The Tunnel of Eupalinos

When the Greek engineer Eupalinos contrived a tunnel in the 6th century B.C. to carry water through Mount Kastro to Samos, he started digging simultaneously from the north and south, hoping that the two tunnels would meet in the heart of the mountain. He arranged this through some timely doglegs: When the two teams could hear one another (meaning they were about 12 meters apart), each deviated from its course in both the horizontal (left) and vertical (right) planes:

https://commons.wikimedia.org/wiki/File:Eupalinos_horizontal.svg — Images: Wikimedia Commons

This ensured that they wouldn’t tunnel on hopelessly past one another on parallel courses.

This worked amazingly well: In fact the vertical alignment, established using levels at the start, had been maintained so faithfully that the two tunnels differed by only a few millimeters, though they’d traversed a combined distance of more than a thousand meters.

This is only the second known tunnel to be excavated successfully simultaneously from both ends, and the first to accomplish this feat using geometric principles, which Euclid would codify only centuries later.

October 16, 2020October 16, 2020 | Science & Math · Technology

Throwing a Curve

In 2009, mathematician Jeff Chyatte and his colleagues at Maryland’s Montgomery College built a mathematical sculpture: An inclined rod is connected at its center to a horizontal arm, which is connected to a rotating vertical axis. As the axis rotates, the rod passes through a vertical plane.

What shape does the rod cut in the plane? Perhaps surprisingly, it’s a hyperbola. See the video above for an explanation. Chyatte’s sculpture was displayed at Washington’s Touchstone Gallery with the title “Theorem.”

(“Just Passing Through,” Math Horizons 16:4 [April 2009], 16.)

October 9, 2020October 8, 2020 | Science & Math

Local Rules

A footnote from T.W. Körner’s The Pleasures of Counting:

It may help to recall the bon mot I heard from a Russian physicist: ‘Proofs in physics follow the standards of British justice and hold the accused innocent until proved guilty. Proofs in mathematics follow the standards of Stalinist justice and hold the accused guilty until proved innocent.’

October 8, 2020October 7, 2020 | Science & Math

Experiment

From Enrico Fermi’s eyewitness report on the first detonation of a nuclear device, July 16, 1945:

About 40 seconds after the explosion, the air blast reached me. I tried to estimate its strength by dropping from about six feet small pieces of paper before, during, and after the passage of the blast wave. Since, at the time, there was no wind I could observe very distinctly and actually measure the displacement of the pieces of paper that were in the process of falling while the blast was passing. The shift was about 2½ meters, which, at the time, I estimated to correspond to the blast that would be produced by ten thousand tons of T.N.T.

Radiochemical analysis of soil samples later indicated that the total yield had been around 18.6 kilotons of TNT.

10/09/2020 UPDATE: Here’s Fermi’s report. (Thanks, Sivaraam.)

October 7, 2020October 9, 2020 | Science & Math

The Burned House Phenomenon

https://commons.wikimedia.org/wiki/File:Burned_House_Horizon_Map.PNG — Image: Wikimedia Commons

The Cucuteni-Trypillian culture of Neolithic Europe left behind a curious puzzle for archaeologists: It appears that, for more than a thousand years, the houses in every settlement were burned. It’s not clear why. Possibly the fires arose accidentally or through warfare, or possibly they were set deliberately. The extent of each fire must have been considerable, because the raw clay in the walls has been vitrified by intense heat, an effect that has not appeared in modern experiments with individual houses. But the reason for the phenomenon, and for its longevity, remains unknown.

October 7, 2020October 6, 2020 | History · Oddities · Science & Math

Turnabout

In the early 20th century, medical students often posed for photographs with the cadavers they were learning to dissect — in some cases even trading places with them for a tableau called “The Student’s Dream.”

John Harley Warner and James M. Edmonson have published a book of these photos, Dissection: Photographs of a Rite of Passage in American Medicine 1880-1930. “What we know with certainty about any particular photograph often is frustratingly meager,” they write. “A dissection room photograph discovered tucked between the pages of an old anatomy textbook or up for auction on eBay is likely to have no indication of where or when it was taken, who took it, or who is in it. The photographs suggest stories that cannot easily be recovered.”

But they say that the images generally were intended not to be entertaining or flippant, but to mark a professional rite of passage for the students. “Privileged access to the body marked a social, moral, and emotional boundary crossing. ‘Know thy Self’ inscribed on the dissecting table, the Delphic injunction nosce te ipsum, could refer to the shared corporeality of dissector and dissected. But it most certainly referred to knowing the new sense of self acquired through these rites. As visual memoirs of a transformative experience, the photographs are autobiographical narrative devices by which the students placed themselves into a larger, shared story of becoming a doctor.”

