Squaring the Circle in a Projective Way

Squaring_Circle

I think it could be possible to explain the area of the circumference in a simple and rational way by projecting the square on the radius through the Z diagonal until the point that touches the circle and adding an additional extension.

In the picture above, the coloured spaces represent the area of the circumference.

You can see a circumference of radius 1. The blue square outside of the the circle is a square of area 4; the red square inside of the circle is a square of area 2; And inside, you also can see a blue square 1, a red square 0,5 and a blue square 0,25; the center of the circumference is the zero point that acts as the center of symmetry of all those squares.

I divided the square 1 in four squares of 0,25 for getting their respective center of symmetry by tracing two diagonals inside of them. I signalled with a blue point the center of symmetry of the upper right 0,25 square.

I also divided the square 2 in four squares of 0,50 for getting their respective center of symmetry, and signalled with a red point the center of the upper right 0,50 square.

Finally I divided the square of area 4 in four squares of area 1 and signalled with a blue point the center of the upper right square 1.

Comparing those centers of symmetry you can see that the distance between the zero point and the center of the square of 0,25 is the same that the distance between the center of the square 0,25 and the square 1. But that distance is different than the distance between the zero point and the center of the square 0.50; and those different distances are not proportionated. I consider a “rational” interval the distance between zero and the center of 0,25 and an “irrational” interval the distance between zero and the center of 0,50.

Those disproportionate intervals represent the two kind of symmetries that are present in the circumference. The difference between them will be constant independently on the length of the radius. (The good news are that those disproportionate intervals – the two kind of symmetries – converge at a point in the Z coordinate).

In the circumference participate two different symmetries because the vertex of the square 2 touches the inner point of the circle in the Z coordinates; and the sides of the square 4 touches the outer point of the circle in the X and Y coordinates.

When we take the square built with the radius (coloured in blue) when can project it by displacing it through the Z coordinate following the rational or the rational interval; following the irrational interval we will make two displacements until the red point that touches the circle. I coloured those two displaced squares in red colour. But that is not enough for measuring the whole area of the circumference because we only have considered the parts of the circle that touches the irrational symmetry, the red points at the Z coordinates, but we have not measured yet the parts of the circle that touches the rational symmetry, the blue points at X and Y coordinates.

For considering the rational symmetry we need to make an extra displacement until the blue point that represents the rational symmetry. I coloured that added extension in blue colour.

The problem with the infinite decimals of Pi would appear when calculating the value of the blue extended area; infinite decimals of irrational numbers appear because we compare numbers that represent two kind of symmetries that are disproportionate. Numbers are not purely abstract entities, they represent symmetry and proportion.

The red square with the blue extension is a complex area formed by the irrational symmetry (in its red part) and the rational one (in its blue extended part). So if we try to calculate the value of the blue extended area considering it as a part of that complex figure (dividing the complex area by the blue area) we will be comparing two different disproportionate symmetries. Also note that the complex figure is not a square because it has an asymmetry in its upper left corner and its bottom right corner.

It seems the blue extended area should be calculated in terms of the blue square. But in any case, it could be possible to save the disproportion between the two symmetries in the convergence point: Those two symmetries – the two intervals – converge in the green point that touches the upper right corner of the extended square. (I think that green point is the critical zero of the Riemann Hypothesis; Prime numbers represent new kinds of disproportionate symmetries).

In this sense it could be possible also to calculate the area of the circumference displacing the square of area 2 (the square inside of the circumference) until the convergence point, where its 0,50 square has the same center of symmetry than the 0,25 square, and then eliminating the parts of the squares of area 2 and the convergent area that are excessive.

In the picture below the squares of area 2 appear in green colour, and the convergent area appears coloured in yellow.

CircumferenceIMG

I think in this way is also possible to explain and rationalize in the same way the disproportion between the perimeter and the diameter.

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