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.)

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.’

Experiment

https://commons.wikimedia.org/wiki/File:Trinity_Detonation_T%26B.jpg

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.)

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.

Turnabout

https://www.reddit.com/r/interestingasfuck/comments/j1iekj/as_photography_became_more_common_an_odd/

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.”

https://collections.countway.harvard.edu/onview/items/show/17964

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.)

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.

https://books.google.com/books?id=i8NJomIVzlgC&pg=PA508

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.)

Pursuit

http://cyclopediaofpuzzles.com/page/216

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?

Click for Answer

Root Words

Square roots:

EIGHTY-ONE has 9 letters.

ONE HUNDRED has 10 letters.

FIVE HUNDRED AND SEVENTY SIX has 24 letters.

Cube roots:

THIRTY-NINE THOUSAND THREE HUNDRED FOUR has 34 letters.

SIXTY-EIGHT THOUSAND NINE HUNDRED AND TWENTY-ONE has 41 letters.

ONE MILLION and ONE BILLION have 10 letters each, making them a sixth root and (in the United States) a ninth root word.

(Dave Morice, “Kickshaws,” Word Ways 30:2 [May 1997], 129-141.)

10/26/2020 UPDATE: Reader Hans Havermann has found many more, including this alarming specimen:

341183 = ONE HUNDRED SIXTY-NINE SENONAGINTILLION, FIVE HUNDRED FIVE QUINONAGINTILLION, SEVENTY-SEVEN QUATTUORNONAGINTILLION, FIFTY-ONE TRENONAGINTILLION, EIGHT HUNDRED SEVENTY DUONONAGINTILLION, SEVEN HUNDRED FIFTY-FOUR UNONAGINTILLION, SEVEN HUNDRED THIRTY-EIGHT NONAGINTILLION, SIX HUNDRED FIFTY-SIX NOVOCTOGINTILLION, TWO HUNDRED SEVENTY OCTOCTOGINTILLION, TWO HUNDRED NINETY-EIGHT SEPTENOCTOGINTILLION, SIX HUNDRED FORTY-SEVEN SEXOCTOGINTILLION, EIGHT HUNDRED FORTY-EIGHT QUINTOCTOGINTILLION, FOUR HUNDRED FOUR QUATTUOROCTOGINTILLION, EIGHT HUNDRED FIFTY-SIX TRESOCTOGINTILLION, SIX HUNDRED THIRTEEN DUOOCTOGINTILLION, EIGHT HUNDRED THIRTY-THREE UNOCTOGINTILLION, NINE OCTOGINTILLION, SEVEN HUNDRED SIXTY-FIVE NOVEMSEPTUAGINTILLION, EIGHT HUNDRED SIXTY-SEVEN OCTOSEPTUAGINTILLION, ONE HUNDRED FIFTY-THREE SEPTENSEPTUAGINTILLION, NINE HUNDRED SEVENTY-TWO SESEPTUAGINTILLION, EIGHT HUNDRED SEVENTY-EIGHT QUINSEPTUAGINTILLION, FIFTY-NINE QUATTUORSEPTUAGINTILLION, SEVEN HUNDRED SIX TRESEPTUAGINTILLION, EIGHT HUNDRED FIFTEEN DUOSEPTUAGINTILLION, TWO HUNDRED THIRTY-ONE UNSEPTUAGINTILLION, FOUR HUNDRED EIGHT SEPTUAGINTILLION, ONE HUNDRED FIFTY-FIVE NOVEMSEXAGINTILLION, FOUR HUNDRED EIGHTY-SIX OCTOSEXAGINTILLION, TWELVE SEPTENSEXAGINTILLION, TWO HUNDRED FORTY-SIX SESEXAGINTILLION, EIGHT