The Hadwiger–Nelson Problem

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Suppose we want to paint the plane so that no two points of the same color are a unit distance apart. What’s the smallest number of colors we need?

Surprisingly, no one knows. In the figure above, the hexagons are regular and have diameters slightly less than 1. Painting them as shown demonstrates that we can do the job successfully with seven colors.

Can we do it with fewer? In 1961 brothers William and Leo Moser showed that it’s impossible with three colors: They devised a graph (the “Moser spindle,” overlaid on the hexagons above) each of whose edges has length 1. The vertices of the spindle can’t be colored with three colors without both ends of some edge having the same color.

So we don’t need more than seven colors, but we need at least four. But whether the minimum needed is 4, 5, 6, or 7 remains unknown.

04/13/2018 UPDATE: By an amazing coincidence, there’s just been some progress on this. (Thanks, Jeff.)

| Science & Math

Applications

In 2004 University of Bristol mathematicians Hinke Osinga and Bernd Krauskopf crocheted a Lorenz manifold. They had developed a computer algorithm that “grows” a manifold in steps, and realized that the resulting mesh could be interpreted as a set of crochet instructions. After 85 hours and 25,511 stitches, Osinga had created a real-life object reflecting the Lorenz equations that describe the nature of chaotic systems.

“Imagine a leaf floating in a turbulent river and consider how it passes either to the left or to the right around a rock somewhere downstream,” she told the Guardian. “Those special leaves that end up clinging to the rock must have followed a very unique path in the water. Each stitch in the crochet pattern represents a single point [a leaf] that ends up at the rock.”

They offered a bottle of champagne to the first person who would produce another crocheted model of the manifold and received three responses in two weeks (and more since).

Of their own effort, Osinga and Krauskopf wrote, “While the model is not identical to the computer-generated Lorenz manifold, all its geometrical features are truthfully represented, so that it is possible to convey the intricate structure of this surface in a ‘hands-on’ fashion. This article tries to communicate this, but for the real experience you will have to get out your own yarn and crochet hook!” Their instructions are here.

(Hinke M. Osinga and Bernd Krauskopf, “Crocheting the Lorenz Manifold,” Mathematical Intelligencer 26:4 [September 2004], 25-37.)

| Science & Math

In a Word

ignoration
n. the state of being ignorant

debarrass
v. to disembarrass; to disencumber from anything that embarrasses

succedaneum
n. a substitute

arride
v. to please, gratify, delight

A ludicrous story is told of a great naval function which took place during the reign of the last Napoleon and the Empress Eugénie. Several American vessels were present, and they were drawn up in line to salute the Empress’s yacht as it passed. The French sailors, of course, manned the yards of their ships, and shouted ‘Vive l’Impératrice!’ The American Admiral knew that it was impossible to teach these words to his men in the time left to him, so he ordered his crew to shout ‘Beef, lemons, and cheese!’ The imperial yacht came on, and as it passed the fleet there was a mighty roar of ‘Beef, lemons, and cheese.’ And the Empress said she had never received such an ovation before.

— Current Literature, August 1893

UPDATE: Swansea poet Nigel Jenkins wrote an English phonetic version (not a translation) of the Welsh national anthem, so that Welsh people who don’t speak Welsh can join in:

My hen laid a haddock on top of a tree,
Glad barks and centurions throw dogs in the sea,
My guru asked Elvis and brandished Dan’s flan,
Don’s muddy bog’s blocked up with sand.
Dad, Dad! Why don’t you oil Aunty Glad?
When oars appear, on beer bottle pies,
Oh butter the hens as they fly.

I’m told this sounds convincing when sung with a Welsh accent in a crowd. Maybe they should just adopt these lyrics outright!

(Thanks, John.)

11/11/2018 In Sweden there’s a story that drought-stricken Scanian peasants assailed Queen Desideria with cries of Vi vill ha regn! (“We want rain!”) — the French-born queen was perplexed to hear “Vive la Reine!” (Thanks, Martin.)

| Language

Correspondence

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In September 1780 Erasmus Darwin and Anna Seward exchanged letters “in the name of their respective cats.” Darwin’s Persian cat Snow wrote to Miss Po Felina at the Bishop’s Palace:

Dear Miss Pussey,

As I sat, the other day, basking myself in the Dean’s Walk, I saw you in your stately palace, washing your beautiful round face, and elegantly brinded ears, with your velvet paws, and whisking about, with graceful sinuosity, your meandering tail. That treacherous hedgehog, Cupid, concealed himself behind your tabby beauties, and darting one of his too well-aimed quills, pierced, O cruel imp! my fluttering heart.

Ever since that fatal hour have I watched, day and night, in my balcony, hoping that the stillness of the starlight evenings might induce you to take the air on the leads of the palace. Many serenades have I sung under your windows; and, when you failed to appear, with the sound of my voice made the vicarage re-echo through all its winding lanes and dirty alleys. All heard me but my cruel Fair-one; she, wrapped in fur, sat purring with contented insensibility, or slept with untroubled dreams. …

Permit me this afternoon, to lay at your divine feet the head of an enormous Norway rat, which has even now stained my paws with its gore.