October 4, 2020October 3, 2020 | Death · Oddities · Science & Math

Bird Brains

Crows are smart. In 2014, University of Auckland psychologist Sarah Jelbert and her colleagues assessed the causal understanding of water displacement in New Caledonian crows by presenting them with a narrow tube in which a reward floated out of reach. To get the reward, a bird had to drop objects into the tube to raise the water level.

“We found that crows preferentially dropped stones into a water-filled tube instead of a sand-filled tube; they dropped sinking objects rather than floating objects; solid objects rather than hollow objects, and they dropped objects into a tube with a high water level rather than a low one.”

Apparently crows read Aesop. And Aesop was right.

(Sarah A. Jelbert et al., “Using the Aesop’s Fable Paradigm to Investigate Causal Understanding of Water Displacement by New Caledonian Crows,” PloS One 9:3 [2014], e92895.)

October 2, 2020 | Science & Math

Magic

A “kinde of Divination” “to tell your friend how many pence or single peeces, reckoning them one with another, he hath in his purse, or should thinke in his minde,” from Robert Recorde’s The Ground of Arts, 1618:

[F]irst bid him double the peeces hee hath in his purse, or the number hee thinketh. … Now after hee hath doubled his number, bid him adde thereunto 5 more, which done, bid him multiply that his number by 5 also: which done bid him tell you the just sum of his last multiplication, which sum the giver thinking it nothing availeable, because it is so great above his pretended imagination: yet thereby shall you presently with the helpe of Subtraction tell his proposed number.

Apparently the section on “divers Sportes and Pastimes, done by Number” was contributed by Southwark schoolmaster John Mellis in 1582. “[T]he fact that this chapter on mathematical games was included in every subsequent edition of The Ground of Artes, save one, indicates that the idea of mathematical games found a receptive audience among arithmetic students.”

(Jessica Marie Otis, “‘Sportes and Pastimes, done by Number’: Mathematical Games in Early Modern England,” in Allison Levy, ed., Playthings in Early Modernity: Party Games, Word Games, Mind Games, 2017.)

October 2, 2020October 1, 2020 | History · Science & Math

Dürer’s Unfolding Problem

https://commons.wikimedia.org/wiki/File:Dodecahedron_flat.svg — Image: Wikimedia Commons

Can every convex polyhedron be “unfolded” into a shape that doesn’t overlap itself, like this dodecahedron?

Surprisingly, no one knows.

The Renaissance artist Albrecht Dürer first wondered about this in the early 1500s, and in the ensuing five centuries no one has found a polyhedron that won’t permit it. But one may yet exist.

September 27, 2020September 26, 2020 | Science & Math

Pursuit

A problem from Sam Loyd’s Cyclopedia of Puzzles, 1914:

Here is the puzzle of Tom the Piper’s Son, who, as told by ‘Mother Goose,’ stole the pig and away he run. It is known that Tom entered the far gate shown at the top on the right hand. The pig was rooting at the base of the tree 250 yards distant, and Tom captured it by always running directly towards it, while the pig made a bee-line towards the lower corner as shown. Now, assuming that Tom ran one-third faster than the pig, how far did the pig run before he was caught?

Intriguingly, Loyd adds, “The puzzle is a remarkable one on account of its apparent simplicity and yet the ordinary manner of handling problems of this character is so complicated that solvers are asked merely to submit approximately correct answers, based upon judgment and common sense, just to see who can make the best guess. The simple rule for solving it, however, which will doubtless be quite new to our puzzlists, is based upon elementary arithmetic.” What’s the answer?

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Loyd’s answer: In the puzzle of Tom the Piper’s Son, who stole the pig, Mother Goose reports that Tom had to run 571 3/7 yards to catch the pig, while the porker ran three-quarters of that distance, which would be 428 4/7 yards. The simple rule for solving problems of this class is to halve the distance the man would have to travel to catch the pig in a straight line added to the distance he would go if they advanced toward each other. Tom is 250 yards from the pig, and as he goes one-third faster, in a straight line, he would catch him in 1,000 yards. If they run toward each other, Tom goes four-sevenths of the distance, viz.: 142 6/7 of the 250 yards, which, added to 1,000, equals 1,142 6/7 yards. Half of this equals 571 3/7 yards, which is the correct answer. Loyd doesn’t really explain his “simple rule,” but in the Pi Mu Epsilon Journal (Volume 6, Number 9, Fall 1978, page 545), Tom Apostol of Caltech shows that Loyd’s method is generally valid. The editors add, “The charm of this problem lies in the insidious way the stated result defies our intuitive notions. At first we suspect that the result, if true, might accidentally apply only to the special case involving the given distances and the relative speeds but then the denouement affirms the generality of Loyd’s method of solution.” In his edited collection of Loyd’s puzzles, Martin Gardner added, “The man’s path forms one of the simplest of ‘pursuit curves,’ the study of which forms an interesting branch of what one might call ‘recreational calculus.'”

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September 26, 2020September 25, 2020 | Puzzles · Science & Math