*Summary: Working with two parallel lines, one of them virtually existent, it can be demonstrated the convergence of two non-parallel lines mentioned on the Euclid’s fifth postulate. Non-Euclidean geometries are not Euclidean because they do not follow the Euclid’s definition of parallels. *

The fifth postulate of the Euclid’s Elements states that *“If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles”*.

Many mathematicians have though that the fifth postulate was not self-evident; more than a simple postulate to accept, it seems a theorem or proposition that need to be demonstrated.

Many unsuccessful attempts were developed through the different centuries trying to demonstrate the validity or invalidity of the Euclidean fifth postulate. Some of them tried to deduce it from the other axioms or from the propositions of the Elements. Some others tried to write a simpler and clearer formulation of the postulate that has been know as the “parallel postulate”, that is the case of the so-called the so-called “Playfair’s axiom” which states that *“there is at most one line that can be drawn parallel to another given one through an external point*”.

At the beginning of the XIX century Nicolai Lobachevsky though the validity fifth postulate could not be demonstrated but he demonstrated that it is not valid when it comes to curved lines on curved spaces, because more than one parallel line can be traced on a point set outside of a n initial line, initiating then the nowadays known as non-Euclidean or hyperbolic geometries.

After the discovering of the non-Euclidean geometries, the discussions on the demonstration of the fifth postulate seem to be a mere historical reminiscence.

Nothing says the fifth Euclidean postulate about how many parallel lines can be traced on a point, it only states that two straight lines will converge at a point when the two internal angles of a same side of a crossing line sum less than 180 degrees.

I have drawn a picture showing how I interpret the parallel postulate. I think that “internal” angles are the angles that are place inside of the space created by the two lines crossed by another one. This crossing line divides the space in different sides, right and left, up or down.

The above picture represents the two parallel when the internal angles of a same side of the crossing line sum 180 degrees. If the sum were less than 180 degrees the two lines should converge at some point of the straight side of the crossing line.

But the fifth postulate seems to be incomplete because it does not mention as a non-parallels case the situation when the two internal angles of a same side of the crossing line sum more than 180 degrees. If the result of that sum is smaller or higher than 180 degrees will depend on the direction towards which we move the parallel lines to make them non-parallels and on what side of the crossing line we consider the internal angles to be measured.

The fourth Euclidean postulate,* “All right angles are equal to one another”*, is related to the case of equal angles that sum 180 degrees.

In this sense, it seems logically coherent to place the fifth postulate after the fourth one because the fifth postulate is related to the case of different angles that sum 180 degrees.

The lines are only parallels when the two internal angles of a same side of the crossing line are equal to 180 degrees.

But is there any way to demonstrate that two non-parallel lines must converge at some point?

I think to demonstrate the necessary existence of a convergence point when the two lines are not parallel – when the two internal angles of a same side are less or higher than 180 degrees – we have to consider firstly the existence of an initial point of convergence from which the existence of the final point of convergence can be logically deduced.

Let’s start from the picture of two parallel lines *a* and *b*. Let *a* to be displaced from its right side toward *b* and let the displaced line *a* to be named *a’*.

Now we have two non-parallel lines, *a’* and *b*, that supposedly must converge at some point of the right side of the crossing line. The sum of the internal angles of the same side is less than 180 degrees.

But we can also consider the already non-existent line *a* as a virtually existent line *a* that we can call *va*. The internal virtual angle that *va* forms with the crossing line, added to the actually existent internal angle that *b* forms with the crossing line has a “semi virtual” value of 180 degrees.

On the picture below we see that *a’* converges at its left side with *va*. The convergence exists necessary – although in a virtual way – because it takes place on the rotational axis of *a’* when it was displaced *a* towards *a’*. So here we have an accredited point of virtual convergence between two non-parallel lines *a’* and *va*. *a’* converges with itself on its rotational axis, or more accurately, with itself in a previous time, that is to say with *va*.

Because we already have found out a point of initial although virtual convergence between *va* and *a’*, we can assume that a final of actual point of convergence must exist between *a’* and *b*, because between the spaces placed inside of the lines *va* and *b* there is a mirror symmetry determined by *e* and

According to the above, I think that working with two parallel lines, being one of them virtually existent, it can be demonstrated the convergence of two non-parallel lines mentioned on the Euclid’s fifth postulate.

