Beatty Sequences

Here’s another interesting source of complementary sequences. Take any positive irrational number, say $\sqrt{2}$ , and call it X. Call its reciprocal Y; in this case $Y = 1/\sqrt{2} = \sqrt{2}/2$ , or about 0.7. Add 1 to each of X and Y and we get

1 + X ≈ 2.4

1 + Y ≈ 1.7.

Now make a table of the approximate multiples of 1 + X and 1 + Y:

beatty sequences

If we drop the fractional part of each number in the table, we’re left with two complementary sequences — every number 1, 2, 3, … appears in one sequence or the other, but never in both.

They’re called Beatty sequences, after Sam Beatty of the University of Toronto, who discovered them in 1926. A pretty proof by A. Ostrowski and J. Hyslop appears in the March 1927 issue of the American Mathematical Monthly and in Ross Honsberger’s Ingenuity in Mathematics (1970).

January 27, 2016January 26, 2016 | Science & Math

A House Afire

If box-office receipts are adjusted for inflation, Gone With the Wind is still the highest-grossing film of all time, with earnings of $3.4 billion in 2014 dollars.

After the film’s 1939 premiere in Atlanta, playwright Moss Hart wired producer David O. Selznick:

OH, ALL RIGHT, GO AHEAD AND HAVE A VULGAR COMMERCIAL SUCCESS!

January 26, 2016 | Entertainment

A Hidden Economy

During the American Civil War, enemy soldiers would sometimes meet to barter. Tobacco was hard to get in the North, and coffee was scarce in the South, so, where it could be done safely, soldiers would meet between the lines to trade.

In some cases this was done across distances. If a river or lake separated the lines, a tiny boat would be laden with commodities and sent to the other side, where it would be unloaded and filled with exchange cargoes, as agreed on by shouting and signaling across the water. On the Rappahannock early in 1863 a group of New Jersey soldiers received a shipment “by miniature boat six inches long.” It carried this note:

Gents U.S. Army

We send you some tobacco by our Packet. Send us some coffee in return. Also a deck of cards if you have them, and we will send you more tobacco. Send us any late papers if you have them.

Jas. O. Parker
Co. H. 17th Regt. Miss. Vols.

Alfred S. Roe, who served in a New York artillery unit, recalled that near Petersburg in the winter of 1864, “a certain canine of strictly impartial sentiments” was “taught to respond to a whistle from either side. Thus with a can of coffee suspended from his neck he would amble over to the Johnnies, and when they had replaced coffee with tobacco he would return in obedience to Union signals, intent only on the food reward both sides gave him.”

(From Bell I. Wiley, The Life of Billy Yank: The Common Soldier of the Union, 1952.)

January 26, 2016January 25, 2016 | History · Society

Unquote

“Faced with the choice between changing one’s mind and proving that there is no need to do so, almost everyone gets busy on the proof.” — John Kenneth Galbraith

January 25, 2016 | Quotations

Podcast Episode 91: Voyage of the Damned

In 1939, an ocean liner carrying 900 Jewish refugees left Nazi Germany seeking sanctuary in North America, but it was turned away by every nation it appealed to. In this week’s episode of the Futility Closet podcast we’ll follow the so-called “voyage of the damned” and the plight of its increasingly desperate passengers.

We’ll also discuss the employment prospects for hermits in Seattle and puzzle over the contentment of a condemned woman.

See full show notes …

January 25, 2016October 19, 2017 | History · Podcast

Point of Interest

https://commons.wikimedia.org/wiki/File:Jungholz_im_Bezirk_RE.png — Image: Wikimedia Commons

The Austrian village of Jungholz is connected to the rest of the country by a single point, the summit of the mountain Sorgschrofen. The summit marks a “quadripoint,” a point that touches four distinct territories — here, Austria to the north and south and Germany to the east and west.

This raises some philosophical questions. Suppose a stubborn, infinitely skinny Austrian leaves Schattwald (in the south) to visit Jungholz at the same moment that an equally stubborn, infinitely skinny German leaves Bad Hindelang (in the west) to visit his grandmother in Pfronten (in the east). What happens? They can’t sidle past one another; the countries meet at a single, indivisible point. If neither will stand aside while the other passes, then the only solution I can see is for one to hop briefly onto the other’s shoulders.

Now suppose a border guard accosts them at this moment. The two occupy the same point on the Earth’s surface, yet one claims to be in Austria while the other is in Germany. Are they both right? And suppose that the lower man is killed by being jumped upon (he is very attenuated, after all). Which jurisdiction has authority over the crime scene? Who should judge the crime?

Further confusing borders: the Belgian town of Baarle-Hertog and, until recently, Cooch Behar in the Indian state of West Bengal.

(Thanks, Dan.)

January 24, 2016January 23, 2016 | Oddities

Enlightenment

Biologist F.W. Went points out that the physical size of human beings was a critical factor in their mastery of fire. Any flame must maintain a certain size in order to sustain the ignition temperature of its fuel, and a wood or coal fire in particular radiates so much heat that it must maintain a fairly large critical mass in order to keep burning; a small fire will go out.

