Hyperbolic Geometry and Daily Life

This story begins with an almost unbearable 10th grade Geometry class — unbearable because of the attitude of the teacher. He shall remain nameless, but if he reads this (unlikely to say the least), I’m hoping he will recognize his part. It was 1963 and I was a good and reasonably devoted student. Our teacher, however, stated often that girls just couldn’t learn math very well, so he walked around the class during exams whispering hints and answers to them. However, the powers of the universe (whatever you believe them to be) provided, as they often do, recompense for this disaster of a learning experience in the form of a Mr. Peter Drees (whose name I shall happily mention). Mr. Drees was, I believe, the math coordinator for the school district, and he got it into his head to discover whether he could teach a group of undistinguished 15-year-olds Hyperbolic Geometry. Mr. Drees was successful at getting a few of us to entertain the concept of a non-Euclidean universe with curved space. He didn’t get many of the details across, but, to my mind he taught something even more important.

The geometric version goes like this (with apologies to any real mathematicians out there): In Euclidean geometry all of the proofs are based on assumptions. There is one major assumption which underlies all proofs and which cannot itself be proven. That is the assumption that states that through a given point there is one and only one parallel to a given line. Or, in more geometry-bookish terms: For each point A and line r, there is just one line through A that is parallel to r (see below).

The details arena so important to this discussion. What is, is that (1) this assumption cannot be proven and (2) it is the foundation for the proofs that follow which tell us that a rectangle, for example, has four right (90-degree) angles. The people who devised non-Euclidean geometries (with names like Einstein and Lobachevesky and Gauss) asked what would happen if one changed this assumption and postulated that through our point there were either no lines parallel to r or an infinite number of lines parallel to r.  The short version is: what you get are rectangles with two right angles and two acute angles or rectangles with two right angles and two obtuse angles or other mind-stretching forms.

Again, the details are not what impressed me. The important point seemed to be that everything was contingent on one underlying assumptions. These are the underlying rules that become so ingrained that one becomes unaware that they exist. They lurk in the foundations of our thinking and form the conclusions that follow our logic. These assumptions lie deep within us and are formed from the day we are born.  Finding them is not always easy — and they often seem quite simplistic.  As simplistic as the thinking of an infant.  Some are built more of emotion than of intellect.  They are things like “the world is safe” / “the world is dangerous.”  Or “my needs will be met” / “my needs will never be met.”  And they drive much of the thinking and reacting that we do.

These kinds of assumptions go a long way to explaining why logical arguments between pacifists and warriors, religious zealots and secular humanists, capitalists and communists, never seem to get anywhere.  They may also explain why, for example, some people find the world to be frightening while others seem to be more comfortable and tend to feel safe.  Perspective is crucial and it is imperative to examine one’s own underlying assumptions when trying to understand ourselves and others.

© 2003 – Chris D. Cooper, PhD

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