# The Plate Trick

Theoretical physicist Paul Dirac offered this example to show that some objects return to their original state after two full rotations, but not after one.

Hold a cup water in one hand and rotate it through 360 degrees (in either direction). You’ll have to contort yourself to accomplish this without spilling any water, but if you continue rotating the cup another 360 degrees in the same direction, you’ll find that you return to your original state.

The same principle can be demonstrated using belts. In the video below, the square goes through two full rotations and we find that the belts have returned to their original state. This would not be the case after a single rotation. (Here two belts are attached to the square, but the trick works with any number of belts.)

# The Eighth Plague

On June 15, 1875, physician Albert Childs was standing outside his office in Cedar Creek, Nebraska, when he saw the horizon darken. At first he was hopeful for some needed rain, but then he realized that the cloud was moving under its own power.

“And then suddenly it was on him, a trillion beating wings and biting jaws,” writes entomologist Steve Nicholls in Paradise Found (2009). It was an unusually huge swarm of Rocky Mountain locusts descended from the mountains. Stunned, Childs set about estimating its size:

Using the telegraph, he sent messages up and down the line and found the swarm front to be unbroken for 110 miles. With his telescope he estimated the swarm to be over half a mile deep, and he watched it pass for ‘five full days.’ He worked out that the locusts were traveling at around fifteen miles an hour and came up with the astonishing fact that the swarm was 1,800 miles long. This swarm covered 198,000 square miles, or, if it was transposed on to the east coast, it would have covered all the states of Connecticut, Delaware, Pennsylvania, Maryland, Maine, Massachusetts, New Jersey, New York, New Hampshire, Rhode Island, and Vermont.

“Albert Childs had recorded the largest ever swarm — the biggest aggregation of animals ever seen on planet Earth,” Nicholls writes. University of Wyoming entomologist Jeffrey Lockwood calls it the “Perfect Swarm.”

# Complementary Sequences

Another interesting item from James Tanton’s Mathematics Galore! (2012):

Write down a sequence of positive integers that never decreases. The list can include duplicates. As an example, here’s a list of primes:

2, 3, 5, 7, 11, 13

Call the sequence pn. Now, a “frequency sequence” records the number of members less than 1, less than 2, and so on. For the list of primes above, the frequency sequence is:

0, 0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6

Pleasingly, the frequency sequence of the frequency sequence of pn is pn. That is, if we take the frequency sequence of the list 0, 0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6 above, we get 2, 3, 5, 7, 11, 13 again.

Now add position numbers to each of the two lists, pn and its frequency sequence — that is, add 1 to the first element of each, 2 to the second, and so on. With the primes that gives us:

Pn: 3, 5, 8, 11, 16, 19 …

Qn: 1, 2, 4, 6, 7, 9, 10, 12, 13, 14, 15, 17, 18, 20 …

These two sequences will always be complementary — all the counting numbers appear, but they’re split between the two sequences, with no duplicates.

# Steps Back

For what it’s worth, here’s a dance from the 1780s:

1. Glissade round (first part of tune).
2. Double shuffle down, do.
3. Heel and toe back, finish with back shuffle.
4. Cut the buckle down, finish the shuffle.
5. Side shuffle right and left, finishing with beats.
6. Pigeon wing going round.
7. Heel and toe haul in back.
9. Changes back, finish with back shuffle and beats.
10. Wave step down.
11. Heel and toe shuffle obliquely back.
12. Whirligig, with beats down.
13. Sissone and entrechats back.
14. Running forward on the heels.
15. Double Scotch step, with a heel Brand in Plase. [sic]
16. Single Scotch step back.
17. Parried toes round, or feet in and out.
18. The Cooper shuffle right and left back.
19. Grasshopper step down.
20. Terre-a-terre [sic] or beating on toes back.
21. Jockey crotch down.
22. Traverse round, with hornpipe glissade.

It’s “A Sailor Hornpipe — Old Style,” by John Durang, George Washington’s favorite dancer. Durang taught it to his son Charles, who reproduced it in a study of theatrical dancing published in 1855, which is how it comes down to us.

The terminology is influenced by French ballet, but already it incorporates innovations such as “shuffles”; in time the hornpipe would evolve into modern tap dancing. In Tap Roots, Mark Knowles writes, “It is believed that the ‘whirligig, with beats down’ is similar to a renversé turn such as the kind later done by the tap dancing film star Eleanor Powell.”

(From Julian Mates, The American Musical Stage Before 1800, 1962.)

# De Gua’s Theorem

French mathematician Jean Paul de Gua de Malves discovered this three-dimensional analogue of the Pythagorean theorem in the 18th century.

