After the customary comments on "This Day in Math History," Dr. Barsky started off by noting that "golden ratios" were considered a thing of beauty by a lot of people throughout history, notably by the Pythagoreans. A "golden ratio" occurs when a line segment AB is cut at a point H in such a way that the ratio of the shorter part to the longer part is equal to that of the longer part to the whole.

The Pythagoreans also had a great interest in pentagons, so much so that a pentagram drawn inside a regular pentagon was their symbol. When drawing diagonals connecting every possible pair of vertices in a regular pentagon, two noteworthy things happen: one is that the center forms another, smaller, regular pentagon, and the other is that golden ratios pop up everywhere.

Dr. Barsky demonstrated that all triangles resulting from the drawing of all the diagonals in a regular pentagon are isosceles triangles with base angles measuring either 72 degrees or 36 degrees. He then showed, by both geometric and algebraic methods, that a side and a diagonal of a regular pentagon are incommensurable. Dr. Barsky wrapped up the lecture by showing how the Babylonians literally completed the square geometrically.

There were two things that sparked my curiosity here. First, why isn't the Babylonian method of completing the square shown in Intermediate Algebra or College Algebra classes? Students tend to grasp concepts much better when they can actually see them, and see how they work. I've seen students struggle mightily with completing the square, and most times when they do grasp it, they do so by memorizing a formula. I think that introducing the dramatization could be helpful, and it certainly couldn't hurt.

The other question I had was why did these people go to such lengths to find facts such as whether or not the side and diagonal of a pentagon are commensurable? A lot of these things don't appear to have been of any use for real world applications. So what purpose did they have for working with them?

Something Katz discussed that I found amusing but haven't yet heard about in class is Zeno's paradoxes. Katz describes one of them as saying that a faster runner can never catch a slower runner if the faster runner starts out behind because first he would have to narrow the gap by half, then do so again, and again infinitely. In theory, this could be applied to any accomplishment. For example, I can never finish the work in my math history class, because in order to do so, I must first finish half of it, then half of what remains, then half again....

This lecture was part three of the Beginnings of Mathematics in Greece. Dr. Barsky picked up where we left off on Tuesday by reminding us a little of what we covered towards the end of class that day. Today's lecture was more about Pythagoras, but this time he presented to us the "mean and extreme ratio". Dr. Barsky did this in a geometric fashion, concluding that the side and diagonal of a regular polygon are incommensurable.

I liked this lecture because, as I have stated before, I enjoy doing geometry. I got a little lost, however, at the first step of substitution when we substituted DE for BC. As compared to the Algebraic version, I liked the Geometric version more because I feel it required a bit more thinking and I could "see" what we were doing. I think some day, for the fun of it, I may try to find that mean and extreme ratio using that proof.

The presentation given today by Dr. Barsky began with some commentary on Montucla, a French mathematician born on the same day in 1725, after which the presentation seemed to take on an overall theme of logic and cutting a line in mean and extreme ratio. The topics covered consisted of Montucla, Pythagoreans, the Pentagram, and cutting a line in mean and extreme ratio.

Among the topics covered, cutting a line in mean and extreme ratio interested me the most, especially the fact that Kepler considered it to be like a precious jewel. According to the lecture, the Pythagoreans made the pentagram their symbol because of its mathematical qualities including cutting a line in mean and extreme ratio. The way they used it to show incommensurability intrigues me even more. According to the lecture, when we look at a pentagram, we can see another pentagram inside of the smaller pentagon which contains an infinite number of pentagrams. We showed that if the unit length L scales the side and diagonal of the larger pentagon, then it can also do so with the smaller one. Since L never changes and we can clearly see that the pentagon size gets infinitely smaller, we come to the conclusion that the side and the diagonal of a (regular) pentagon are incommensurable. In my interpretation, the use of the line between mean and extreme ratio applies to all the pentagrams in this example and the Pythagoreans have found a mathematical wonder in their symbol. This must have taken a lot of thought and time to figure out. I believe that the Pythagoreans spent the majority of their time on mathematical thought such as this, especially since they believe in numbers so religiously. They probably did this as a ritual and they may have performed it everyday. They certainly must have performed it a lot to come up with these answers.

