Aesthetic Curiosity – The Root of Invention

This 1975 New York Times article by Cyril Stanley Smith left an indelible impression over the years to the point that I wanted to capture it in digital format. I found a 1996 copy online, formatted it more closely to the original article, and found more recent images of the illustrations used in the original article. It is a powerful argument for the benefits of basic research vs. directed research. The pursuit of pure mathematics often is accused of being a product of imagination run rampant with no practical purpose. It is argued that Government expenditures of public moneys for research should be applied more to directed research that has specific practical goals with explicit criteria for success. It has always been difficult to argue otherwise. Smith’s article, however, goes a long way toward a rebuttal, as well as showing the benefits of play and artistic creativity for its own sake. See the Aesthetic Curiosity – The Root of Invention.

Nahin Triangle Problem

This article is basically a technical footnote without wider significance. At the time I had been reading with interest Paul J. Nahin’s latest book Number-Crunching (2011). Nahin presents a problem that he will solve with the Monte Carlo sampling approach.

“To start, imagine an equilateral triangle with side lengths 2. If we pick a point ‘at random’ from the interior of the triangle, what is the probability that the point is no more distant than d = √2 from each of the triangle’s three vertices? The shaded region in the figure is where all such points are located.”

Nahin provided a theoretical calculation for the answer and said that it “requires mostly only high school geometry, plus one step that I think requires a simple freshman calculus computation.” This article presents my solution without calculus. See the Nahin Triangle Problem.

Math and Religion

This was a catchy, misleading title that I could not resist, since my essay is not about math vs. religion as one might expect from the title, but rather about math helping religion. Back in 2016 I was reading Dr. Bart D. Ehrman’s blog that he was writing in preparation for his eventual book, The Triumph of Christianity, in which he was considering Rodney Stark’s purely mathematical analysis of the growth of Christianity in the first three centuries. Neither Rodney Stark nor Bart Ehrman described explicitly the underlying mathematical models of exponential growth that they were using and exactly what was meant by a rate of growth. Given the natural audience for the subject, these omissions were not surprising. So I thought I would clarify the math and also offer some variations on the models, which eventually reflected the actual situation more faithfully. See Math and Religion.

Angular Momentum

I have always had a tenuous relationship with the concept of angular momentum, but recently my concerns resurfaced when I did my studies on Kepler, and in particular his “equal areas law” and Newton’s elegant geometric proof. I love the fact that a simple geometric argument, seemingly totally divorced from the physical situation, can provide an explanation for why the line from the Sun to a planet sweeps out equal areas in equal time as the planet orbits the Sun, solely under the influence of the gravitational force between them. However, modern physics books invariably cite the conservation of angular momentum as the “explanation.” I indicated before in my “Kepler’s Laws and Newton’s Laws” essay that this “explanation” irritated me. In this essay I go into detail about my reservations concerning this line of argument. See Angular Momentum.

Complex Numbers – Geometric Viewpoint

This may be a futile attempt at an elementary introduction to complex variables by emphasizing their geometric properties. The elementary part is probably undermined by an initial discussion of field extensions and a necessary reference to trigonometry. Hopefully, the suppression of the explicit use of complex powers of Euler’s constant e until the very end will allow the geometric ideas to have center stage. A primary goal of the essay is to realize that complex polynomials involve sums of circles in the plane. The image of real polynomials as wavy curves in the plane is misleading for an understanding of complex behavior. See Complex Numbers – Geometric Viewpoint.

Meditation on “Is” in Mathematics II – Mathematical Reality

This post continues a meditation on the nature of mathematics begun in Part I. It involves the perennial question about whether mathematics is invented or discovered, and consequently evokes questions about mathematical reality. This subject is probably of little interest to most people, and even most mathematicians. But the extremely heavy involvement of mathematics in the descriptions of quantum mechanics, and the even more mathematically abstruse excursions into ideas such as string theory in an effort to wed quantum mechanics to general relativity, force us to confront the central place mathematics has in “explaining” our physical reality. Of course, this essay has no definitive answers, and leaves the situation as a mystery.  See Meditation on “Is” in Mathematics II – Mathematical Reality.

Meditation on “Is” in Mathematics I – Zeno’s Paradox

This post is the first on a meditation on the nature of mathematics as I see it. I have been thinking about this for some time, and my thoughts were again stimulated by a March 2014 article I read in Slate by Brian Palmer that attempted a popularized explanation of the mathematical concepts associated with Zeno’s Paradox. It was a laudable effort that I applaud. So it is a bit churlish of me to critique it, but I felt its misconceptions got at the heart of some fundamental ideas about mathematics that I wanted to clarify.

The key idea exemplified in this article is the role “making it up” plays in math. That is, the general impression seems to be that math is dealing with things as they actually are if we can just be brought to see it. Whereas the idea that mathematicians make things up or define things is given little credence. For example, 0 x 2 “is” 0 doesn’t make any sense if you arrive at multiplication inductively from the intuitive idea of its being repeated addition. That is, 2 x 0 = 0 + 0 = 0 makes sense, but 0 x 2 = 0 does not. So mathematicians just say let’s define 0 x 2 = 0. If we do, it will be consistent with the other rules we have abstracted from the repeated addition idea, such as the commutative and distributive rules – that is, nothing breaks. (Try defining 0 x 2 to be any other number than 0 and see what breaks.) To put it another way, the reason we want to have 0 x 2 = 0 is for a different reason than we originally thought was meant by multiplication. We have extended the original idea into new territory. A similar thing happens with the advent of negative numbers. This is a very sophisticated idea and a challenge to present at an elementary stage.

In Part I, I will first present the article, heavily annotated with my critique. Then in Part II I will try to explain in more depth the admittedly philosophical concepts I am trying to get at.  See Meditation on “Is” in Mathematics I – Zeno’s Paradox.

Polygon Altitude Problems II

James Tanton has provided further elaborations on the polygons and the sum of perpendicular distances from interior points. Again I approached the solutions with a mix of areas and vectors. It is rather impressive to see the number of variations that can be rung on the Viviani Theorem theme. See Polygon Altitude Problems II

Polygon Altitude Problems I

I found this collection of related problems by James Tanton on Twitter. Even though all these problems do not involve perpendiculars, they have a common solution approach – a sort of theme and variations idea. In a later tweet Tanton refers to a Viviani Theorem associated with these types of problems. I did not recall that theorem explicitly or by name. I also have not looked it up yet, in order to solve these problems on my own. I am guessing there is a more classical Euclidean geometry proof, but I like my vector approach for its clarity. I also throw in a bit a calculus at the end for fun. See Polygon Altitude Problems I