The Geometry of Hyperparallelism

One of the most common families of non-Euclidean geometry is hyperbolic geometry – a self-consistent geometry of “obtuse” curvature.

While “normal,” foundational Euclidean geometry was founded upon five fundamental principles (Euclid’s five axioms), it is the fifth and final principle – the so-called parallel postulate – which is used to differentiate between this and any non-Euclidean space.

This principle defines, simply enough, that two lines are parallel to each other when a third line drawn between them will create angles which add up to exactly 180 degrees on each side. When any sort of space with curvature is considered, however, this postulate no longer holds true (necessarily), and therefore, an entirely new form of geometry must be instated.

A Saddle-Shaped Space

Hyperbolic Geometry (which is really one of just two “main” forms of non-Euclidean geometry, the other being spherical geometry) is one which is more often than not described as being “saddle” shaped. That is, it curves outward in all directions, which is simple enough to describe visually in these two or three dimensional terms, but becomes quite nearly impossible to describe in four or more dimensions without resorting to pure mathematics.

One of the chief results of this hyperbolic curvature – and one of the defining features of the resulting geometry – is that triangles no longer have angles which add up to 180 degrees. In fact, any triangle drawn on a hyperbolic surface will, by definition, have an area of less than 180 degrees. Furthermore, any two triangles with the same area have the same sum total of angles, and vice versa. So any triangle whose angles add up to 150 degrees will have exactly the same area as every other triangle with a similar sum of angles (this amount, however, is dependent on the degree to which the space is curved).

Hyperparallelism

Another feature of hyperbolic geometry, and one which relates directly to Euclid’s fifth axiom, is the nature of parallel lines in such a space.

Consider a single straight line, marked X, in a flat, Euclidean space. Just beside line X is a point, Y. In Euclidean geometry, for every such line, there is one and only one parallel line which runs through point Y. No more, no less.

In hyperbolic geometry, on the other hand, that is not necessarily the case, although one is required to rethink their entire definition of “parallel” in such a world. Here, for any line, X, there are an infinite number of parallel lines running through Y. The reason for this is that lines will only run parallel in such a space at a single point, and then curve away from each other. So while they are parallel in that the two lines never touch, they are not exactly parallel in a normal Euclidean sense.

While it is difficult for most people to fully wrap their brains around non-Euclidean geometry, the fact is that it is often times this sort of mathematics which best describes the universe around us. Non-Euclidean geometry didn’t come about just to make things difficult – it arose because the universe itself, according to Einstein, is essentially non-Euclidean. So while all of this may seem a bit superfluous, it is certainly not. This is important, and it is really rather interesting.

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