One thing I have always been curious about, but never got around to investigating, is how hard is it to see that the Lambert Equal-Area Projection of a sphere onto a cylinder in fact preserves areas? This 2012 essay attempts to provide an answer. The essay was recently updated to provide a link to the fabulous Youtube site by Grant Sanderson at 3blue1brown. He shows a different way of looking at the problem also without explicitly resorting to calculus. All his videos are spectacular and provide unparalleled insights into mathematics. What I wouldn’t give to have had such videos available when I was studying math. How much more quickly would I have been able to learn. See Lambert Equal Area Projection.
Having immersed myself in studying Kepler’s discovery that the planetary orbits were ellipses, I was immediately aware of how the British mathematician, Katie Steckles, justified her technique to cut an elliptical pizza into equal slices in her video of 14 March 2017. In her video Katie makes the claim that the result of any affine transformation of the circular pizza cut into equal sectors will also be a set of equal area slices. I made an attempt to substantiate these remarks. See Cutting Elliptical Pizza.
I have long been fascinated by Newton’s proof of Kepler’s Equal Areas Law and wanted to write about it. Of course, others have as well, but I wanted to emphasize an aspect of the proof that supported my philosophy of mathematics.
Before I get to Newton, however, I wanted to discuss how Kepler himself justified this law, since his approach has a number of fascinating historical aspects to it. I have previously discussed Kepler’s ellipse and in the process of doing that research, I came across a number of articles about how Kepler arrived at his equal areas law. One notable result is that even though now we call the idea that a planet orbits the Sun in an elliptical path with the Sun at one focus, Kepler’s First Law, and the idea that the line from the Sun to the planet sweeps out equal areas in equal times, Kepler’s Second Law, Kepler actually discovered these laws in reverse order. See Kepler’s Equal Areas Law
I had been exploring how Kepler originally discovered his first two laws and became fascinated by what he did in his Astronomia Nova (1609), as presented by a number of researchers. Among the writers was A. E. L. Davis. She mentioned that the characterization of the ellipse that Kepler was using was the idea of a “compressed circle,” that is, a circle all of whose points were shrunk vertically by a constant amount towards a fixed diameter of the circle. I did not recall ever hearing this idea before and tried to track down its origin together with a proof — futilely, Davis’s references notwithstanding. I then tried to prove it myself. It was easy to do with analytic geometry. But in the spirit of the Kepler era (before the advent of Fermat’s and Descartes’s beginnings at fusing algebra and geometry) I tried to prove it solely within Euclid’s plane geometry. Some critical steps seemed to come from the great work of Apollonius of Perga (262-190 BC) on Conics. But for me a final elegant proof was not evident until 1822 when Dandelin employed his inscribed spheres. See Kepler’s Ellipse.
In the process of exploring the compressed circle idea I acquired an immense appreciation and regard for Kepler and his perseverance in the face of the dominant paradigm of his era, namely, the 2000 year old idea that the celestial motions were all based on the most perfect motion of all, that of circles. The kinds of extremely laborious calculations he went through (just prior to the invention of logarithms by John Napier) were daunting, especially considering the trials he was undergoing in his personal life (trying to survive the religious destruction between Catholics and Protestants, along with defending his mother against charges of witchcraft).
Years ago (1963) I got the paperback The Calculus:A Genetic Approach, by Otto Toeplitz, which presented the basic ideas of the differential and integral calculus from a historical point of view. One thing Toeplitz did at the end of his book that I had not seen in other texts was to show the equivalence of Kepler’s Laws and Newton’s Law of Gravity. (Since 1963 David Bressoud has developed this theme in his excellent 1991 text.) I thought I would try to emulate Toeplitz’s approach with more modern notation (vectors) and arguments in hopes of extracting the essential ideas from the clutter.
A by-product of this effort was to reveal strongly the different paths that physics and mathematics follow in understanding physical reality. The mystery is that the mathematics ends up describing the physics so well. I will return to this theme a number of times in other posts. See Kepler’s Laws and Newton’s Laws.
This is another problem from the Futility Closet website. It turned out to be pretty simple. The idea is to show the length of BC remains the same no matter where A is chosen on its arc of C1.
(Update 7/1/2020) There is more to this problem than I realized, thanks to a revisit prompted by a question from Deb Jyoti Mitra. See the revised Keyhole Problem.
Years ago (1967) I read about an interesting solution to the three jugs problem in a book by Nathan Court which involved the idea of a billiard ball traversing a skew billiard table with distributions of the water between the jugs listed along the edges of the table. The ball bounced between solutions until it ended on the desired value. I thought it was very clever, but I really did not understand why it worked. Later I figured out an explanation, which I present here. See the Three Jugs Problem.
Another good source of problems is the Futility Closet site. This puzzle involved finding the line of maximal length passing through the intersection of two circles. I solved it before looking at the Futility Closet solution. Their solution of course was short, sweet, and elegant. Mine was more like the old adage of cracking a walnut with a sledge hammer. Still, I thought there were some unexplained parts to the elegant solution that justified the effort on mine. At least my solution provided an interesting, though convoluted, alternative. See the Two Circles Puzzle.
This is one of Alex Bellos’s Monday Puzzles in the Guardian. I basically found the same solution as Bellos and his commenters, but wrote it up with what I thought were more explanatory graphics. The idea is that there is a bunch of ants on a stick who all walk a the same speed of 1 centimeter per second. When an ant runs into another ant, they both turn around and go the opposite direction. “So here is the puzzle: Which ant is the last to fall off the stick? And how long will it be before he or she does fall off?” See the Ant Problem.
This was one of my more satisfying essays. Several years ago I gave some thought to what it meant for the earth to be considered a magnet. More recently in 2012 an article in the magazine BirdWatching brought it all back when I saw its diagram of the earth as a magnet for guiding migratory birds. Knowing that magnets have north and south poles, where should we expect to find the earth’s north and south magnetic poles? See Earth as Magnet.