We can see the mirror symmetry dividing the virtual parallelogram in two equal parts by tracing a horizontal line *e*, parallel to *b* and *va*, and tracing a line *f* perpendicular to *va* and *b* that passes through the point *p* of intersection between *a’* and *e*. The mirror symmetry exists between both up and down sides of the virtual parallelogram formed by *va*, *e*, and *b*, and between both left and right sides formed by the perpendicular line *f* inside of the virtual parallelogram formed by *va* and *b*.

But if the validity of the Euclid’s fifth postulate can be demonstrated, how is it possible then that the non-euclidean geometers can trace more than one parallel through a point external to a line using curved lines on curved spaces? Do Quadratic straight lines on flat spaces follow the fifth Euclidean postulate while curved lines following a same straight direction on curved spaces are independent of it?

I think to answer this question it is necessary to clarify the notion of irrationality.

Mathematical irrationality or incommensurability appears when tracing a diagonal inside of the square existent on the XY coordinates, or when rotating a segment existent on the X or Y coordinates to create a circle and its circumference.

I think we cannot consider the Z diagonal traced on a flat space as an independent axis on the flat quadratic plane we are working on. The Z diagonal is the same Y or X coordinate displaced toward the right or left sides. We have no metric problem by displacing the XY coordinates if we displace the whole flat plane at the same time, but if we want to preserve our original XY referential coordinates and introducing a new Z coordinate on that flat space, we need to be aware that will be creating a new plane – the one represented by Z – on our working space, which can not be measured by using the referential metrics that we use on our XY plane. The Z plane is a new referential plane, displaced with respect to the initial position of our original referential plane XY. If we want to work on our space with these two planes as if they were a unique plane, the actual effect will be the same as if we had modified our physical space, expanding it when displacing Y toward Z, and contracting it when displacing Z towards Y.

Think for a moment about how human beings could have started to measure linear lengths and areas. I guess to measure a linear length for the first time it was necessary to make an abstraction. We needed to start from a referential segment measured from one point to another point, based on a primary referential number of quantity; In that sense, we can suppose we created our referential segment of length 1 based on our primary number 1, one step, one thumb, one elbow, or whatever other one that represented a linear limited distance. With that referential segment 1 we already can measure any linear distance on our flat space.

Our primary segment 1 is of course an abstraction, we did not measure how many points or lines are inside of that segment, we simply accepted it was right because it was very practical to consider that distance as our referential unity to measure linear distances, our initial referential metric; But that abstraction we performed is not a total abstraction because of inside of that referential segment there is a central point which divides the segment in to two equal segments; that central point determines the inner and perfectly proportionated symmetry of our referential segment 1.

Later, we also could create a referential segment of length 2 based on our referential segment of length 1, simply by repeating (or extending) the segment 1 two consecutive times. We now have a larger referential segment which also is a very practical thing to measure large linear distances. The original inner symmetry of 1 is perfectly respected because we can divide the segment 2 into to two equal segments of length 1. Here we place the intermediate point – which is a zero point – between the two segments 1 instead of being placed in the middle of a segment 1. Until now the symmetry is still perfect.

The first problem arises when we want to combine the referential segments 1 and 2 to measure another distance because then the original symmetry is lost: the segment 2 is larger that the segment 1, and we cannot set our central point of symmetry in the middle of those segments to save the symmetry. But that disproportion can be saved by creating a new referential segment based again on our originary referential segment 1. So we can create the referential segment 3: The symmetry is saved again, now by setting the originary segment 1 (with its central point) in the middle of a right segment 1 and a left segment 1, naming the whole distance our new referential segment 3. That is the way I think we can imagine how prime numbers appeared in geometry. Prime numbers appear each time that our originary referential symmetry is lost and only can be fixed by creating a new reference based on our originary referential unity.

Continuing our invented but plausible story about the prehistory of maths, we also can imagine how we were able to measure areas for the first time. For that we necessary needed to create a referential area based on our referential segment 1. So we took our referential segment 1 and built a square. And we agreed that the space inside of that square has the value 1. One square area. It is also an abstraction because we didn’t measure that inner space in any way but the central point of that square of area 1 and its inner symmetry is not any abstract convention, it is a concrete reality.

