# Flexagons

Create a strip of 19 triangles like the one above (printable version here) and fold the left portion back successively at each of the northeast-pointing lines to produce a spiral:

Fold this spiral backward along line ab:

Then fold the resulting figure backward at cd. You should be left with one blank triangular tab that can be folded backward and pasted to another blank panel on the opposite side. The resulting hexagon should have six 1s on one side and six 2s on the other.

With some adroit pinching this hexagon produces some marvelous effects. Fold down two adjacent triangles so that they meet, and then press in the opposite corner to join them. Now the top of the figure can be prised open and folded down to produce a new hexagon — this one with 1s on one face and a surprising blank on the second. What has become of the 2s?

Exploring the properties of this “hexahexaflexagon” offers an intuitive lesson in geometric group theory:

When Martin Gardner wrote about these bemusing creatures in his first column for Scientific American in 1956, he received two letters. The first was from Neil Uptegrove of Allen B. Du Mont Laboratories in Clifton, N.J.:

Sirs:

I was quite taken with the article entitled ‘Flexagons’ in your December issue. It took us only six or seven hours to paste the hexahexaflexagon together in the proper configuration. Since then it has been a source of continuing wonder.

But we have a problem. This morning one of our fellows was sitting flexing the hexahexaflexagon idly when the tip of his necktie became caught in one of the folds. With each successive flex, more of his tie vanished into the flexagon. With the sixth flexing he disappeared entirely.

We have been flexing the thing madly, and can find no trace of him, but we have located a sixteenth configuration of the hexahexaflexagon.

Here is our question: Does his widow draw workmen’s compensation for the duration of his absence, or can we have him declared legally dead immediately? We await your advice.

The second was from Robert M. Hill of The Royal College of Science and Technology in Glasgow, Scotland:

Sirs:

The letter in the March issue of your magazine complaining of the disappearance of a fellow from the Allen B. Du Mont Laboratories ‘down’ a hexahexaflexagon, has solved a mystery for us.

One day, while idly flexing our latest hexahexaflexagon, we were confounded to find that it was producing a strip of multicolored material. Further flexing of the hexahexaflexagon finally disgorged a gum-chewing stranger.

Unfortunately he was in a weak state and, owing to an apparent loss of memory, unable to give any account of how he came to be with us. His health has now been restored on our national diet of porridge, haggis and whisky, and he has become quite a pet around the department, answering to the name of Eccles.

Our problem is, should we now return him and, if so, by what method? Unfortunately Eccles now cringes at the very sight of a hexahexaflexagon and absolutely refuses to ‘flex.’

# Flash Mob

When Wilhelm Kieft tried to outlaw smoking in New Amsterdam in the 1630s, he brought on a unique protest. Washington Irving writes:

A mob of factious citizens had … the hardihood to assemble before the governor’s house, where, setting themselves resolutely down, like a besieging army before a fortress, they one and all fell to smoking with a determined perseverance, that seemed as though it were their intention to smoke him into terms. The testy William issued out of his mansion like a wrathful spider, and demanded to know the cause of this seditious assemblage, and this lawless fumigation; to which these sturdy rioters made no other reply, than to loll back phlegmatically in their seats, and puff away with redoubled fury; whereby they raised such a murky cloud, that the governor was fain to take refuge in the interior of his castle.

Wilhelm finally gave in — people could smoke, he said, but they had to give up long pipes. “Thus ended this alarming insurrection, which was long known by the name of the pipe plot, and which, it has been somewhat quaintly observed, did end, like most other plots, seditions, and conspiracies, in mere smoke.”

(Thanks, Dan.)

# Black and White

By Alexander Yarosh. The position above was reached in a legal game, except that one piece has been knocked off the board. What was it?

# Saving Face

“The Joker,” a picture-preserving geomagic square by Lee Sallows. The 16 pieces can be assembled in varying groups of 4 to produce the same picture in 16 different ways, without rotation or reflection.

The outline need not be a joker — it can take almost any shape.

# In a Word

titivil

n. “Name for a devil said to collect fragments of words dropped, skipped, or mumbled in the recitation of divine service, and to carry them to hell, to be registered against the offender.” [OED]

# Low Profile

In 2008 L.A. Innes of Jamestown, Saint Helena, auctioned a collection of images taken during the Boer War. This one shows a prisoner standing next to a tortoise on the island. The tortoise was mature at the time of the photograph, which was taken in 1900, and investigators were surprised to find that he’s still alive — “Jonathan” lives on the grounds of the governor’s residence, blind in one eye but still active and mating with other tortoises.

If he was 70 at the time Innes’ photograph was taken, then he’s 184 today — the oldest living reptile on earth.

# Small World

I don’t think this was ever built — in 1904 engineer Hiram Stevens Maxim designed an amusement with a rotating parabolic floor “for producing illusionary effects”:

With such a contrivance when persons enter the hollow sphere they will not be able to tell whether it is revolving or standing still and by reason of the parabolic floor, persons near the outer edge would, to the persons standing near the centre, appear to be walking with their heads directed inward. When the sphere revolves some curious phenomena will be obtained in walking outward and inward on such a floor, and the throwing of a ball from the centre outward and vice versa will move in an unexpected direction that will be very puzzling to the people in the sphere.

