n. the action of carrying
adj. relating to gifts
n. a pair of things joined
n. the action of carrying
adj. relating to gifts
n. a pair of things joined
In 1836, Indians abducted a 9-year-old girl from her home in East Texas. She made a new life among the Comanche, with a husband and three children. Then, after 24 years, the whites abducted her back again. In this week’s episode of the Futility Closet podcast we’ll tell the story of Cynthia Ann Parker, caught up in a war between two societies.
We’ll also analyze a forger’s motives and puzzle over why a crowd won’t help a dying woman.
From reader Isaac Lubow:
In 2008 a Learjet operated by Kalitta Air was en route from Manassas, Va., to Ypsilanti, Mich., when the air traffic controller noted that the pilot’s microphone button was being pressed continuously. When he contacted the plane, the pilot told him in slow, slurred words, over the sound of audible alarms, that he was unable to maintain altitude, speed, or heading but that everything else was “A-OK.”
Euphoria is a sign of hypoxia. With the help of the pilot of a nearby aircraft, the controllers were able to understand that the Learjet had become depressurized. It turned out that the first officer had been completely unconscious, and his flailing arm had both disengaged the autopilot and keyed the microphone. The open microphone had alerted the controllers, and the need to hand-fly the plane had kept the pilot conscious and able to respond to their commands.
The pilot managed to descend from 32,000 feet to 11,000, where the crew recovered, and the plane landed safely at Detroit’s Willow Run Airport. Controllers Jay McCombs and Stephanie Bevins were awarded the Archie League Medal of Safety, and the episode is now used as a classroom teaching aid at the Civil Aerospace Medical Institute in Oklahoma City.
(From Fear of Landing. Thanks, Isaac.)
A triangular number is one that counts the number of objects in an equilateral triangle, as above:
1 + 2 = 3
1 + 2 + 3 = 6
1 + 2 + 3 + 4 = 10
1 + 2 + 3 + 4 + 5 = 15
1 + 2 + 3 + 4 + 5 + 6 = 21
Some of these numbers are palindromes, numbers that read the same backward and forward. A few examples are 55, 66, 171, 595, and 666. In 1973, Charles Trigg found that of the triangular numbers less than 151340, 27 are palindromes.
But interestingly, every string of 1s:
… is a palindromic triangular number in base nine. For example:
119 = 9 + 1 = 10
1119 = 92 + 9 + 1 = 91
11119 = 93 + 92 + 9 + 1 = 820
111119 = 94 + 93 + 92 + 9 + 1 = 7381
The pattern continues — all these numbers are triangular.
02/12/2017 UPDATE: Reader Jacob Bandes-Storch sent a visual proof:
“Given a number n in base 9, if we tack a 1 on the right, the resulting number is 9*n + 1. (By shifting over one place to the left, each digit becomes nine times its original value, and then we add 1 in the ones place.) So given a triangular number, there’s probably a way of sticking together 9 copies of it with a single additional unit to form a new triangle. Sure enough:”
From my notes, here’s a paradox he offered at a Copenhagen self-reference conference in 2002:
Have you heard of the LAA computing company? Do you know what LAA stands for? It stands for ‘lacking an acronym.’
Actually, the above acronym is not paradoxical; it is simply false. I thought of the following variant which is paradoxical — it is the LACA company. Here LACA stands for ‘lacking a correct acronym.’ Assuming that the company has no other acronym, that acronym is easily seen to be true if and only if it is false.
In 1927, Chicago investment bank Halsey, Stuart & Co. published a prospectus inviting its clients to consider investing in the burgeoning motion picture business. Among the illustrations was this Paramount Studios map of international shooting locations in California.
“It was not mere chance that established the motion picture industry in Southern California,” notes the booklet. “The actual localization of production there came through a process of pure competitive selection in which the geographical advantages favorable to producers in that region literally forced competing directors and their companies to come to California — and Hollywood.
