Benardete’s String Paradox

benardete's string paradox

Let us take a piece of string. In the first half minute we shall form an equilateral triangle with the string; in the next quarter minute we shall employ the string to form a square; in the next eighth minute we shall form a regular pentagon; etc. ad infinitum. At the end of the minute what figure or shape will our piece of string be found to have assumed? Surely it can only be a circle. And yet how intelligible is that process? Each and every one of the polygons in our infinite series contains only a finite number of sides. There is thus a serious conceptual gap separating the circle, as in the limiting case, from each and every polygon in the infinite series.

— Jose Amado Benardete, Infinity: An Essay in Metaphysics, 1964

| Science & Math

Math Notes

Turn each of these palindromes “inside out” and their sum remains the same:

13031 + 42024 + 53035 + 57075 + 68086 + 97079 = 31013 + 24042 + 35053 + 75057 + 86068 + 79097

Remarkably, this holds true even if you square or cube them:

130312 + 420242 + 530352 + 570752 + 680862 + 970792 = 310132 + 240422 + 350532 + 750572 + 860682 + 790972

130313 + 420243 + 530353 + 570753 + 680863 + 970793 = 310133 + 240423 + 350533 + 750573 + 860683 + 790973

From Albert Beiler, Recreations in the Theory of Numbers, 1964.

| Science & Math

Menagerie

How many pets do I have if all of them are dogs except two, all are cats except two, and all are fish except two?

Click for Answer
| Puzzles

Running On

http://books.google.com/books?id=L_QPAAAAYAAJ

Cushing Biggs Hassell’s thousand-page History of the Church of God (1886) is notable for a single sentence — this one, on page 580, beginning “The nineteenth is the century …”

It’s six pages long, with 3,153 words, 360 commas, 86 semicolons, and six footnotes. Many regard it as the longest legitimate sentence ever published in a book.

Essentially it’s one long indictment of the 19th century, proving for Hassell that “after all our progress, this is still a very sinful and miserable world.” Why he felt he had to show this in a single sentence is not clear.

| Language · Literature

A Moveable Feast

http://books.google.com/books?id=J68vAAAAMAAJ&printsec=frontcover&source=gbs_v2_summary_r&cad=0#v=onepage&q&f=false

On March 28, 1903, industrialist and horse lover C.K.G. Billings hosted a one-of-a-kind dinner at Sherry’s Restaurant in New York. He covered the floor of the restaurant’s grand ballroom with turf and brought in 36 horses via the freight elevator. The diners passed the evening on horseback, eating from tables on their pommels and drinking champagne from chilled bottles in the saddlebags.

The bill for this came to $50,000, but that was nothing to Billings, who had just retired as president of the People’s Gas Light and Coke Company in Chicago and was celebrating the opening of a new stable in Manhattan. The horses got oats.

See Feeder of the Pack and Black Tie Optional.

| Oddities

Post Production

In 1929, aspiring actor Charles Loeb had a friend pack him into a box labeled “Statue–Handle With Care” and ship him from Chicago to the Pathé motion picture studios in Culver City, Calif.

He arrived four days later, nearly dead, but told police he was pleased he’d finally made it through the gates of a major studio.

See Special Delivery.

| Entertainment

A Valuable Oversight

http://www.google.com/patents/about?id=MiNeAAAAEBAJ

Patent examiners are busy people, and when this application arrived at the U.S. Patent Office in 1881 it seemed innocuous enough — the inventor, John Sutliff, had called it simply “motor.” So they issued the patent.

It is, in fact, a perpetual motion machine. When ball L rolls to the left, it depresses the bellows, which fills the submerged bulb, raising the lever and turning cogwheel F. This pivots the box, which sends the ball back to the right, drawing air into the bellows and submerging the bulb again, “and so on alternately.”

Thus the cogwheel turns forever, driving shaft H, which you can hook up to anything you like. A convenient source of endless free energy, and it’s been under our noses all this time.

| Technology

Alphabet Soup

A story has been told of a graceless scamp who gained access to the Clarendon printing-office in Oxford, when the forms of a new edition of the Episcopal prayer book had just been made up and were ready for the press. In that part of the ‘form’ containing the marriage service, he substituted the letter k for the letter v in the word live, and thus the vow to ‘love, honor, comfort,’ etc., ‘so long as ye both shall live,’ was made to read, ‘so long as ye both shall like.’ The change was not discovered until the whole of the edition was printed off.

— Ballou’s Monthly Magazine, October 1870

When Teddy Roosevelt was sworn in in 1901, the New York Times printed a B in place of an O in one story and recorded that “surrounded only by a few friends, Theodore Roosevelt took his simple bath to defend and carry out our Constitution.”

“The most amusing feature of the above,” reported the Bookman, “is due to an English newspaper which quoted the paragraph, did not recognise the misprint and went on to comment upon it with perfect seriousness.”

| Language

A Rather Long Resume

In 1867, the finance department of the city of Madrid employed a man named Don Juan Nepomuceno de Burionagonatotorecagageazcoecha.

If he had moved to northern Bohemia and become “deputy-president of the Food-Rationing-Winding-Up-Commission,” his job title would have been Lebensmittelzuschlusseinstellungskomissionsvorsitzenderstellvertreter.

In a perfect world he would then moonlight as a Gesundheitswiederherstellungsmittelzusammenmischungsverhältniskundiger, Bismarck’s term for an apothecary.

In his library he would keep the Antipericatametanaparbeugedamphicribrationes Toordicantium, mentioned by Rabelais in Gargantua and Pantagruel, and Thomas Love Peacock’s Headlong Hall, in which the phrenologist Mr. Cranium describes anatomical structures as osseocarnisanguineoviscericartilaginonervomedullary and osteosarchaematosplanchnochondroneuromuelous.

And he would vacation in Bristol, whose spa waters were described by the English medical writer Edward Strother (1675-1737) as aequeosalinocalcalinoceraceoaluminosocupreovitriolic.

Or perhaps he should just stay in Madrid.

| Language