I don’t know whether this is contrived or whether a student offered it on an actual exam — Ed Barbeau presented “this little beauty of a howler” in the January 2002 College Mathematics Journal, citing Ross Honsberger of the University of Waterloo in Ontario.
In his Harmonices Mundi of 1619, Johannes Kepler wrote, “The heavenly motions are nothing but a continuous song for several voices, to be perceived by the intellect, not by the ear; a music which, through discordant tensions, through syncopations and cadenzas as it were, progresses toward certain pre-designed six-voiced cadences, and thereby sets landmarks in the immeasurable flow of time.” In 1979 Yale geologist John Rodgers and musician Willie Ruff scaled up the frequencies of the planetary orbits into the range of human hearing so that Kepler’s “harmony of the world” could become audible:
Mercury, as the innermost planet, is the fastest and the highest pitched. It has a very eccentric orbit (as planets go), which it traverses in 88 days; its song is therefore a fast whistle, going from the E above the piano (e′′′′′) down more than an octave to about C# (c#′′′′) and back, in a little over a second. Venus and Earth, in contrast, have nearly circular orbits. Venus’s range is only about a quarter tone, near the E next above the treble staff (e′′′); Earth’s is about a half tone, from G (g′′) to G# at the top of that staff. … Next out from Earth is Mars, again with an eccentric orbit … it ranges from the C above middle C (c′′) down to about F# (f#′) and back, in nearly 10 seconds. The distance from Mars to Jupiter is much greater than that between the inner planets … and Jupiter’s song is much deeper, in the baritone or bass, and much slower. It covers a minor third, from D to B (D to B1) just below the bass staff. Still farther out and still lower is Saturn, only a little more than a deep growl, in which a good ear can sometimes hear the individual vibrations. Its range is a major third, from B to G (B2 to G2), the B at the top being just an octave below the B at the bottom of Jupiter’s range. Thus the two planets together define a major triad, and it may well have been this concord … that made Kepler certain he had cracked the code and discovered the secret of the celestial harmony.
(The outer planets, discovered after Kepler’s time, are represented here with rhythmic beats.) “The Earth sings Mi, Fa, Mi,” Kepler wrote. “You may infer even from the syllables that in this our home misery and famine hold sway.”
(John Rodgers and Willie Ruff, “Kepler’s Harmony of the World: A Realization for the Ear,” American Scientist 67:3 [May-June 1979], 286-292.)
A curious puzzle by Dartmouth mathematician Peter Winkler: You’ve just joined the Coin Flippers of America, and fittingly the amount of your dues will be decided by chance. You’ll name a head-tail sequence of length 5, and then a coin will be flipped until that sequence appears in five consecutive flips. Your dues will be the total number of flips in U.S. dollars; for instance, if you choose HHHHH and it takes 36 flips to produce a run of five heads, then your annual dues will be $36. What sequence should you pick?
At first it seems that it shouldn’t matter — any fixed sequence should have the probability (1/2)5, or 1/32. But “Not so fast,” Winkler writes. “Overlapping causes problems.” It is true that in an infinite sequence of random flips, the average distance between one occurrence and the next of any fixed sequence is 1/32. But if you choose HHHHH (for example), one occurrence of this outcome gives a huge head start to the next — if the next flip is a tail, then you’re starting over cleanly, but if it’s a head then you’ve already produced the next occurrence.
“If X is the average time needed to get HHHHH starting fresh, the average of 1 + X and 1 is 32,” Winkler writes. “Solving for X yields a startlingly high 62 flips.” To get your expected dues down to $32, you need to pick a sequence where this “head start” effect doesn’t obtain. There are 10 such sequences; one is HHHTT.
(Peter Winkler, “Coin Flipping,” Communications of the ACM 56:11 [November 2013], 120.)
An observation by Oxford University mathematician Nick Trefethen:
A student leaves university in America with a transcript full of information. Even with grade inflation, there are thirty marks of A or A- or B+ or B to look at, each one attached to a different course like Advanced Calculus or 20th Century Philosophy or Introduction to Economics. Grade-point averages are constructed from these transcripts and reported to three digits of accuracy.
An Oxford graduate finishes with no transcript, just a degree result which may be a First, a II.1, a II.2, a Third, a Pass, or a Fail. Failures are more or less nonexistent, and the numbers last year [2000] for the other degrees were 691, 1925, 374, 39, and 3, respectively. The corresponding probabilities are 23%, 63%, 12%, 1%, and 0.1%.
If you add up these probabilities times their base 2 logarithms, all times minus one, you find out how much information there is in an Oxford degree. The result is: 1.37 bits of information.
(From Trefethen’s Index Cards, 2011.)
Early experimenters in electricity sometimes dealt in frogs’ thighs. Dissecting a frog creates an “injury potential” in its muscles, which can then be arranged in series to produce a kind of biological battery. Carlo Matteucci strung together 12 to 14 half-thighs to make a “frog battery” strong enough to decompose potassium iodide; he was able to induce some effect even with living frogs.
Matteucci did similar work with eel, pigeon, and rabbit batteries. In 1803 Giovanni Aldini used a galvanoscope made of frogs to detect current in a circuit that ran from an ox’s tongue to its ear through Aldini’s own body. The mechanisms underlying these results weren’t always clearly understood, but they formed important early strides in bioelectrochemistry.