The next problem arouse when we traced the diagonal inside of our referential square of area 1. And that problem could not be saved in the same way that we solved the disproportion that appeared when it came to measuring linear distances, because now, with the diagonal, the disproportion appears inside of our referential square area 1 built on our originary referential segment 1. The disproportion appears inside of the originary unity itself. So, we cannot use the segment 1 to fix that disproportion created with the diagonal. We cannot create a new referential segment with the distance of the diagonal based on our referential segment 1. And so we see horrified – as it is said ancient Greek people saw – that the hypotenuse of the equilateral triangle is irredeemably disproportionate with respect to the side 1 (the square root of 1) of our primary square area 1. The infinite decimals of incommensurable quantities appear, I think, because we are trying to compare two quantities and one of them is not related to (has not been derived from) our originary reference of metric.

It occurs the same when we try to measure the area of the circle of radius 1 by using our referential square of side 1. We get the unsolvable disproportion between the perimeter and the diameter because they respond to different and disproportionate referential originary (or primary) metrics, referential originary (or primary) linear segments, referential originary (or primary) square areas: the rational and the “irrational” ones.

In this sense I think the circumference of radius 1 is a complex area formed by two kind of different squares that carry and are derived from two different kind of disproportionate originary references of metric that have two kind of different symmetries: the square of area 4, referenced to our “rational” (or irrational) referential square 1, and the square of area 2, related to an “irrational” (or rational) square of area 1″i”. (I said “i” by expressing that it is incommensurable with respect to 1). The square 2 is inside of the perimeter of the circumference touching with its four corners the circle of radius 1, and the square 4 is outside of that perimeter touching with its four sides the circle of radius 1.

It is clearer when we divide the exterior square of area 4 in to four squares of area 1, and we divide the interior square of area 2 in to four squares of area 0,50, and then we set the central point of each of those different squares. Measuring through any diagonal the distances from the central point of the circumference – the zero point to us – until the central point of those different squares we see that we need necessarily two different kind of referential intervals or referential segments of measuring, two different kind of referential metrics based on different originary referential segments, to reach the central points of symmetry of the squares of 0.25, 1 and 4, and the central points of symmetry of the squares 0,50 and 2. 0,25, 1 and 4 are ruled by the same referential segment, while 0,50 and 2 are ruled by another kind of referential segment.

I think what I’m calling referential segments or referential metric intervals, are the same thing that already was considered by Bernard Riemann when he spoke about “quanta” of varieties (or manifolds) and “multiply extended magnitudes”, in an abstract way; Also i think those ideas are similar to the “gauges” considered by Hermann Weyl as different distance scales.

If we repeat those two kind of referential intervals or segments, extending (or “projecting”) them in a consecutive way through any the diagonal from the center of the circumference, we see that the two kind of disproportionate intervals concur or converge at a specific point of the diagonal: at the 7th and 5th intervals, the points which represent the two kind of referential centers of the two kind of disproportionate symmetries, converge.

In current mathematics geometric diagrams are not considered a valid mathematical probe of anything, but let’s continuing analizing them:

I draw them using two different colours, red and blue. You can see there are 7 blue intervals with 8 blue points and 5 red intervals with 6 points. I’m considering the first and last points – the centers of the circumferences – as complex zeros formed at the same time by blue and red points.

Following the diagonal we can see:

Zero blue an red points

1 blue point (blue interval 1)

1 red point (red interval 1)

2 blue point (blue interval 2)

2 red point (red interval 2)

3 blue point (blue interval 3)

XXXXXX (no red point)

4 blue point (blue interval 4)

3 red point (red interval 3)

5 blue point (blue interval 5)

4 red point ( red interval 4)

6 blue point (blue interval 6)

XXXXXX (no red point)

Zero blue and red points where the blue 7th interval and the red 6th interval converge at their respective end.

I think the intervals comprehended inside of the two rows marked with XXXX are what is known in musical terms as the “tritone”. There, the periodical alternation between the two kind of intervals is altered and it is perceived by our senses as something unexpectedly inharmonic.

