Sender: Robert
Subject: Non-Euclidean Geometry
Date: Mon, 8 Feb 1999 19:03:15 -0500

Is non-euclidean geometry necessary for the college bound student? I have students that are inerested in teaching math one day. My school is restricted to Euclidean Geometry.

Hi Robert
Here are two replies to your question

It is GOOD to be familiar with non-euclidean geometry, but not necessary. If, in your area, non-euclidean geometry is required, it would be covered at college.
   I teach geometry at university, to a class which is primarily people who wish to become teachers. The primary non-euclidean geometry I cover is SPHERICAL geometry. This is easy to illustrate, has lots of applications, and help give perspective on plane geometry. As a formal 'non-euclidean geometry' this is often done with opposite points identified, so any two 'lines' (great circles) meet exactly once.
   Of course, there is also affine and projective geometry, which MIGHT be viewed as 'non-euclidean' or simply as aspects of euclidean geometry. These are important in the learning of geometry, as they highlight the role of transformations in geometry (the 'modern approach' as of 1870's). To learn this approach seems to require some sense of 'choices' and 'alternatives'. In that sense, it is valuable for students to realize that there are choices of geometry. This is pretty important in lots of applications (computer graphics, design, physics, ... ). For example, if we do STATICS correctly, we move to affine and projective geometry. The fact that we do not teach much of this these days, means that even leading engineering texts have to misrepresent what is atually going on in some basic structures in buildings (and robotic manipulators). The top experts certainly know, but many students do not, because they do not have the vocabularly or the background.

Walter Whiteley
York University

I will defer to the opinions of the University people but I have the following feeling about your question. I don't feel that it is necessary to study non-Euclidean geometry in high school. It is my view that it should be introduced at university as new material. I suspect that, even at university, it is possible to have some pretty solid courses in mathematics (Calculus, Numerical Analysis, etc) without any reference to non-Euclid geom. The only way that I would introduce it in high school would be if I were teaching a unit on the nature of reasoning. One could then explore what happens if you choose a different postulate than Euclid did. Essentially Euclid postulated, but could not prove, that 2 intersecting lines cannot both be parallel to a 3rd line. If you postulate that they can be then you develop a different, logical system of geometry. However, it is my opinion that one can explore the role of If, Then by other means. It is of more than passing interest that the Mad Hatter in Alice in Wonderland makes perfect sense if you accept his postulates. This is not so surprising since Lewis Carroll (really --- Dodson) was a math prof. Someone, I think Martin Gardner, produced a (mathematicaly) annotated text of Alice to support this. I personally think that the future math teachers should receive a love of solving new and different problems so that they can pass this on to their students. One obvious source is the various mathematics contests and journals. (AHSME, Math League, etc in US; Can Math Competitions, Crux Mathematicorum, etc in Can).

Jack LeSage

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