Fig. 3, below, shows the perspective from the edge of the floor as it rotates. If mirrors were positioned overhead, as in Fig. 4, “people could then be made to appear to be walking all over the inside of the sphere with their heads pointing inward and their feet pointing outward.”

# The Just World Phenomenon

In 1966 University of Kentucky social psychologist Melvin Lerner asked 72 undergraduate women to observe a peer working on a learning task. When the learner made an error she appeared to receive a painful electric shock. In describing her suffering, the observers tended to reject and devalue her when they thought they would continue to see her suffer in a later session.

Lerner suggested that we come to terms with the suffering we see around us by deciding that the world is just — that those who are unfortunate somehow deserve their fate, and thus that we can avoid such a fate ourselves. This is reflected in figures of speech such as “You reap what you sow” and “He got what was coming to him.”

“If people did not believe that they could get what they want and avoid what they abhor by performing certain appropriate acts, they would be virtually incapacitated,” Lerner wrote. “If this is true, then the person who sees suffering or misfortune will be motivated to believe that the unfortunate victim in some sense merited his fate.”

(Melvin J. Lerner and Carolyn H. Simmons, “Observer’s reaction to the ‘innocent victim’: Compassion or rejection?”, Journal of Personality and Social Psychology 4(2) [August 1966], 203-210.)

# Seeing Double

In 1944, as the Allies were preparing to invade France, British Intelligence sought a way to confuse the Germans as to their plans. They hired Meyrick Clifton James (right), an Australian-born lieutenant in the Army Pay Corps who bore a striking resemblance to Field Marshal Bernard Montgomery, who would be commanding the Allied ground troops during the invasion. David Niven, then a colonel in the Army Film Unit, invited James to London under cover as a journalist, and James set about studying the general’s speech patterns and mannerisms. Then he was conspicuously dispatched, as Monty, to Gibraltar and then to Algiers, watched by avid German spies.

It seemed to work. The plot went through “from start to finish without a hitch,” MI5 reported, “and we knew that the main feature of its story had reached the Germans.” The real Monty led the successful landings at Normandy while James recovered from the ordeal in a safe house in Cairo. “He was under terrible pressure and strain,” reported the wife of an intelligence officer detailed to look after him. “Coming out of that part was very difficult for him.” But he had some consolation: Under army rules, he would receive the equivalent of a general’s pay for every day he had impersonated Monty.

# Podcast Episode 16: A Very Popular Sack of Flour

In 1864 Nevada mining merchant Reuel Gridley found a unique way to raise money for wounded Union soldiers: He repeatedly auctioned the same 50-pound sack of flour, raising \$250,000 from sympathetic donors across the country.

In this episode of the Futility Closet podcast we’ll discover the origins of Gridley’s floury odyssey. We’ll also hear H.L. Mencken’s translation of the Declaration of Independence into American English and try to figure out where tourism increases the price of electricity.

Sources for our story on Reuel Gridley and the flour auction:

Ralph Lea and Christi Kennedy, “Reuel Gridley and a Sack of Flour,” Lodi, Calif., News-Sentinel, Sept. 30, 2005.

Mark Twain, Roughing It, 1872.

Here’s his monument, in the Stockton Rural Cemetery in California:

The empty flour sack is in the collection of the Nevada Historical Society.

“The Declaration of Independence in American,” by H.L. Mencken, from The American Language, 1921:

When things get so balled up that the people of a country have to cut loose from some other country, and go it on their own hook, without asking no permission from nobody, excepting maybe God Almighty, then they ought to let everybody know why they done it, so that everybody can see they are on the level, and not trying to put nothing over on nobody.

All we got to say on this proposition is this: first, you and me is as good as anybody else, and maybe a damn sight better; second, nobody ain’t got no right to take away none of our rights; third, every man has got a right to live, to come and go as he pleases, and to have a good time however he likes, so long as he don’t interfere with nobody else. That any government that don’t give a man these rights ain’t worth a damn; also, people ought to choose the kind of goverment they want themselves, and nobody else ought to have no say in the matter. That whenever any goverment don’t do this, then the people have got a right to can it and put in one that will take care of their interests. Of course, that don’t mean having a revolution every day like them South American coons and yellow-bellies and Bolsheviki, or every time some job-holder does something he ain’t got no business to do. It is better to stand a little graft, etc., than to have revolutions all the time, like them coons and Bolsheviki, and any man that wasn’t a anarchist or one of them I. W. W.’s would say the same. But when things get so bad that a man ain’t hardly got no rights at all no more, but you might almost call him a slave, then everybody ought to get together and throw the grafters out, and put in new ones who won’t carry on so high and steal so much, and then watch them. This is the proposition the people of these Colonies is up against, and they have got tired of it, and won’t stand it no more. The administration of the present King, George III, has been rotten from the start, and when anybody kicked about it he always tried to get away with it by strong-arm work. Here is some of the rough stuff he has pulled:

He vetoed bills in the Legislature that everybody was in favor of, and hardly nobody was against.

