The Hidden Rationality of the Pythagorean Theorem, the Square Root of 2, and the Pi number.

We construct the square areas of the legs a  and b  in the Pythagorean theorem
a^2+b^2=c^2   placed on and related to the specific spatial coordinates x  and y  .

When the value of the leg  is =  1 , the square area constructed is our primary square area 1. To say that the space that exists inside of a square area with sides of length 1 measures also 1, (1 square area), is a pure abstraction. To be able to establish a reference for measuring square areas, we make the abstract convention of considering that our primary square area constructed with our primary lineal length 1 (1 step, 1 elbow, 1 thumb, 1 food, 1 meter… whatever that is our primary length for measuring linear lengths which is a also an agreed abstraction) has the value 1.

That primary square area 1 is not an abstract number 1, it’s a concrete and specific square area, our primary square area 1 with an abstract covenanted value inside, represented as 1^2 .

When we trace the hypotenuse inside of our primary square area 1 and then we construct its own square area, we are not using the xy  coordinates but two new and different coordinates x_{2} y_{2}  created by displacing xy  toward x_{2} y_{2}  .

The effect of such displacement of xy  toward x_{2} y_{2}  has the same consequence than introducing an expansion on the space upon we constructed our Pythagorean squares. It’s like to create two different planes on a same space and to separate them. The referential length 1f(xy)  we used for creating our referential square area 1 has been transformed in its own inner essence, inside the square area of 1 itself. So we need to use a new referential length given by the hypotenuse for creating the new referential square area of 1f(x_{2} y_{2})

In this sense, the Pythagorean areas are multiplanar, multidimensional if you will, areas related to different referential lengths xy  and x_{2} y_{2}  on a shared space. To be able to compare them it will be necessary to take account of the transformation introduced by such displacement.

(That’s in fact what we make when comparing different referential magnitudes as centimetres and metres, we can not compare directly different things).

To say that a^2+b^2 =c^2  is an incorrect statement because we are comparing different things.

For comparing them it will be necessary to say that (a^2+b^2)f(xy) = c^2 f(x_{2} y_{2})

To be more accurate, those compared areas are coincident (which is not the same as saying that they are equal). And the problem here comes when we try to formulate their quality of being coincident in a mathematical way and then derive abstract consequences of that formulation.

If a = 1 and a = 1 , we say that 1^2+1^2 =c^2 , then, we logically say that because 1^2 = 1 , it results that \sqrt[2] {1+1} = c

And we know that \sqrt[2] {2} has infinite decimals which is an irrational magnitude.

But 1^2 \not= 1 , actually 1^2 = 1(1^2)

In the same way, 1^2+1^2 \not= 2 , actually 1^2+1^2 = 2(1^2)

It’s not the same, because our square areas are not abstract things, they are concrete things with abstract values inside.

In this sense, the square area created with the hypotenuse is 1^2 f(x_{2} y_{2}) , which is not the same as saying 1^2f(xy) because 1 f(xy) \not= 1 f(x_{2} y_{2})

The square area created with the hypotenuse is also a primary square of 1 related to the new referential length of c . The value of its inner space is an abstraction that is disproportionate to the abstract space we agreed to consider as 1 square area of 1 related to the coordinates xy .

To be able to express 1^2 f(x_{2} y_{2}) in terms of xy and to know its abstract value inside of it in terms of our primary referential abstraction 1^2 f(xy) , to be able in fact to compare c^2 (x_{2} y_{2}) with a^2 f(xy) and b^2 f(xy) , we will need to consider the transformation of the space introduced by displacing xy toward x_{2} y_{2} .

In this sense it results that \sqrt[2]{1f(xy) +1f(xy)} = 1+(d1-d2)


Something similar could be said about the disproportion that appears when comparing the perimeter and the diameter of a circumference.

If we draw a larger square sA outside of the circumference and a smaller square sb inside of the circumference we will have two squares ruled by different referential lengths and planes, the sA related to the irrational x_{2} y_{2} , and Sb related to the rational plane xy .

The perimeter is measured in terms of sA but the diameter is measured in terms of sb .

What I mean is that length of the radius is always the same independently on the coordinate it is, but it’s only in appearance because we are two different square areas with different referential lengths to measure it creating an inconsistency.

So, when the radius R is on y_{2} (45 degrees), it’s on the irrational coordinate of the perimeter; we are going to measure it in terms of the smaller square of sb because it’s on the corner of sb , while for measuring the diameter we are going to measure until the corner of y_{2}f(sA).

When R is on y (90 degrees) we still should measure it in terms of f(sb) , but by the contrary here we measure its length in terms of f(sA) . So here we are changing or referential length measuring the radius in terms of the perimeter.

The consequence of that is that we finally have a discordant length between the diameter and the perimeter. The tricky question here is that when the radius is on x (0 degrees) or y (90 degrees) it’s not being measured in terms of the the referential length of the rational coordinates xy but in terms of the referential lengths of x_{2} y_{2} .

In this sense I think it could be said that the circumference is, as the square area of the hypotenuse, a multiplanar area. To be able to sty it in a rational way it seems to me it should be necessary to study it taking account we are working with different planes.


As far as we know, In fact I know vey few about maths, complex numbers were find out as a mere instruments or tools for solving real problems. But without having a conceptual idea about what they are and why they appear.

I don’t know if complex planes are being used for explaining complex areas or even if there’s is a concept of “complex area” but it seems very symptomatic that something so absurd as the idea of infinite decimals in limited segments has endured so far without being questioned and rationally clarified. Maybe has the assumption about irrationality hidden a closed door avoiding people thinking and being aware about the evident idea that those figures are formed but combining different planes with different referential lengths that cannot be compared directly, that they are “complex areas”?

I think Squaring the circle should have be possible considering a complex variable area formed by a square that varies periodically, a pulsating square that expands and contracts periodically. In this sense, considering the circle from the square perspective, the diameter will formed by a segment r1 that decreases and increases periodically; and the perimeter is formed by a segment r_{2}  that varies with an opposite phase increasing and decreasing periodically.

So, at 0 degrees, the radius of the circumference will act as r_{2}  (it will correspond to the larger expanded square that rules the perimeter); But at 45 degrees the radius will work as r_{1}  (and it will correspond to the smaller contracts square that rules the diameter).

The contracted square would correspond to one of the rational square areas formed by the legs on the Pythagorean theorem, and the expanded square would correspond to the irrational square formed with the hypotenuse on the Pythagorean theorem.

It also could be thought that to be able to express the radius or the diameter in terms of the perimeter, it should be necessary to work with a rotational square, rotating the square sb to reach the irrational corners i that rule the perimeter:


When it comes to calculating the area of the circumference multiplying 3,14159 x r^2 we are considering that the radius does not vary in any case. That would be true if we considered the rotation of that r^2 , but if we are considering a static square then we will have to consider the displacement of xy toward x_{2} y_{2} and x_{2} y_{2} toward xy , periodically, what creates the effect that the radius changes its length periodically.


But it’s not enough with considering the different squares, to be able to compare them by addition it’s necessary to transform them to the same referential length, which here is the length of the radio, the resultant area of the circumference will be expressed in terms of the square formed with the length of the radius.


The key issue is the change on the reference center of the squares when we displace the coordinates. That’s what creates the disproportions between the two (or n) squares of 1 and the areas that we calculate basing on them.