This lecture continued our discussion of mean and extreme ratio and incommensurability. A geometric demonstration showing that the side of a pentagon and a diagonal connecting two vertices of a pentagon are incommensurable was presented as a possible basis for this problem in early Greek mathematics because of the Pythagorean's relationship with the pentagon.

Dividing the pentagon by drawing lines between opposite vertices and using the concept of similarity between various triangles created in the process, a golden section relationship can be developed with the cutting of the diagonal by another diagonal. A geometric proof was given showing the side and diagonal of a pentagon are incommensurable was given, but I have to admit that I did not fully understand it. It had to do with creating smaller and smaller pentagons by connecting the vertices of each successive pentagon created by drawing the diagonals of the previous pentagon.It was shown that there did not exist a length l that ruled both the sides and the diagonal.

An algebraic discussion showing how the golden section can be found by solving a quadratic equation was also presented. It was also shown how the Babylonians had used a geometric method of completing the square to arrive at the same solution.

Lots of interesting concepts were talked about in this lecture. We discussed the fact the Pythagoreans used a special sign. This sign happened to be a pentagram. They took this pentagram and divided it. Each time a diagonal was cut by another diagonal, it was cut into "extreme and mean ratio."

We saw that if you draw a rectangular pentagon, when the five diagonals are drawn, the diagonals make a smaller pentagon. If you draw them again, you get another pentagon which is even smaller. This will continue until the pentagons get infinitely smaller. It was concluded that the ratio of a diagonal to a side is irrational.

When we looked at the equation a(a-x)=x^2 (algebraically), we realized that the Babylonians already knew how to do this. They basically completed the square when calculating the golden ratios, the end result is (sqrt(5)-1)/2. The same equation was done by geometrically completing the square. The geometric version paints a clear picture. It is difficult to interpret the wording from the book, but when someone is walking me through it, the blurry picture becomes clear...to an extent!

This lecture continued the discussion of the topics of the golden ratio and incommensurability and their it's relation to the pentagon. We used principles of the angular measure of polygons to establish relative lengths of segments of a pentagram to produce the golden ratio. Then we used some of these results to prove that the side and diagonal of a pentagon are incommensurable. Finally, we used an algebraic approach to discover a formula to find the golden section of any length segment.

Although it was hard to discern the point of our earlier machinations with the angles and segments of a pentagram, I did get the general idea of how to prove the incommensurability of the side and diagonals. I don't know if it would be proper to call this a "recursive proof", but that word does come to mind. I know we do not have time in this course to rigorously develop our own epistemology of math as opposed to casually comparing those of the evolving communities we are surveying, but I had to wonder to myself whether I REALLY knew enough to be confident that there are no holes in the logic that no small, basic unit "l" that could measure the side could also measure the diagonal on the basis of the argument that you would then HAVE to use this same "l" in the next recursion. I see the logic intuitively, but I had doubts that I could be certain there is no way to get around it, for instance, by allowing a different "l" at every recursion, or by some other concept in modern geometry of which I am ignorant.

The presentation itself was a bit difficult to follow as I was not sure of motivation for all of our discussion of angles and segment lengths. I really do appreciate the algebraic proof and the mention of how the Babylonians knew how to solve quadratic equations centuries before the Greek Golden Age. I noted, but did not relate well, to this statement in an earlier lecture, but this discussion of completing the square as an alternative to the quadratic formula to find the golden section helped to bring it all together. Personally, I enjoy more the generalized discussions of paradigm shifts and epistemology more than the detailed analysis of certain proofs as I am also a "history buff", but then again, we were warned that this is a "math course more than a history course".