Revisiting Euclidean Geometry

Geometry. When I go back and think about the subject in high school that was the most tedious, especially in math, geometric proofs are one of the first things that come to mind. Even though I have always enjoyed studying math, Euclidean geometry is very different from the rest of the math that is studied in high school, which is more computational in nature. It doesn’t deal with common notions like numbers and algebra, but relies mainly upon an arcane system of postulates, definitions, and theorems. The only way knowledge is derived is via proof, which does not seem mathematical to most 14 or 15 year-olds. Even after coming to the realization that the theorems studied in geometry provide a framework which allows us to study coordinate geometry and trigonometry, I still did not have a high regard for this branch of mathematics.

Why has this forgettable topic suddenly come back? This semester, I am taking Math 163, which is the seminar course that all math majors are required to take. This class is meant for underclassmen who are interested in pursuing mathematics, and its purpose is to introduce us to mathematical rigor and reasoning, as well as to show us how to construct solid mathematical proof.The theme of the class is geometry, and we are currently working with the first book of¬†Euclid’s Elements, which includes all the familiar congruence theorems, characterizes parallel lines, and ends with a proof of the Pythagorean Theorem. A recurring theme that our professor reminds us about constantly is the all-important Fifth Postulate, which deals with the angles formed by two lines and a line that falls across them. The axiom is used mainly in dealing with parallel lines. It is also interesting to view the Pythagorean Theorem in a purely geometric context, rather than the common algebraic equation, a^2+b^2=c^2.

Although I was originally complaining about having to look at Euclid again, the methods of proof learned will come in handy in higher-level mathematics courses, where I am already having to construct proofs for homework on a weekly basis. Maybe it is because I now see a purpose for it, but I am glad I was forced to revisit geometry, and I’m looking forward to moving on to more modern geometry and applying my knowledge acquired from studying Euclid.

Professor John Conway Visits Lehigh

Last week John Conway, Professor of Mathematics at Princeton University came to Lehigh for a discussion on his many accomplishments. The talk was arranged by Lehigh’s Professor Donald Davis, who teaches a course based off of Conway’s work called “Popular Mathematics.” The beginning of the discussion was about the Game of Life, arguably Conway’s most famous accomplishment. The Game of Life is one of the first examples of cellular automata, and is based on a grid of uniform square cells that are either alive or dead. There is also a set of rules that defines whether the cells will die, be reborn, or stay the same. After going into a lengthy ramble about this, Conway went on to describe one of his most recent and favorite accomplishments, the free will theorem. This theorem in the field of quantum mechanics is based on the premise that if humans have free will, so should elementary particles. Although Conway spoke very softly and was hard to hear for most of the talk, the event was interesting enough, and I will surely look into both the Game of Life and the free will theorem during my free time.