By Gabriel Xu, VI Form
What if math students no longer had to study similar triangles because they simply don’t exist?
What if you could draw as many lines parallel as you want to a given line from only one point?
What if angle-angle-angle was enough to prove congruence of triangles?
Would these changes to our known geometric system finally make it easier, or would they further contribute to its fascinatingly intricate nature?
Studying Non-Euclidean Geometry aims to answer these questions.
The class began by learning the first two books of Euclid’s Elements, where the Greek mathematician lays the foundation of (Euclidean) geometry from five postulates. With a straight-edge (a ruler without a scale) and a compass, Euclid managed to develop a world of geometry from two simple shapes — lines and circles. It has become clear to us that Euclid must have paid particular attention to the order of his constructions and proofs as most of which were built upon previously constructed shapes and proven theorems. From the congruence of triangles to the Pythagorean Theorem, the system of geometry we were ever so familiar with expanded in front of us in its purest and most fundamental form.
And then, things got interesting.
After we grew accustomed to Euclid’s way of constructions and proofs, we were introduced to a much stranger world of Non-Euclidean geometry. The fifth and last postulate in the Elements is equivalent to the notion that, from a given point, there is one and only one line parallel to any given line. Generations of mathematicians have devoted themselves to prove this seemingly obvious postulate, but all have failed. Coincidentally or not, Euclid did not make use of the last postulate in his work until the later part of Book I in Elements. His reluctance to take advantage of the parallel postulate when it could have simplified his proofs has sparked questions to his own certainty of said postulate as well.
Non-Euclidean geometry has been developed assuming that the fifth postulate is false. Rather ironically, this system was first built in order to indirectly prove the fifth postulate — by trying to find an error or contradiction that would result from such an assumption. However, as the system evolves, the much anticipated contradiction never appears, and all the work done under the indirect proof can therefore be seen as the cornerstone of the newly-developed structure. The class was surprised at how much we took for granted collapsed once one postulate was taken out. For example, the constant angle sum of 180 degrees (or two-right-angles, as there is no concept of “degree” in Non-Euclidean geometry) for any triangle, which seems perfectly irrelevant to the parallel postulate, will no longer hold true outside the Euclidean plane. In the hyperbolic plane (where there can be more than one parallel line drawn from any given point to a given line), the angle sums of triangles are distinctive and all under 180 degrees. The domino effect continues, with the external angle of a triangle no longer equal to the sum of the two alternative angles; the concept of similar triangles collapses completely; and suddenly, angle-angle-angle is enough to prove congruence in triangles.
In addition to losing most of the established theorems, we often had to operate against our intuitions as well. We will not be far from the truth if we are to say that we live in a Euclidean world — all of our mathematical tools are Euclidean (most of which developed from straight edges and compasses), and consequently the constructions we make with these tools are Euclidean as well. We cannot draw two intersecting lines both parallel to a third on paper—we probably cannot even imagine them properly. We cannot locate the “ideal point” where all the relatively parallel lines approach but never meet. We cannot even draw a triangle with angle sum less than 180 degrees.
Muddling through these strange, even absurd, theorems and constructions, I have found myself on many occasions pondering a rather philosophical question — what is the definition of “wrong”? In primary school, I would be “wrong” for saying the angle sum of a triangle is less than 180 degrees; in middle school, I would be “wrong” for trying to prove triangles congruent using angle-angle-angle. But now, these seemingly ridiculous propositions are merely components of an alternative possibility that we cannot eliminate, and in that sense, a possibility as legitimate as the one we used to assume. We cannot prove a construction to be false simply because we cannot construct it with the tools we have. Similarly, we cannot prove a theorem wrong simply because we fail to prove it right. With almost all known propositions and techniques stripped away, we are left with the most primitive, yet most powerful skill in mathematics — logic. Instead of constantly going back to my old knowledge which I was so dependent upon and finding conclusions no longer legitimate, I have begun to adopt a much clearer look at the problems at hand — starting with what is given and what needs to be found, and coming up with the logical steps in between to bridge the two ends. One might ask what is the point of learning about Non-Euclidean Geometry if we live in a “Euclidean world.” A tempting answer would be to develop one’s logical ability and to cultivate the mindset that everything is possible until proven illogical.
I sincerely hope that this essay could trigger interest for the readers to further investigate this fascinating alternative geometric system. However, if everything above doesn’t make any sense to you (or if you can’t even bear to read it and only scroll down to see how long this thing drags out), I wish you can still take away from this article an appreciation for Euclid — the Greek mathematician in 300 BCE who made geometry much more manageable for students in the thousands of years ever since.
Gabriel Xu is a VI Former from Beijing, China. He enjoys playing tennis and is an avid fan of his mother’s cooking.