I think the periodical points of convergence of the rational and irrational intervals on the center of the drawn circumferences, the zero points, are the the “non-trivial” or “relevant” zeros considered by Riemann when it comes to determining the periodicity of the appearance of prime numbers.

When it comes to measuring the area of the circle, if we project three times the square area 1 through the diagonal following one of the mentioned intervals, from the zero point until the point placed where the perimeter is touched by one of the corners of the square area 2, we get a resultant area of value 1 + 1 + 1 = 3. We know the area of the circle of radius 1 is not 3, it is an irrational quantity, 3,14… with infinite decimals. But I think we should be able to measure the area of the circle without considering irrational magnitudes by taking account of the two kind of mentioned symmetries, considering the circumference as a complex area.

So, to measure the area of the circle by projecting our referential square 1 N consecutive times through the diagonal, we should project (displace) that square not only 3 times until the place where the perimeter is touched by the corner of the square 2 but also displacing the referential square – built on the radius of circle – until the next point which is related to the square 4 that touches with its sides the perimeter of the circumference. On the above picture, the 0,14 (without infinite decimals) would be represented as the last blue stripe placed after the three consecutive red colored squares.

I think at that extended point is not possible to accept the linear infinite decimals because there is where the irrational and rational referential intervals converge.

On this view, the circumference is a complex area that cannot be measured by using a unique and static referential square (attempts that historically were known as the “quadrature of the circle”, which impossibility has been already demonstrated but not by using different referential square metrics) but considering the two kind of referential originary squares built on the two kind of originary referential segments that participate on its area.

Incommensurability also appears in the Pythagorean theorem. When we trace the diagonal inside of a square built on a segment of length 1, our originary referential segment for measuring linear distances, we transform that rational referential metric creating a new one, the irrational metric. That occurs because we displace Y to create the diagonal.

If a^2+b^2 is equal to c^2, why a^n+b^n is Not = c^n when n is higher than 2?. And why a^2 + b^2 is equal c^2, if c^2 is built on a segment that is not commensurable with respect to the roots of the squares a^and b^2? Are the square areas following a same referential metric when linear segments are ruled by different referential magnitudes?

I think the area c^2 is coincident with the areas a^2+ b^2 because the external elements of the squares a and b (the sides of those squares) are equivalent to the length of the internal elements of the square c^2 (the two hypotenuses placed inside of the square c^2); and the internal elements of the squares a^2 and b^2 (the two hypotenuses – four in total – of the squares a^2 and b^2) are equivalent to the external elements of the square c^2 (the sides of the square c^2). The sides of the squares and the hypotenuses of those squares carry the two kind of different originary symmetries, the rational and the irrational ones. (I say “symmetries” but it could be said the two kind of referential and originary metric magnitudes).

In this sense, the squares a^2, b^2 or c^2 carry internally and externally the two kind of disproportionate referential metric symmetries. a^2+b^2 is Not equal to c^2, those square areas are coincident or equivalent but their inner and outer symmetries are different. Square areas are not abstract entities that can be managed operatively in an abstract way by using arithmetically purely abstract symbols. Their symmetries are real and concrete and can be distributed in a different although equivalent way.

I think that Euclidean geometry works on a static and quadratic space ruled only by the rational coordinates XY existent on a same plane and its unique referential rational metric, while non-Euclidean geometries work, being or without being aware of it, with variably spaces that expand and contract, or with curved static spaces that are complex because of the concurrence of the rational and irrational planes on them.

Non-Euclidean geometries in this sense, are not Euclidean because their parallels are not on a same plane, and so they do not follow the Euclide’s definition of parallels as *“straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction”*.

That definition is respected by the Euclidean parallels and non-parallels. If the irrational diagonal appears when the two internal angles of a same side are smaller or higher than 180 degrees, the lines are not parallel.

Note: For the Euclid’s texts I’ve used the “The Thirteen Books of The Elements” by Sir Thomas L. Heath. Second Edition.

Pdf version of the post: 1virtual-and-mirror-convergences-on-the-demonstration-of-the-euclids-fifth-postulate