Dan Pearcy’s Twitter feed highlights a pretty link between Pascal’s triangle and the tangent angle formulae:
Author: Greg Ross
Early Warning
In 2010, as the Colombian government was preparing to rescue 16 soldiers held by armed FARC guerrillas, it looked in vain for a way to alert the soldiers without tipping off their captors. Finally Colonel Jose Espejo arranged to have local radio stations broadcast a pop song that contained a message in Morse code, which the soldiers had learned in basic training but that the guerrillas likely wouldn’t recognize.
The lyrics run, “In the middle of the night / Thinking about what I love the most / I feel the need to sing … About how much I miss them.” And hidden at three points in the song (1:30, 2:30, 3:30), in Morse code, is the message “19 people rescued. You are next. Don’t lose hope.”
“The hostages were listening to our own stations, so we made sure the song was played,” Espejo told The Verge. “The code message said, ‘you’re next’ because the hostages thought if they ran away, they would die in the jungle. We let them know that our troops were nearby.”
It worked. “We know of hostages who heard the message,” Espejo said, “and were able to escape and provide information that led to the release of more hostages.”
Aiming High
“Reynolds said that ‘[Samuel] Johnson always practised on every occasion the rule of speaking his best, whether the person to whom he addressed himself was or was not capable of comprehending him. ‘If,’ says he, ‘I am understood, my labour is not lost. If it is above their comprehension, there is some gratification, though it is the admiration of ignorance;’ and he said those were the most sincere admirers; and quoted Baxter, who made a rule never to preach a sermon without saying something which he knew was beyond the comprehension of his audience, in order to inspire their admiration.'” — James Boswell, Life of Samuel Johnson, 1791
But:
“To write well, express yourself like the common people, but think like a wise man.” – Aristotle
Elegance
This transit map of Stuttgart’s rail network, adopted around 2000, was unique: By omitting horizontal and vertical lines and setting all diagonals at 30 degrees, the designers produced the appearance of three dimensions.
“This diagram is the only one of its type in the world,” wrote Mark Ovenden in Transit Maps of the World, “although Harry Beck did experiment briefly with a 60/120-degree variation of the London map in 1940.” Alas, it’s since been superseded.
Unquote
“It is indeed a desirable thing to be well descended, but the glory belongs to our ancestors.” — Plutarch
“He who boasts of his ancestry praises the merits of another.” — Seneca
“A man who makes boast of his ancestors doth but advertise his own insignificance.” — Benjamin Franklin
“The man who has not anything to boast of but his illustrious ancestors is like a potato — the only good belonging to him is under ground.” — Sir Thomas Overbury
Podcast Episode 309: The ‘Grain of Salt’ Episode
Sometimes in our research we come across stories that are regarded as true but that we can’t fully verify. In this week’s episode of the Futility Closet podcast we’ll share two such stories from the 1920s, about a pair of New York fruit dealers and a mythologized bank robber, and discuss the strength of the evidence behind them.
We’ll also salute a retiring cat and puzzle over a heartless spouse.
Top Drawer
Introduced by Eberhard Faber in 1934, the Blackwing 602 premium writing pencil was stamped with the words “Half the Pressure, Twice the Speed”: Compared to an ordinary pencil, its core contained more graphite, less clay, and wax, so that it wrote like a pencil of 4B hardness but with a unique gliding feel.
It has attracted an impressive roster of creative admirers, including Walt Disney, Stephen Sondheim, Quincy Jones, Vladimir Nabokov, and John Steinbeck, who wrote, “I have found a new kind of pencil — the best I have ever had. Of course it costs three times as much too, but it is black and soft but doesn’t break off. I think I will always use these. They are called Blackwings and they really glide over the paper.”
Steinbeck would use a Blackwing pencil right down to the ferrule (pencil devotees now call this “Steinbeck stage”) and then pass them on to his son, another writer. “Writing with a Blackwing 602, more than any other pencil, feels like an event — something like a rite of passage for a pencil obsessive,” writes Caroline Weaver in The Pencil Perfect: The Untold Story of a Cultural Icon (2017). “When they are sold in my shop I always encourage the customer to sharpen it at least once and to use it for special occasions, because most of the pleasure of owning it comes from knowing what it feels like to write with it as much as it comes from the history.”
Clearance
In the 1928 film Steamboat Bill, Jr., a falling facade threatens to flatten Buster Keaton, but he’s spared by the fortunate placement of an open attic window. “As he stood in the studio street waiting for a building to crash on him, he noticed that some of the electricians and extras were praying,” writes Marion Meade in Cut to the Chase, her biography of Keaton. “Afterward, he would call the stunt one of his greatest thrills.”
It’s often said that the falling wall missed Keaton by inches. Is that true? James Metz studied the problem in Mathematics Teacher in 2019. Keaton was 5 feet 5 inches tall; if that the “hinge” of the facade is 5 inches above the surface of the ground, the attic window is 12 feet above that, and the window is 3 feet high, he finds that the top of the window came only within about 1.5 feet of Keaton’s head.
“The window was tall enough to allow an ample margin of safety, so the legend about barely missing his head cannot be true,” Metz writes. “Apparently, Keaton had more headroom than was previously suspected.”
(James Metz, “The Right Place at the Right Time,” Mathematics Teacher 112:4 [January/February 2019], 247-249.)
All for One
In 1988, Florida International University mathematician T.I. Ramsamujh offered a proof that all positive integers are equal. “The proof is of course fallacious but the error is so nicely hidden that the task of locating it becomes an interesting exercise.”
Let p(n) be the proposition, ‘If the maximum of two positive integers is n then the integers are equal.’ We will first show that p(n) is true for each positive integer. Observe that p(1) is true, because if the maximum of two positive integers is 1 then both integers must be 1, and so they are equal. Now assume that p(n) is true and let u and v be positive integers with maximum n + 1. Then the maximum of u – 1 and v – 1 is n. Since p(n) is true it follows that u – 1 = v – 1. Thus u = v and so p(n + 1) is true. Hence p(n) implies p(n+ 1) for each positive integer n. By the principle of mathematical induction it now follows that p(n) is true for each positive integer n.
Now let x and y be any two positive integers. Take n to be the maximum of x and y. Since p(n) is true it follows that x = y.
“We have thus shown that any two positive integers are equal. Where is the error?”
(T.I. Ramsamujh, “72.14 A Paradox: (1) All Positive Integers Are Equal,” Mathematical Gazette 72:460 [June 1988], 113.)
Mirror Therapy
When a limb is paralyzed and then amputated, the patient may perceive a “phantom limb” in its place that is itself paralyzed — the brain has “learned” that the limb is paralyzed and has not received any feedback to the contrary.
University of California neuroscientist V.S. Ramachandran found a simple solution: The patient holds the intact limb next to a mirror, looks at the reflected image, and makes symmetric movements with both the good and the phantom limb. In the reflected image, the brain is now able to “see” the phantom limb moving. The impression of paralysis lifts, and the patient can now move the phantom limb out of painful positions.
A 2018 review called the technique “a valid, simple, and inexpensive treatment for [phantom-limb pain].”