“The advantages of dependable sunshine, permitting outdoor production without delays, and of great variety in scenery at close range (the ocean on one side, the deserts, mountains, and forests in other directions), so that the sequence of almost any picture can be suitably filmed with but little cost for travel — all have militated to established Hollywood as the center of the motion picture world.”
(“The Motion Picture Industry as a Basis for Bond Financing,” reprinted in Tino Balio, ed., The American Film Industry, 1985.)
Suppose we have a finite set of points in the plane, no three of which are collinear. A line drawn through one of them pivots around that point until it encounters another point, when it adopts that point as the new pivot. Call this line a “windmill”; it continues indefinitely, always rotating in the same direction. Show that we can choose an initial point and line so that the resulting windmill uses each point as a pivot infinitely many times.
Vernet relates, that he was once employed to paint a landscape, with a cave, and St. Jerome in it; he accordingly painted the landscape, with St. Jerome at the entrance of the cave. When he delivered the picture, the purchaser, who understood nothing of perspective, said, ‘the landscape and the cave are well made, but St. Jerome is not in the cave.’ ‘I understand you, Sir,’ replied Vernet, ‘I will alter it.’ He therefore took the painting and made the shade darker, so that the saint seemed to sit farther in. The gentleman took the painting; but it again appeared to him that the saint was not in the cave. Vernet then wiped out the figure, and gave it to the gentleman, who seemed perfectly satisfied. Whenever he saw strangers to whom he shewed the picture, he said, ‘Here you see a picture by Vernet, with St. Jerome in the cave.’ ‘But we cannot see the saint,’ replied the visitors. ‘Excuse me, gentlemen,’ answered the possessor, ‘he is there; for I have seen him standing at the entrance, and afterwards farther back; and am therefore quite sure that he is in it.’
— Thomas Byerley and Joseph Clinton Robertson, The Percy Anecdotes, 1821
In December 2013 a U.S. District Court decided that copyright in the fictional characters Sherlock Holmes and Dr. Watson had expired, but only for the characters as they’re depicted in the earlier novels by Arthur Conan Doyle. Aspects of the characters that are mentioned only in the later novels — such as Dr. Watson’s athletic background, first described in a 1924 short story — are considered new “increments of expression” of those characters, and remain protected.
That makes eminent sense for writers and lawyers, but what about poor Dr. Watson, anxiously stirring the fire at 221B Baker Street? Does he have an athletic background or doesn’t he? The copyright law seems to apply to a version of him that does, and not to one that doesn’t. Should we say there are two Dr. Watsons? That doesn’t seem right.
Worse, “If an author now wants to write a new Holmes novel, but is prohibited from mentioning almost everything pertaining to Professor Moriarty (who only rose to prominence in the later work Valley of Fear), how can we say that he is still writing about the ‘the same’ Holmes, given how much his character was formed through the interaction with his nemesis?” ask legal scholars Burkhard Schafer and Jane Cornwell. “Does this not render any new Holmes necessarily ‘incomplete,’ that is lacking character traits and memories Holmes is ‘known to’ possess, according to the canonical work?”
Even the “public domain” Holmes seems to multiply in this light. We learn that Holmes has an older brother, Mycroft, in “The Adventure of the Greek Interpreter,” published in 1893. But if Mycroft is older than Sherlock, then surely he’s been Sherlock’s brother ever since Sherlock’s birth in 1854. Does the early Sherlock (in, say, A Study in Scarlet) have a brother?
(Burkhard Schafer and Jane Cornwell, “Law’s Fictions, Legal Fictions and Copyright Law,” in Maksymilian Del Mar and William Twining, eds., Legal Fictions in Theory and Practice, 2015.)
In Pascal’s triangle, above, the number in each cell is the sum of the two immediately above it.
If you “tilt” the triangle so that each row starts one column to the right of its predecessor, then the column totals produce the Fibonacci sequence:
That’s from Thomas Koshy’s Triangular Arrays With Applications, 2011.
Bonus: Displace the rows still further and they’ll identify prime numbers.