Euclidean and non-Euclidean Parallel lines on Lobachevsky’s Imaginary Geometry.   Leave a comment

Non-Euclidean or hyperbolic geometry started at the beginning of the XIX century when Russian mathematician Nicolai Lobachevsky demonstrated that the fifth Euclid’s postulate – the parallel postulate – was not applicable when it comes to curved lines and so that more than one parallel can be traced through a point external to another line.

As you know, 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”.

More than a simple axiom that can be accepted without being demonstrated, the fight postulate seems to be a theorem and so many mathematicians, from the beginning, tried to demonstrate its validity by deducing it from other axioms or by mean of the propositions of the Elements through the different centuries, without finding out any successful solution. Other mathematicians tried to formulate that postulate in a simpler and self evident way, such is the case of 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”.

I tried a demonstration of the fifth postulate on previous posts. The Euclidean parallels and nonparallels are expressed in this diagram:


But how parallels and non parallels can be represented when it comes to Lobachevsky’s imaginary geometry?

Lobachevsky developed his geometry in an axiomatic way without having any visual representation of it; that is because he named the new geometry “imaginary” in the sense that it seemed it did not represent reality in a descriptive way, he did not find a model to represent the geometry that he demonstrated was consistent on itself and with the Euclidean geometry except the fifth Euclid’s postulate.

In one of his last books, “Geometrical researches on the theory of parallels” (1840), translated to English by George Halsted in 1914 – so you can get this book free of charge on the internet, it’s a short book of 45 pages – Lobachevsky presented some descriptive representations of his imaginary geometry. But the Lobachevsky’s figures are symbolic representations of a non-Euclidean geometry by using the Euclidean geometrical elements. His aim was to make more intuitively understandable the non-Euclidean geometry; actually it was after this book that the non-Euclidean geometry started to be accepted and understood in Europe; before it had been rejected in Russia because of its high abstract and counterintuitive nature.


This is a diagram of Lobachevsky’s parallels. To be able to represent more than one parallel passing through the same point external to another line he needed to represent this another line as curved line. Euclid’s Elements did not specified thoroughly the definition of straight line and it seems a curved line could be accepted to be straight line.

Actually all non-Euclidean geometry is limited to the realm of curved geometries.

Later, some other mathematicians developed different models based on the Lobachevsky’s non-Euclidean geometry, like the Riemann’s Sphere or other well known representations like the pseudo sphere. Today there are many non-Euclidean representations, but are they the faithful representation of the original imaginary geometry of Lobachevsky, or are they representations of the non-Euclidean geometry passed through the filter of other ulterior models?

I think a natural way of representing the Lobachevsky’s Imaginary geometry is by considering two intersected fields – two longitudinal waves – that periodically, expanding and contracting.

Lobachevsky already spoke about the intersection of two spheres; he also considered his “imaginary” geometry as a “theory of spatial relations”.

In this way, the Lobachevsky parallels are derived from curved spaces but the parallel lines are not curved, they are straight lines in the Euclidean sense, they are the on the flat plane of the different fields created by the first intersection and they are actually straight, not curved.

I think to consider straight lines as curved is away to evidently force concepts to fit them in an instrumental way in our models, to force them to make they serve to what we feel and know are a demonstrated – axiomatically and algebraically and arithmetically demonstrated – true.


On these first pictures, the intersected fields vary with the same phase, they expand and contract at the same time. The three different figures represent three different moments of the periodical variation of the intersected fields. When both fields contract, there is an upward displacement, and when they both expand there is a downward displacement. At the time of maximum contraction, the top figure, the vertical lines a and b (in red colour) are not parallel; at this moment the intersected fields start to expanding and there is point at a moment where the two vertical lines are parallel (in blue color on the middle figure); Later the expansion continues and the two vertical lines become nonparallel again (in red colour on the last figure) until the maximum expansion is reached and the contraction starts again.

The process is repeated periodically but the phases of variation synchronize and desynchronize also periodically. On the picture below they have become desynchronised and the fields vary with opposite phases, when one of the expands the another one contracts and vice versa.


In this case the displacement takes place toward left and right instead of upwards and downwards. When the displacement is toward the left side (when the left intersected field contracts and the right one expands) the vertical lines are parallel. and they remain parallel when moving toward the right side but there is a moment or a strip of moments when the vertical lines become nonparallel (in red colour on the middle figure). That time of non parallelism divide the realm of the negative parallel lines (at the left side) and the positive parallel lines ( tat the right side).

But negative and positive parallel lines are the same lines in different times, when moving left and right in a pendular way. When the symmetry is displaced toward the left the right positive lines exist only virtually, potentially in a next time, and vice versa.

So when the phases of variation are equal, the division between the up positive realm of non parallelism and the negative down realm of non parallelism is given by a moment of parallelism. Between the positive nonparallel lines and the negative nonparallel lines there is a mirror inverted symmetry.

And when the phases of variation are opposite, the division between the left negative realm of parallelism and the positive right realm of parallelism is given by a moment or a strip of moments of non parallelism. Between the negative parallel lines and the positive parallel lines there is a mirror symmetry.

You can also see when the other lines existent on the Euclidean plane are parallel and nonparallel. Euclidean and non-Eculidean parallels or non parallels coexist in this model of spatial interactions.

The multiple Lobachevsky’s non-Eculidean parallels passing through a same point should be looked for then on different times and that same point would be a point with a pendular motion. It also possible to consider the intersection of more than two fields. The pictures below would be a representation looking at the interescted fields form above, when they change with the same phase of variation:



Although Lobachevsky did not find a model of representation of his geometry, only isolated elements, he was convinced that the new non-Euclidean geometry was the true way on which Nature was organized.

I did not elaborate this model starting from the Lobachevsky’s ideas, I did not have any idea about mathematics or geometry when I started to develop it now 8 years ago; I made the model of two intersected fields looking for a mechanical explanation of anomalous cell divisions in a non conventional way, thinking about gravity like a force of pressure and gravitational fields as intersected fields which curvature vary periodically, expanding and contracting. At the beginning it was a representation of a non conventional solar system model.

Later I used the same model as the representation of an atomic nucleus; the fields created in and by the intersection of the two fields are the subatomic particles of the shared atomic central nucleus:




When I showed this images to physicists I saw that they did not understand them. They always said that current physics is only understandable through mathematics and that those pictures could be artistic but never mathematical representations; I understood they did not have any visual representation of the currently atomic model and so to them it is impossible understand or discuss anything conceptually in a consistent way outside of their mathematical formulas; the atomic model, quantum mechanics have been developed in highly abstract – and also instrumental way – because the mathematics that is based on is highly abstract and do not have clearly visual references outside of their algebra and arithmetic.

Since the extreme formalization of arithmetic and geometry that started at the XIX century, descriptive geometry – the geometry of the ancient Greece based on the visual representation of geometric figures – has been progressively discredited and today it is consider as an heuristic help which does not have a true mathematical value.

I think the extreme formalization and the contempt of the value of descriptive elemental geometry, as if the only true mathematics were arithmetic and algebra, is only a passing fad whose excess and their consequences on the progressive problematization of the development of the today very stuck physics will be corrected in the future, maybe when the always recurrent question is pronounced one more time, again:

How did we not realize something so simple before?


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