values.
In this case we displace the referential square 1 through the left and right diagonals, creating two unfolded or separated squares of 1 with a new symmetry. But we also can fold them again into a merged square 1.
So here we have actually created another tensor (a vertical one) displacing the rational square 1 through the Y coordinate, taking as primary referential zero the center of the intersection of the two circumferences. And we can combine in an interleaved way the ortogonal tensor and the diagonal one.
Riemann distinguished between trivial and non trivial zeros. The real part of every non-trivial zero of the Riemann zeta function is 1/2. (Note that I do not distinguish between “real” and “imaginary” but between “rational” and “irrational” depending on if its follows the referential (rational) symmetry or it doesn’t).
So in the Riemann’s non trivial zeros converge two parts, the rational (“real”) and the irrational (“Imaginary”) ones.
In this sense, a non trivial zero should be what I named the Do note when the red and blue points converge at the octave in the diagonal projection. And for getting only the real part we should separate the two squares of 1, the rational and the irrational 1.
But when it comes to the vertical projection taking as referential zero the center of the intersection between the two circumferences, the non trivial zero is formed by two irrational (imaginary) symmetries that converge in the same point of rational symmetry; So, for getting the real part of the non trivial zero we need to multiply the displaced symmetries of the two irrational squares of 1, the + and the – ones.
On the other hand we also could perform the same procedure by rotating the squares around the central point(s) of the circumference(s), and playing with their diagonals. So, the lines which have the zero points will follow a descendent path in some of those cases.
In this sense it seems that for explaining the whole periodicity of the rational and irrational zeros placed through the diagonal it’s necessary to consider the square areas that are derived from the other diagonals traced from the center of the circle, the ones that pass through the 4 points marked as “i” (from “irrational”) in the circumference drawn in the above picture.
Looking at that circumference you can see that there are four irrational points i, i2, i3, i4, that do not coincide with the rational symmetry but follow the irrational one. That creates a disproportion when we try to measure the area of the circumference in terms of our rational square without being aware that we are mixing two different kinds of prime symmetry, the rational and the irrational ones. The disproportion is expressed in the appearing of infinite decimals. That is also what happens when comparing the diameter and the perimeter of the circumference.
It is also necessary to consider other dispositions, for example having a central circumference, and 4 other circumferences intersected with the central one, and so on.
On the other hand, if we can create disproportions and new kind of groups of symmetry by displacing the prime squares through the diagonal, we also can restore the lost rational symmetry creating new transformations between prime and non prime areas depending on the length of the segment (the interval or tone) we take as reference for making the displacements. I guess it could be consider “covariant” or “contravariant” transformation of symmetry.
Considering a tensor the displacement or projection of the prime squares and zero points through the Z diagonal, then the Z coordinate should be consider as a rotational tensor in itself, because it is created by displacing the coordinate X toward X2(Z) and X3(Y) and so on, describing a circle. In this sense the only real dimension should be the X coordinate and all the other should be projection of X.
A tensor can be understood as a projection but it represents an expansion or contraction of the space. The essential point is that it can imply a transformation of the center of the referential symmetry and proportionality that we were using.
We can measure areas taking as reference a square area 1. The square area 1 that we build with our referential length 1 is an abstraction; we do not measure all the points or lines that are inside that space; we attribute it the vale 1. But its center of symmetry is a concrete and real reference that we have created and we need to be aware of the transformations that we introduce working with it.
Modern mathematics has found out some elements of a mathematical landscape that yet has not been able to observe as whole because of its lack of comprehension of irrationality. This landscape is not static, it contains all kind of possible variations, periodicities and transformations, and it’s the expression of primary or primordial times or “tempos” inherent to the numbers that appear through the geometrical space.
Mathematics is the look for proportion and symmetry and the explanation of any apparent asymmetry. Numbers express proportionality. Whatever notation or symbol we use should be always considered from the point of view of the symmetry it represents.
Feb 15 Update:
When it comes to projecting through the Z coordinate, it seems more natural to displace through the diagonal the square of 1 (with the rational symmetry, in blue color) and the square of 1/2 (with the irrational symmetry, in red colour).
You can see the two entangled symmetries moving from the central circumference toward the upper right circumference. To showing it clearly, I also draw each symmetry in a separated way in the projections traced from the central circumference toward the two descendent diagonals, the irrational symmetry toward the right side and the rational one toward the left side.
The Riemann Zeros appear in the poles of the Polar Rose where converge the different symmetries.
Update Feb 26.
The Riemann critical zeros in the orthogonal axis, rotating the square 1 when projecting it through the Y coordinate. For getting the center of symmetry of the square of 0,25 and the square of 0,50 it’s necessary to rotate the square of 1 and the square of 2.