I thought it might be interesting to explore the mathematics of a common problem with a store-bought HO model train set that contains a collection of straight track segments and fixed-radius curved track segments that form a simple oval. Invariably an initial run of the train has it careening off the track when the train first meets the curved segment after running along the straight track segments.
Why is that? Well of course the train is going too fast. But even if it slows down enough not to fall off the curve, it still jerks unstably and may derail when it first reaches the beginning of the curve. What is going on?
See the Train Wreck Puzzle
Twitter comments to the recently released GDP growth numbers for the Third Quarter reminded me of an old trap regarding percentages. The financial and technical articles were accurate, but the comments by Twitter users often reflected the pitfall.
After a Second Quarter annualized US GDP fall of about 33%, the Third Quarter showed a gain of about 33%. So some commenters thought the gain canceled the previous loss.
Being born on February 29 I have always had an interest in the calendar and the mechanics of Leap Year. Since I am sure everyone knows about Leap Year, I will just rattle off a few trivia questions to stimulate the memory. Why was I excited about my birthday in 2000 when everyone knew it was a Leap Year, being 4 years after 1996? When I lived in Brazil, everyone referred to Leap Year as bissextile. What was that all about? After the Gregorian reform in 1582, how come George Washington’s mother recorded his birth in their family bible as 11 February 1731 when we say it is 22 February 1732 (whereas Abraham Lincoln’s mother recorded 12 February 1809 for her son, which we agree with)? See February 29.
Update (2/29/2020) Continue reading
Tanya Khovanova’s recent blog post “The Annoyance of Hyperbolic Surfaces” about crocheting a hyperbolic surface added to the numerous examples of such activity, usually from knitting. Somehow this post caught my attention, in particular about the exponential growth of each added row and the fact that the resulting “surface” had constant negative curvature. I explored the exponential growth in this article and saved the mathematical exploration of the constant negative curvature for a later essay. See Exponential Yarn.
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.
Reading Axios on Christmas Eve day 2017, I was struck by what appeared at first to be a strange graph showing preferences for Christmas movies divided between men and women. The thing that struck me as strange was the computation for the total votes: the percentages were the average of the men and women percentages. This, of course, is not how you average percentages. What was going on? See Strange Statistics.
Recently I viewed a startling video by Matt Parker about the Tupper Self-Referential Formula. It is a formula that visually represents itself when graphed at a specific location in the (x, y) plane. I found it difficult to fathom, so I looked it up on Wikipedia and Google. After reading different explanations, I finally think I have the idea. So thought I would add my version to the mix. See Tupper Self-Referential Formula.
I came across the following entry in the Futility Closet website that cried out for justification. “An arrangement of three mutually perpendicular planes, like those in the corner of a cube, have a pleasing property: They’ll reflect a ray of light back in the direction that it came from.” So the question is, why is this reflection property true? See Corner Reflectors.
All too frequently I come across the usual statements questioning why non-technical folks should bother studying math. A typical example is the Pythagorean Theorem. People say, “What good is that? I’ll never use it. So why bother?” Ah, the famous “utility” argument – as if everything worthwhile must be “useful.” I thought I would take this “useless” math example par excellence and show that, in fact, it harbors many of the best aspects of mathematics that anyone should find appealing. See the Pythagorean Theorem
Paul Krugman in his 26 July 2013 New York Times column “The Conscience of a Liberal” provided an analysis about why we are always being caught in the slow lane of a congested highway. I tried to fill in the gaps in his laconic explanation. See the Slow Lane Problem.