The Pythagorean Theorem in the Complex Plane.

The square 1 that we build with the referential segment of length 1, is an abstraction: we do not measure the lines and points there inside of it; We convey that the space inside of the square 1 has the value 1, 1 square, and we are going to use it as reference for measuring square areas.

But the square 1 is not a pure abstraction, there is something real and concrete inside of it, its centers of symmetry. The center of symmetry of the square 1, and the centers of symmetry of the squares that are proportionated with it, as it is the case of the 4 squares of 0,25 (inside of the square 1).

The center of symmetry determines the physical properties of things. We can imagine if we will that we have a pure abstract object that is not in our physical reality but when it comes to the our material worlds, all things have physical properties and they comes determined by their centers of symmetry.

In this sense, if we have two squares a^2 = 1 and b^2 = 1, we cannot say they are = c^2. Because c^2 has its centers of symmetry displaced in a way that is disproportionate with respect to 1^2. It is evident that there is a disproportion when it comes to comparing the sides of the squares a^2+b^2 with c^2; but it´s not so evident that the areas a^2+b^2 are also disproportionate with respect to c^2 because they are coincident. They seem equal or equivalent but the actually are not.

The disproportion between the centers of symmetry is evident in the picture below; the squares 1, 2 and 4 have the same center of symmetry, the center of the circumference; the 4 squares of 0,25 (inside of the square 1) and the 4 squares of 1 (inside the square 4) have their centers of symmetry placed in a proportionated way, they respect the same interval; Until here all is proportionated. But the centers of symmetry of the 4 squares of 0,50 (inside the square 2) do not follow that interval, they are displaced in a different and disproportionate way.

Centers_Symmetry_1

The square 2 has a different nature than the squares 1 and 4, and because if that it cannot be compared directly. They only could be compared if we are able to transform them in such a way that fixes the existent disproportion. The disproportion created by displacing Y toward Z when drawing the diagonal inside of the referential square 1 can be saved through the projective plane by considering that the square 2 is a complex area.

In the picture below I projected in a consecutive way the squares 1 and 2 following two different kind of intervals determined by the distance between the center of the circumference and the center of the squares of 0,25 or the center of the squares of 0,50. The radius of the circle is 1.

PythagoreanT_0001

In this sense I think the Pythagorean theorem only can be considered true if it is expressed in terms of the complex plane in the cases that c^2 is a prime number.

Pythagorean_Theorem

Pyth_Complex

I draw the root square of two in its irrational form, that is to say in terms of the real (not projected) square of 1, but I think it must be possible to express it in terms of the complex areas related to the projected square 1 (the square 1 displaced 6 times from the center of the circumference).

The squares 1 and 2 are decompose by projecting them through Z, and there we compare de Pythagorean areas in terms of 6 squares of 1 (so a total of 12 squares of 1 in the Pythagorean Theorem) and 3 squares of 2. So, the comparison must be made in terms of 12 / 6.

In other previous post I suggested that the points where the disproportionate intervals converge, should be critical Riemann Zeros. The Riemann Z function should actually different convergent Z functions. The non prime square would be formed with four different zero points.

Centers_Symmetry_4

I think prime numbers represent areas whose centers of symmetry are disproportionated with respect to the centers of symmetry of our referential square 1 or the centers of symmetry of any other prime area.

If the disproportion between the Phytagorean areas can be saved in this projective way, then, when it comes to comparing cubic areas, I guess it could be possible to save in a projective way the disproportion that reflects that a^3 + b^3 is not = c^3, which would imply to refute the Last Fermat Theorem that states that a^n + b^n is not = c^n when n is greater than 2.

I think the picture actually represents quintic functions.

Quintic_Equations

Update Mar 24, 2016:

In the picture below I’ve coloured the projected squares of 1 and 2 with different colours to show it clearly:

Quintic_Z_Fucntions

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