I think it could be told that there is a rational number and an irrational number .
For drawing the picture above I followed the next steps:
1. Draw a circumference with a radius 1 (or )
2. Draw its exterior square. Each of its sides represent the
3. Draw another circumference outside of the square area of
4. Trace a new square are outside of that new circumference. The sides of this new square are the
5. Inside of one of the the square areas of trace a circumference.
6. Inside of that traced circumference draw a square area. This square (draw in red colour in the picture above) is a new square of 1. The square of 1 is our abstract and primary reference for measuring square areas.
Then, here we have a primary and rational square of 1 (that is our primary reference for measuring square areas) build with the radius 1 (which is our primary reference for measuring linear lengths), and an irrational square of 1 that has a different center of symmetry that our primary square 1.
The essential question here is to be aware that we are working with different symmetries that have different centers. When we trace our primary square area of 1 with our primary linear segment of 1 we are making an abstraction and we convey that the space existent inside of that square is 1 (1 square area); but that square area has a concrete center of symmetry, the center of symmetry of our referential square area is not an abstraction.
7. Now, we can trace a circle outside of our irrational square of 1. Tracing an square outside of that circle we will find again the
8. Now, trace a new circle outside of the square formed with the . This circle will be traced from the center of symmetry of the irrational 1.
9. Trace a new square outside of the circle traced in the previous step. The side of that square is the , but here we are considering the square area of in terms of the rational symmetry of the rational square 1. We are considering as the sides of that square only until the rational coordinates x and y.
So, the square formed with the is a multidimensional shape in the sense that it’s formed by combining two different symmetries the rational and the irrational ones.
10. Trace the diagonals inside of the square formed with the considering its corners the rational axes X and y.
11. Now trace a new circumference outside of that square, taking as center of symmetry the center of the square formed with the
I think that in this way, playing with the different centers – the rational and irrational ones – of symmetry, we should be able to draw any irrational square root.
Irrationality points toward disproportion, an asymmetry that arises when comparing things with different nature; and here for different nature I think we should consider the existence of a different center of symmetry.
In previous posts I considered these asymmetries as the consequence of comparing different referential lengths, as if it we were comparing for example decimetres and centimetres as if they both were the same thing without transforming them. That different referential lengths should appear when displacing the rational coordinates xy toward x2y2 after tracing the irrational diagonal inside of our rational square of 1. But the idea was not clear enough.
Here I think that the irrational asymmetry created by displacing the referential coordinates arises from displacing the center of rational symmetry.
We are comparing things of different nature because they have different symmetries.
In this sense, I think that asymmetry that appears by tracing the diagonal in the Pythagorean theorem should also explain Pi.
The circumference follows 4 point of irrational symmetry – the points – which are asymmetric and so incommensurable from a rational point of view.
If we trace another square of 1 having as center of symmetry the center of our original circumference of radius 1, we will see that this another square of 1 has also a different symmetry than our rational square of 1. But actually now our rational square of 1 will be irrational with respect of the new square of 1 that shares its axis of symmetry with the center of our circumference of radius 1.
I think it’s clearer here:
And how many new irrational squares of 1 can we create, how many new prime symmetries are there? Infinite
To see it we only need to displace the square 1 through the referential points that appear in the diagonal; We can displace it until the next rational point (in blue colour) or until the next irrational point (in red colour). Then, we count the length of the hypothenuse but counting from the point that appears in the picture at the center of the circumference, until the up right corner of the new displaced square. With that new hypothenuse we can create a new square from the center of the circumference, that will an irrational square area with a new prime and irrational symmetry related to the newly displaced irrational square 1. That irrational square area will be prime, and its square will be an irrational number. In that way, knowing the kind of symmetries we are following and if we are combining or mixing them and how, we can predict from a geometrical perspective what numbers will be prime and what will be irrational.
In this sense you could consider the left down corner of each irrational / displaced square of 1 like a Riemann Zero, a displaced or irrational zero that we move through the diagonal line.
Form that view, in a R2 space, the Riemann hypothesis is right in the sense that its zeros are on the same line, but ot’s incorrect because they also can be represented at the same time on the line that follows the another hypothenuse in our primary / rational square 1.
On the other hand, we can form a square that shares the rational and irrational symmetries taking two sides of the irrational square of 1 and two sides of the rational square of 1 . This new square will have a new center of symmetry that can be considered as a mixture of the rational and the irrational symmetries. I’ve coloured that new “shared” square in yellow in the first picture and one of its “shared” axis in green colour.
Do two parallel lines intersect in the infinite? really not. But I think they actually intersect when they respond to different symmetries in the sense described above. Their intersection takes place when mixing their different symmetries by creating a shared and multidimensional or multi-symmetric space (the yellow square) which will have a new shared (rational and irrational) symmetry. The points of that intersection will take place when the shared axis (the green line) of that multi symmetric space meet each one of the parallel lines.
In this sense it could be said a rational and an irrational parallel lines intersect in a projective way when the line that shares their rational and irrational symmetries meet them in two different points.
Now imagine that here we are speaking about planetary orbits.
How could our physicists explain the asymmetries of our solar system model if their mathematics are not able to explain the asymmetries expressed by irrational numbers and areas? If they only work with irrational magnitudes without comprehend their meaning.
It seems incredible that with the enormous and highly complicated developments mad by our mathematicians they yet have not developed a simple view based only in the symmetries and asymmetries of the square and circular areas, but it seems that door has remained closed and hidden because of the lack of comprehension about the irrational disproportion found out in the Pythagorean theorem and the acceptance of irrational magnitudes as a mere instrumental tool.
Currently there is no mechanical explanation about the asymmetries that we appreciate and measure in our solar system, because yet our physicists have not had a rational comprehension of the irrational symmetries.
In this sense, I think the yet misunderstood asymmetries expressed by the planetary orbits of our solar system are very clearly pointing toward the existence of different centers of symmetries in that orbits, what implies different orbited gravitational fields.
At the beginning of the XXI century we yet believe in the validity of a solar system model that is exactly the same geocentric one with the only difference that we set the Sun instead of the Earth in the center of that system and that we are able to detect that planetary orbits are not circular. A solar system model that is full of unexplained asymmetries that are supposed to be created by randomness. Randomness, my dears, does not exist.