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
Given the aggravating times, I thought I would vent my frustration by ranting on a somewhat nonsensical topic: “The fact that the earth revolves around the sun, rather than the sun around the earth.” This assertion is often used to separate the supposed dunces from the enlightened. It is put on the same level as “the fact that the earth is round (a sphere) and not flat” with the dunces labeled “flat-earthers.”
However, I take umbrage with the use of the word “fact” to conflate these two instances as examples of what “is” or what is “true.” I claim the earth is spherical (more or less: a better approximation is an oblate spheroid) or certainly “curved” rather than “flat.” Whereas the statement that the “sun is at the center of the solar system” is not a fact but an arbitrary convenience.
See Meditation on “Is” in Mathematics III – Heliocentrism
This essay began as an effort to prove Tanya Khovanova’s statement in her article “The Annoyance of Hyperbolic Surfaces” that her crocheted hyperbolic surface had constant (negative) curvature. I discussed Khovanova’s article in my previous essay “Exponential Yarn”. What I thought would be a fairly straight-forward exercise turned into a more concerted effort as I concluded that her crocheted surface did not have constant curvature. However, I found additional references that supported her statement, so I was becoming quite confused. I looked at other, similar surfaces to try to understand the whole curvature situation. This involved a lot of tedious computations (with my usual plethora of mistakes) that proved most challenging. But then I realized where I had gone astray. To cover my ignorance I claimed my error stemmed from a subtle misunderstanding. Herewith is a presentation of what I found. See Bugles, Trumpets, and Beltrami.
(Update 4/6/2019) 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.
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
This 2011 article gives some thoughts I had after reading Michael Dirda’s review in the Washington Post of Larrie D. Ferreiro’s Measure of the Earth. The book described the 1735 Geodesic Mission, whose purpose was to resolve the question of the shape of the earth, that is, whether it was a sphere, or like an egg with the poles further from the center than the equator, or like an oblate spheroid with the equator further from the center than the poles, as Newton averred due to centrifugal force. In the review Dirda said, “A team, sympathetic to Newton’s view, would travel to what is now Ecuador and measure the exact length of a degree of latitude near the equator. This would then be compared with the same measurement taken in France. If the latter was larger, Newton was right.” I wondered at first if Dirda got it right. It turned out my confusion stemmed from a mistaken definition of a degree of latitude. See Degree of Latitude.