Parallel Lines

I learned the parallel postulate in middle school. The best known equivalent of the postulate is attributed to Scottish mathematician John Playfair, and it says that “in a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.”

The reason that I have a special impression on this postulate may be probably due to a popular metaphor in my middle school period. That metaphor related the parallel lines with the mutual feelings between girls and boys: when a girl and a boy cannot stay together, or they do not develop a mutual affection, we say that they are like two parallel lines. No matter what the two parallel lines “do”, they cannot have an interaction. Similarly, for the two unlucky people, no matter what they do, they can never fall in love with each other. I have to say this metaphor describes a tragic situation and sometimes I do not feel satisfied with the “tragic” destinies of the two parallel lines. Fortunately, as my mathematical knowledge grows, I do find that in some other branches of geometry, the seemingly unbreakable law in Euclidean geometry no longer holds. Among the new branches are hyperbolic geometry and elliptic geometry, which will be the main topic of this blog.

Before we talk about non-Euclidean geometry, let me have a brief introduction to the differences between non-Euclidean geometry and Euclidean geometry. The fundamental difference between them lies in the parallel postulate. We already stated a widely adopted equivalent of parallel postulate in the beginning of this article. For two thousand years after Euclid’s work was published, many mathematicians either tried to prove this “fifth postulate” (in Euclid’s Element) or tried to show that it’s not necessarily true. Actually, even in Euclid’s own book, this parallel postulate was left unproved; Also, unlike the first four postulates, the fifth postulate — the “tragic” parallel postulate, was not being used to prove his following theorems in the book. A breakthrough in this topic came out in the 18th century. A Russian mathematician,  Nikolai Lobachevsky, developed the hyperbolic geometry. His most famous contributions are in two aspects: he convincingly showed that Euclid’s fifth postulate cannot be proved, and he presented hyperbolic geometry to the world.

Multiple parallel lines in hyperbolic geometry. Image: Vladimir0987, via Wikimedia Commons.

In the original parallel postulate, we said for any given line R and point P, there is exactly one line through P that does not intersect R; i.e., parallel to R. In hyperbolic geometry there are at least two distinct lines through P which do not intersect R, rendering the parallel postulate invalid. Hyperbolic geometry may be against common sense at first glance, because usually, our recognition about the shape of a space is limited to Euclidean space. However, hyperbolic geometric space does exist, one example is the saddle space with a constant negative Gaussian Curvature. Hyperbolic space is possible in dimensions that are larger than or equal to two. It is curved — the reason why it differs from Euclidean planes — and is characterized by a constant negative curvature. Euclidean spaces are always with zero curvature. To make it more vivid in my own words (which very likely will not be so rigorous), if we observe a small region in the hyperbolic plane, it looks like just a concave plane. And when you draw a triangle in this concave plane, the sum of its inner angles is always less than 180 degrees. This is also a proved theorem in hyperbolic geometry.

In elliptic geometry, we have the following conclusion: “Given a line L and a point p outside L, there exists no line parallel to L passing through p, and all lines in elliptic geometry intersect.” This means we can never find any parallel lines in elliptic geometry. This kind of geometry together with hyperbolic geometry, perfectly form a counter example of the parallel postulate’s assumption “there is one and only one parallel line…”: in elliptic geometry, there is more than one parallel line, and in hyperbolic geometry, there are none. Examples of elliptic geometry are more common in our real life than hyperbolic geometry. One example is the surface of Earth. A line in such a space becomes a great circle (a circle centered at earth’s core). When you draw a line through point P and if P is away from line (great circle) L, the new line you get will be a new great circle, and it will always have two intersections with great circle L, because any two great circles on the surface of sphere will have two intersections.

Here we have three pictures visualizing the relationship between Euclid’s geometry, hyperbolic geometry and elliptic geometry.

Image: Joshuabowman and Pbroks, via Wikimedia Commons.

The establishment of non-Euclidean geometry is the outcome of many generations’ collective endeavors. For example, classical era’s scholar Proclus commented some attempts to prove the postulate, esp. Those attempts tried to deduce it from the previous four postulates; Arab mathematician Ibn al-Haytham in the 10th century, tried to prove the theorem by contradiction; in the Age of Enlightenment Italian mathematician Giordano Vitale and Girolamo Saccheri both contributed new approaches to this problem although they finally failed; Gauss and Nikolai Lobachevsky (we already mentioned him above) also joined the sequence — the latter finally finished this task by establishing a new geometric branch. This mansion was built over such a long time and I am fortunate to feel part of its grandeur and beauty.

So for those suitors who complain their misfortune that their dream lovers and they are like two parallel lines, I think you are too pessimistic. You can imagine yourself being in a elliptic geometric space. Then as long as you try your best, you will always have an intersection with the other line. I am not sure whether this will convince those guys and give them confidence. For me, I am now feeling happy and believe that everything is possible in our real world, just like that everything is possible in mathematics. The story about seemingly very simple parallel lines do make me feel the power and beauty of mathematics.

References:

  1. http://en.wikipedia.org/wiki/Non-Euclidean_geometry
  2. http://en.wikipedia.org/wiki/Elliptic_geometry
  3. http://en.wikipedia.org/wiki/Parallel_postulate
  4. http://en.wikipedia.org/wiki/Hyperbolic_geometry
  5. H. S. M. Coxeter(1942) Non-Euclidean Geometry, University of Toronto Press, reissued 1998 by Mathematical Association of AmericaISBN 0-88385-522-4.
  6. Hazewinkel, Michiel, ed. (2001), “Elliptic geometry”Encyclopedia of MathematicsSpringerISBN978-1-55608-010-4
  7. Weisstein, Eric W.“Hyperbolic Geometry”MathWorld.
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