If a tetrahedron has a right-angled corner (such as the corner of a cube), then the square of the area of the face opposite that corner is the sum of the squares of the areas of the other three faces.

Above,

$A_{ABC}^{2} = A_{ABO}^{2} + A_{ACO}^{2} + A_{BCO}^{2}$

# Short-Timer

The shortest jail sentence ever served in Washington state is one minute. From the Seattle Daily Times, Jan. 20, 1906:

[Joe] Munch is a soldier, on leave of absence. On the thirteenth day of August he found garrison life dull and proceeded to get drunk. A policeman found him in this condition and he was hustled off to the police station. In Judge Gordon’s court he was sentenced to thirty days for being drunk and disorderly, but his case was taken to the higher court.

Judge Frater decided that while the soldier’s crime was not enough to merit punishment, for the looks of things he ought to be sent to jail, and have a lesson taught him. Consequently Munch was sentenced to an imprisonment of one minute, something which the clerk who makes out the sentence documents never heard of before and which caused much merriment in court house circles.

“Those who heard the decision were inclined to take it as a joke of the judge’s, until Munch was hustled off to jail and kept there until the second hand of the jailer’s watch had completed the circle of sixty seconds. Munch was so surprised that he hardly knew what was going on and when released decided that the best thing for him to do was to get away for fear the sight of him should cause the judge to inflict a heavier penalty.”

(From the Washington State Library blog.)

“It is a curious thing that every creed promises a paradise which will be absolutely uninhabitable for anyone of civilized taste.” — Evelyn Waugh

“I have read descriptions of Paradise that would make any sensible person stop wanting to go there.” — Montesquieu

“In heaven, all the interesting people are missing.” — Friedrich Nietzsche

“Of the delights of this world man cares most for sexual intercourse, yet he has left it out of his heaven.” — Mark Twain

“I should have no use for a paradise in which I should be deprived of the right to prefer hell.” — Jean Rostand

# Moving Violation

A revealing anecdote from Mank, Richard Meryman’s 1978 biography of Herman J. Mankiewicz, co-writer of Citizen Kane:

Herman was a mischievous child. One day after some misdemeanor, Herman was confined to the house by his mother. To keep him there during her absence, she hid the long stockings he needed for his knickers. Herman went to his mother’s room, put on a pair of her stockings, got on his bike, and rode off to the Wilkes-Barre public library, where he loved to browse among the shelves and to read for hours. When he came out, the precious bike was gone — stolen. Herman’s punishment was permanent. His father never bought him another bike. His mother answered Herman’s pleas by telling him it was all his own fault.

Meryman concludes, “Rosebud, the symbol of Herman’s damaging childhood, was not a sled. It was a bicycle.”

# Words and Numbers

If you write out the numbers from 1 to 5000 in American English (e.g., THREE THOUSAND EIGHT HUNDRED SEVENTY-THREE), it turns out that only one of them has a unique number of characters. Which is it? Spaces and hyphens count as characters.

# Podcast Episode 88: Mrs. Wilkinson and the Lyrebird

Almost nothing was known about Australia’s elusive lyrebird until 1930, when an elderly widow named Edith Wilkinson encountered one on her garden path one February morning. In this week’s episode of the Futility Closet podcast we’ll follow the curious friendship that evolved between Wilkinson and “James,” which led to an explosion of knowledge about his reclusive species.

We’ll also learn how Seattle literally remade itself in the early 20th century and puzzle over why a prolific actress was never paid for her work.

Please consider becoming a patron of Futility Closet — on our Patreon page you can pledge any amount per episode, and all contributions are greatly appreciated. You can change or cancel your pledge at any time, and we’ve set up some rewards to help thank you for your support.

You can also make a one-time donation via the Donate button in the sidebar of the Futility Closet website.

Sources for our feature on Edith Wilkinson and James:

Ambrose Pratt, The Lore of the Lyrebird, 1933.

Nicolae Sfetcu, The Birds’ World, 2014.

Jackie Kerin, Lyrebird! a True Story, 2012.

“A.P.”, “A Miracle of the Dandenongs,” The Age, Feb. 13, 1932.

Anna Verona Dorris, “The Proud Aristocrat of Birdland,” New Outlook, July-August 1956.

Here’s the full lyrebird video we excerpted on the show:

More lyrebirds mimicking human technology on Futility Closet.

Listener mail:

“The Colorful Front-Gabled Italianate Homes at Damen and 33rd,” Chicago Patterns (accessed Jan. 1, 2016).

John McCarron, “Pilsen Comes Together to Preserve and Build,” LISC Chicago’s New Communities Program, May 3, 2007 (accessed Jan. 1, 2016).

Down to Earth: 9 Stories Above Pilsen (accessed Jan. 1, 2016).

A spite mound.