HUNDRED EIGHTY-FIVE QUINSEXAGINTILLION, TWO HUNDRED FIFTY-SIX QUATTUORSEXAGINTILLION, FOUR HUNDRED NINETY-TWO TRESEXAGINTILLION, SEVEN HUNDRED FORTY-THREE DUOSEXAGINTILLION, FIVE HUNDRED FIFTY-FIVE UNSEXAGINTILLION, ONE HUNDRED FORTY-ONE SEXAGINTILLION, EIGHT HUNDRED TWENTY-FIVE NOVEMQUINQUAGINTILLION, SEVEN HUNDRED FORTY-SIX OCTOQUINQUAGINTILLION, ONE HUNDRED SEVENTY-ONE SEPTENQUINQUAGINTILLION, NINE HUNDRED THIRTY-TWO SEXQUINQUAGINTILLION, ONE HUNDRED SIXTY-SIX QUINQUINQUAGINTILLION, FOUR HUNDRED THREE QUATTUORQUINQUAGINTILLION, THIRTY-TWO TRESQUINQUAGINTILLION, SIX HUNDRED TWENTY-FOUR DUOQUINQUAGINTILLION, ONE HUNDRED FORTY-SIX UNQUINQUAGINTILLION, ONE HUNDRED TWENTY-FIVE QUINQUAGINTILLION, EIGHT HUNDRED TWO NOVEMQUADRAGINTILLION, EIGHT HUNDRED FIFTY-EIGHT OCTOQUADRAGINTILLION, ONE HUNDRED SEVENTY-NINE SEPTENQUADRAGINTILLION, ONE HUNDRED FIFTY-SIX SEXQUADRAGINTILLION, ONE HUNDRED TWENTY-FOUR QUINQUADRAGINTILLION, NINE HUNDRED FIFTY-FOUR QUATTUORQUADRAGINTILLION, NINE HUNDRED ELEVEN TRESQUADRAGINTILLION, TWO HUNDRED FIFTEEN DUOQUADRAGINTILLION, NINE HUNDRED SIXTY-SIX UNQUADRAGINTILLION, NINE HUNDRED TWELVE QUADRAGINTILLION, EIGHT HUNDRED THREE NOVEMTRIGINTILLION, SIX HUNDRED SIXTY-EIGHT OCTOTRIGINTILLION, SEVEN HUNDRED SIXTY-THREE SEPTRIGINTILLION, ONE HUNDRED EIGHTY-THREE SEXTRIGINTILLION, SEVEN HUNDRED NINETY-SIX QUINTRIGINTILLION, NINETY QUATTUORTRIGINTILLION, SEVEN HUNDRED TRESTRIGINTILLION, FIVE HUNDRED EIGHTY-SIX DUOTRIGINTILLION, ONE HUNDRED FORTY-ONE UNTRIGINTILLION, SIX HUNDRED FIFTY-FOUR TRIGINTILLION, SIX HUNDRED EIGHT NOVEMVIGINTILLION, SEVEN HUNDRED FIFTY-THREE OCTOVIGINTILLION, SIX HUNDRED FOURTEEN SEPTENVIGINTILLION, THREE HUNDRED FOUR SEXVIGINTILLION, SEVEN HUNDRED NINETY-FIVE QUINVIGINTILLION, SIX HUNDRED FORTY-FOUR QUATTUORVIGINTILLION, THREE HUNDRED SIXTY-FIVE TREVIGINTILLION, EIGHT HUNDRED NINETY-SEVEN DUOVIGINTILLION, EIGHT HUNDRED SEVENTEEN UNVIGINTILLION, FIVE HUNDRED EIGHTY-TWO VIGINTILLION, FOUR HUNDRED FIFTY-FIVE NOVEMDECILLION, FOUR OCTODECILLION, SIX HUNDRED FORTY-FIVE SEPTENDECILLION, NINE HUNDRED SEVEN SEXDECILLION, FOUR HUNDRED EIGHTY QUINDECILLION, EIGHT HUNDRED THIRTY-SEVEN QUATTUORDECILLION, ONE HUNDRED FORTY-FOUR TREDECILLION, TWO HUNDRED TWENTY-FIVE DUODECILLION, EIGHT HUNDRED EIGHTY-ONE UNDECILLION, SIX HUNDRED SEVEN DECILLION, THREE HUNDRED FIFTY-TWO NONILLION, FIVE HUNDRED FORTY-THREE OCTILLION, FOUR HUNDRED NINE SEPTILLION, TWO HUNDRED SEVENTY-NINE SEXTILLION, ONE HUNDRED EIGHTY-EIGHT QUINTILLION, SEVEN HUNDRED SEVENTY-SIX QUADRILLION, FORTY-EIGHT TRILLION, THREE HUNDRED NINETY-FOUR BILLION, SEVEN HUNDRED SEVENTY-ONE MILLION, EIGHT HUNDRED EIGHT THOUSAND, THREE HUNDRED THIRTY-ONE

The name of that number contains 3,411 letters.

(Thanks, Hans.)