“Interestingly enough,” Went writes, “a wood or coal fire above the critical size produces just the right amount of heat to warm man in a cave, or a room, or a camping site. But ants or small rodents would have to keep too far away to make a fire economical, or rather, they would be unable to bring up enough wood to keep the fire going. Therefore in an ant society fire is not an economical possibility, and they have developed without its benefits, by operating only while outside temperatures are within the physiological range. Man on the other hand has been able to move into very cold areas by using fire.”

“Man, with his remarkable brain, developed the use of fire, but … only a creature of man’s size could effectively control that fire,” writes Peter S. Stevens in Patterns in Nature (1974). “It happens that a small campfire is the smallest fire that is reliable and controllable. A still smaller flame is too easily snuffed out and a larger one too easily gets out of control. Prometheus was just large enough to feed the flames and to keep from getting burnt.”

(F.W. Went, “The Size of Man,” American Scientist, 56:4 [Winter 1968], 400-413.)

January 23, 2016January 22, 2016 | Science & Math · Technology

The Qattara Depression Project

In 1957 the CIA proposed a novel way to bring peace to the Middle East: Flood Egypt’s Qattara Depression with seawater from the Mediterranean. The depression is below sea level and currently an arid waste; connecting it to the sea with giant tunnels or canals would transform it into a lagoon that would be renewed perpetually as the water evaporated, creating a continuous flow would produce endless hydroelectricity for the region.

The idea had begun with John Ball, English director of the Survey of Egypt, in 1927. In presenting it to Dwight Eisenhower, the CIA observed that the lagoon would be “spectacular and peaceful,” that it would “materially alter the climate in adjacent areas,” and that it would “provide work during construction and living areas after completion for the Palestinian Arabs.”

Eisenhower turned it down, but the project has never fallen entirely off the drawing board. In the early 1970s German hydraulic engineer Friedrich Bassler proposed detonating more than 200 nuclear bombs to excavate the canal, which would have meant displacing 25,000 evacuees. That proposal was turned down for ecological reasons, but another solution might yet be found.

See Atlantropa and Dam Ambitious.

January 22, 2016January 21, 2016 | Technology

Something New

The eminent tragedian [William Charles Macready] opened in Lear, our property-man received his plot for the play in the usual manner, a map being required among the many articles (map highly necessary for Lear to divide his kingdom). The property-man being illiterate, read ‘mop’ for ‘map.’ At night the tragedy commences; Macready in full state on his throne calls for his map, a ‘super’ noble, kneeling, presents the aging king a white curly mop. The astounded actor rushed off the stage, dragging the unfortunate nobleman and his mop with him, actors and audience wild with delight.

— Edmund Stirling, Old Drury Lane, 1881, quoted in Ralph Berry, ed., The Methuen Book of Shakespeare Anecdotes, 1992

January 21, 2016January 21, 2016 | Literature

A Pitched Battle

In The Book of the Harp (2005), John Marson mentions a musical oddity — in 1932, a committee devoted to equal temperament was so incensed at the Royal Schools of Music that it hauled them before London’s Central Criminal Court for obtaining money under false pretenses. From The Music Lover magazine, April 30, 1932:

There is a touch of knight errantry about the action of Lennox Atkins F.R.C.O., hon. sec. of the Equal Temperament Committee, in applying at Bow Street for process against the Associated Board of Examiners in Music on the grounds that they were not qualified to know whether the music was being played in tune or not, and that therefore their diplomas were valueless. It certainly savours of the ‘ingenious gentleman’ of La Mancha who tilted at windmills. The temperament question seems to have upon those who take it up an effect similar to that which temperament produces in a prima donna. They become, to say the least, unreasonable. Happily Mr Fry, the magistrate, decided that this was not a matter for a criminal court, so that Sir John B. McEwan and Sir Hugh Allen are not to be shot at dawn, as was at first feared.

McEwan headed the Royal Academy of Music and Allen the Royal College of Music at the time. I find a bit more in the Musical Times, June 1, 1932:

Candidates were allowed to pass off the tuner’s scale as their own, and to obtain certificates to which, the E.T.C. claimed, they were not in equity entitled. Every sound produced was the tuner’s and not the candidate’s. Famous examiners, such as the late Sir Frederick Bridge, had wrongly passed thousands of candidates in keyed instrument examinations. From the point of view of the E.T.C., the candidates were not really examined at all.

The magistrate added that if it was thought that the examiners’ knowledge was insufficient then civil proceedings might be undertaken.

“We have only once before heard of the Equal Temperament Committee — a long while ago — and we were, and are still, vague as to its aims,” noted the Musical Times. “We had imagined it to be a learned Society that met from time to time to exchange light and airy chat about ratios, partials, mesotonics, and other temperamental details. But it seems that it is a body with a Mission, though we are not clear what that Mission is. Judging from the Bow Street evidence, the Committee’s aim is to make ‘Every Musician His Own Tuner’ — which seems rather rough on real tuners.”

January 21, 2016January 20, 2016 | Art · Crime · Oddities