He received this response:

I am but too sensible of the charms of Mr. Snow; but while I admire the spotless whiteness of his ermine, and the tyger-strength of his commanding form, I sigh in secret, that he, who sucked the milk of benevolence and philosophy, should yet retain the extreme of that fierceness, too justly imputed to the Grimalkin race. Our hereditary violence is perhaps commendable when we exert it against the foes of our protestors, but deserves much blame when it annoys their friends. …

Marry you, Mr Snow, I am afraid I cannot; since, though the laws of our community might not oppose our connection, yet those of principle, of delicacy, of duty to my mistress, do very powerfully oppose it. …

The still too much admired Mr Snow will have the goodness to pardon the freedom of these expostulations, and excuse their imperfections. The morning, O Snow! had been devoted to this my correspondence with thee, but I was interrupted in that employment by the visit of two females of our Species, who fed my ill-starred passion by praising thy wit and endowments, exemplified by thy elegant letter, to which the delicacy of my sentiments obliges me to send so inauspicious a reply.

(From Seward’s Memoir of the Life of Dr. Darwin, 1804.)

| Literature

Diminuendo

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At his death in 1794, Czech composer František Xaver Pokorný had written more than 160 symphonies, concertos, serenades, and divertimentos. But more than half of them were then reattributed to other composers. The culprit was apparently Theodor von Schacht, a competing Regensburg composer who may have been jealous of Pokorný’s large output. After Pokorný’s death it appears that Schacht went through more than half his compositions, systematically removed Pokorný’s name, and inserted the name of another composer who he thought might not find out. He assigned most of the pieces to composers whose names began with A or B, which suggests that Schacht might have intended to eradicate Pokorný’s name entirely.

It wasn’t until the early 1960s that musicologists Jan la Rue and J. Murray Barbour uncovered this strange crime and Pokorný was given proper credit. “It would seem as if Baron von Schacht had been galled by the fact that his lowly colleague (the name ‘Pokorny’ means ‘humble’) had written six or seven times as many orchestral works as he had himself,” Barbour wrote. “So, after Pokorny’s death, he had tried to bring him down to his level by falsifying … 109 works. This is a most extraordinary piece of jealousy and arrogance. But, after all, he almost did get away with it!”

(J. Murray Barbour, “Pokorny Vindicated,” Musical Quarterly 49:1 [January 1963], 38-58.)

| Art

Triangle

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Violinist Georges Enesco was saddled with a poor pupil who eventually wanted to give a recital. Enesco agreed to accompany him on the piano but realized at the last minute that he needed a page turner. He prevailed on Alfred Cortot, who was sitting in the audience. A review the next morning read:

“There was a most remarkable concert last night at the Salle Pleyel. The man who should have been playing the violin was playing the piano, the man who should have been playing the piano was turning the pages, and the man who should have been turning the pages was playing the violin.”

(Likewise: “It is a maxim among practical statisticians that ‘The data you need are not the data you have, the data you have are not the data you want, and the data you want are not the data you need.'” — T.W. Körner, The Pleasures of Counting, 1996)

| Art

Unreason

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As an exercise at the end of his 1887 book The Game of Logic, Lewis Carroll presents pairs of premises for which conclusions are to be found:

  • No bald person needs a hair-brush; No lizards have hair.
  • Some oysters are silent; No silent creatures are amusing.
  • All wise men walk on their feet; All unwise men walk on their hands.
  • No bridges are made of sugar; Some bridges are picturesque.
  • No frogs write books; Some people use ink in writing books.
  • Some dreams are terrible; No lambs are terrible.
  • All wasps are unfriendly; All puppies are friendly.
  • All ducks waddle; Nothing that waddles is graceful.
  • Bores are terrible; You are a bore.
  • Some mountains are insurmountable; All stiles can be surmounted.
  • No Frenchmen like plum-pudding; All Englishmen like plum-pudding.
  • No idlers win fame; Some painters are not idle.
  • No lobsters are unreasonable; No reasonable creatures expect impossibilities.
  • No fossils can be crossed in love; Any oyster may be crossed in love.
  • No country, that has been explored, is infested by dragons; Unexplored countries are fascinating.
  • A prudent man shuns hyaenas; No banker is imprudent.
  • No misers are unselfish; None but misers save egg-shells.
  • All pale people are phlegmatic; No one, who is not pale, looks poetical.
  • All jokes are meant to amuse; No Act of Parliament is a joke.
  • No quadrupeds can whistle; Some cats are quadrupeds.
  • Gold is heavy; Nothing but gold will silence him.
  • No emperors are dentists; All dentists are dreaded by children.
  • Caterpillars are not eloquent; Jones is eloquent.
  • Some bald people wear wigs; All your children have hair.
  • Weasels sometimes sleep; All animals sometimes sleep.
  • Everybody has seen a pig; Nobody admires a pig.

He gives no solutions, so you’re on your own.

| Literature · Science & Math