He wouldn’t allow no law to be passed without it was first put up to him, and then he stuck it in his pocket and let on he forgot about it, and didn’t pay no attention to no kicks.

When people went to work and gone to him and asked him to put through a law about this or that, he give them their choice: either they had to shut down the Legislature and let him pass it all by him-self, or they couldn’t have it at all.

He made the Legislature meet at one-horse thank-towns out in the alfalfa belt, so that hardly nobody could get there and most of the leaders would stay home and let him go to work and do things as he pleased.

He give the Legislature the air, and sent the members home every time they stood up to him and give him a call-down.

When a Legislature was busted up he wouldn’t allow no new one to be elected, so that there wasn’t nobody left to run things, but anybody could walk in and do whatever they pleased.

He tried to scare people outen moving into these States, and made it so hard for a wop or one of them poor kikes to get his papers that he would rather stay home and not try it, and then, when he come in, he wouldn’t let him have no land, and so he either went home again or never come.

He monkeyed with the courts, and didn’t hire enough judges to do the work, and so a person had to wait so long for his case to come up that he got sick of waiting, and went home, and so never got what was coming to him.

He got the judges under his thumb by turning them out when they done anything he didn’t like, or holding up their salaries, so that they had to cough up or not get no money.

He made a lot of new jobs, and give them to loafers that nobody knowed nothing about, and the poor people had to pay the bill, whether they wanted to or not.

Without no war going on, he kept an army loafing around the country, no matter how much people kicked about it.

He let the army run things to suit theirself and never paid no attention whatsoever to nobody which didn’t wear no uniform.

He let grafters run loose, from God knows where, and give them the say in everything, and let them put over such things as the following:

Making poor people board and lodge a lot of soldiers they ain’t got no use for, and don’t want to see loafing around.

When the soldiers kill a man, framing it up so that they would get off.

Making us pay taxes without asking us whether we thought the things we had to pay taxes for was something that was worth paying taxes for or not.

When a man was arrested and asked for a jury trial, not letting him have no jury trial.

Chasing men out of the country, without being guilty of nothing, and trying them somewheres else for what they done here.

In countries that border on us, he put in bum goverments, and then tried to spread them out, so that by and by they would take in this country too, or make our own goverment as bum as they was. He never paid no attention whatever to the Constitution, but he went to work and repealed laws that everybody was satisfied with and hardly nobody was against, and tried to fix the goverment so that he could do whatever he pleased.

He busted up the Legislatures and let on he could do all the work better by himself.

Now he washes his hands of us and even declares war on us, so we don’t owe him nothing, and whatever authority he ever had he ain’t got no more.

He has burned down towns, shot down people like dogs, and raised hell against us out on the ocean.

He hired whole regiments of Dutch, etc., to fight us, and told them they could have anything they wanted if they could take it away from us, and sicked these Dutch, etc., on us without paying no attention whatever to international law.

He grabbed our own people when he found them in ships on the ocean, and shoved guns into their hands, and made them fight against us, no matter how much they didn’t want to.

He stirred up the Indians, and give them arms ammunition, and told them to go to it, and they have killed men, women and children, and don’t care which.

Every time he has went to work and pulled any of these things, we have went to work and put in a kick, but every time we have went to work and put in a kick he has went to work and did it again. When a man keeps on handing out such rough stuff all the time, all you can say is that he ain’t got no class and ain’t fitten to have no authority over people who have got any rights, and he ought to be kicked out.

When we complained to the English we didn’t get no more satisfaction. Almost every day we warned them that the politicians over there was doing things to us that they didn’t have no right to do. We kept on reminding them who we were, and what we was doing here, and how we come to come here. We asked them to get us a square deal, and told them that if this thing kept on we’d have to do something about it and maybe they wouldn’t like it. But the more we talked, the more they didn’t pay no attention to us. Therefore, if they ain’t for us they must be agin us, and we are ready to give them the fight of their lives, or to shake hands when it is over.

Therefore be it resolved, That we, the representatives of the people of the United States of America, in Congress assembled, hereby declare as follows: That the United States, which was the United Colonies in former times, is now free and independent, and ought to be; that we have throwed out the English Kings and don’t want to have nothing to do with him no more, and are not in England no more; and that,being as we are now free and independent, we can do anything that free and independent parties can do, especially declare war, make peace, sign treaties, go into business, etc. And we swear on the Bible on this proposition, one and all, and agree to stick to it no matter what happens, whether we win or we lose, and whether we get away with it or get the worst of it, no matter whether we lose all our property by it or even get hung for it.

Sources for the gruesome story of the Smalls lighthouse:

Douglas Bland Hague, Lighthouses of Wales: Their Architecture and Archaeology, 1994.

Christopher Nicholson, Rock Lighthouses of Britain, 1983.

Nicholson writes that Howell’s “ordeal had affected him so greatly it was said that some of his friends did not recognize him on his return.”

You can listen using the player above, download this episode directly, or subscribe on iTunes or via the RSS feed at http://feedpress.me/futilitycloset. The show notes are on the blog. Many thanks to Doug Ross for the music in this episode.

If you have any questions or comments you can reach us at podcast@futilitycloset.com. Thanks for listening!