On a dry summer day in California, physicist Julius Sumner Miller was driving slowly near the desert when a friend overtook him on the left. The friend’s wife, in the passenger seat, reached out to hand him a package of gum. Their hands were no less than 3 inches apart when “a terrific discharge took place which possessed the classical physiological effects. The shock was momentarily disabling, as a three-inch spark in air can well be.”
Miller published an inquiry about this in the American Journal of Physics and received a reply from R.F. Miller of B.F. Goodrich in Ohio. The motion of the cars had built up charges of different amounts; Goodrich had found that the accumulated charges can (or could) increase greatly as the wheel rotates, and “as soon as the tread charges are far enough removed, they will find a lower resistance path through the rim to ground rather than around the tread,” charging the vehicle.
Even at the time the phenomenon was well known; in his original letter Miller noted that gasoline trucks were required by law to carry a dragging chain or strap. But “the question as to how great a charge may accumulate is difficult to answer.”
(Julius Sumner Miller, “Concerning the Electric Charge on a Moving Vehicle,” American Journal of Physics, 21:4 [April 1953], 316.)
The ACLU found John Scopes by running a newspaper ad seeking a teacher willing to test the law about teaching human evolution in the classrooms of Tennessee. From the May 4, 1925, edition of the Chattanooga Times:
We are looking for a Tennessee teacher who is willing to accept our services in testing this law in the courts. Our lawyers think a friendly test case can be arranged without costing a teacher his or her job. Distinguished counsel have volunteered their services. All we need now is a willing client.
Scopes wasn’t a biology teacher but had filled in for one using a textbook that accepted evolution, and that was enough to set the “monkey trial” moving forward.
Strangely, the disputed textbook was the one that Tennessee required its high school teachers to use that year. Clarence Darrow later quipped in his autobiography, “It seems strange that the Dayton school board did not adopt the first and second chapters of Genesis as a modern textbook on biology.”
Carved into the brickwork of a cylindrical tower at Cambridge University’s New Museums Site is a great crocodile. It was commissioned by Pyotr Kapitza, who had moved to Cambridge from Russia expressly to work with Ernest Rutherford, the father of nuclear physics. Kapitza called his mentor “crocodile,” a title that Russians traditionally confer on great men (and also, Kapitza said, because Rutherford’s thunderous voice announced his approach, just as the crocodile in Peter Pan was announced by the ticking watch in its belly).
Eric Gill carved the animal into the side of the Mond Laboratory, which was erected in 1933 with Rutherford’s backing to support Kapitza’s work in low-temperature physics. Unfortunately, after a holiday in Russia the following year, Kapitza was barred from leaving the country, and he never returned to Cambridge.
A few quotations by Rutherford:
Paul Langevin and Rutherford served together as research assistants at Cavendish Laboratory. Asked afterward whether they were friendly, Langevin said, “One can hardly speak of being friendly with a force of nature.”
Amazingly, the notion of a black hole was first posited in 1783, by the English natural philosopher John Michell. In a paper read before the Royal Society that November, he wrote:
Let us now suppose the particles of light to be attracted in the same manner as all other bodies with which we are acquainted; that is, by forces bearing the same proportion to their vis inertiae (or mass), of which there can be no reasonable doubt, gravitation being, as far as we know, or having any reason to believe, an universal law of nature. … [I]f the semi-diameter of a sphere of the same density as the Sun were to exceed that of the Sun in the proportion of 500 to 1, a body falling from an infinite height towards it, would have acquired at its surface greater velocity than that of light, and consequently supposing light to be attracted by the same force in proportion to its vis inertiae, with other bodies, all light emitted from such a body would be made to return towards it by its own proper gravity.
“From these quotations it is clear that Michell in 1783 understood many of the basic principles of black hole physics which are in daily use almost 200 years later,” writes Cambridge physicist Gary Gibbons. Indeed, Michell’s talent doomed him to obscurity: His breakthroughs were lost on his contemporaries and forgotten by the time the world could appreciate them. His notion of a “dark star” was rediscovered only in the 1970s. The American Physical Society says, “[H]e remains virtually unknown today, in part because he did little to develop and promote his own path-breaking ideas.”
(Gary Gibbons, “The Man Who Invented Black Holes,” New Scientist, June 28, 1979.) (Thanks, Alejandro.)
Biologist and mathematician D’Arcy Thompson advanced a strange new idea in his 1917 book On Growth and Form: He found that if you draw the outline of an animal or plant on an ordinary Cartesian grid, and then you put the grid through some mathematical transformation (stretching it, for example, so that its squares become rhombuses), very often the resulting shape is that of a related real creature.
What can that mean? Thompson doesn’t really say. He thought that the biologists of his day overemphasized evolution in explaining the form and structure of living things; he preferred to look for physical and especially mathematical laws. But he didn’t present his ideas as principles that might be tested, so his book has (so far) remained only a notable curiosity.
“This theory cries out for causal explanation, which is something the great man eschewed,” writes zoologist Wallace Arthur. “Perhaps the time is close when comparative developmental genetics will be able to provide such an explanation.”
(Wallace Arthur, “D’Arcy Thompson and the Theory of Transformations,” Nature Reviews Genetics, May 2006, 401-406.)
