Zipf’s Law

In natural language, the most frequent word occurs about twice as often as the second most frequent word, three times as often as the third most frequent word, and so on.

In the Brown Corpus, a text collection of a million words, the most frequent word, the, accounts for 7.5% of all word occurrences, and the second most frequent, of, accounts for 3.5%. A mere 135 vocabulary items account for half the corpus, and about half the total vocabulary of about 50,000 words are hapax legomena, words that occur once only.

Similar distributions are found in data throughout the physical and social sciences; the law is named after the American linguist George Kingsley Zipf.

Stolpersteine

https://commons.wikimedia.org/wiki/File:Stolperstein_of_Frau_Liebermann.JPG
Image: Wikimedia Commons

The streets of Europe are studded with thousands of brass plates, each marking the last residence of an individual before their extermination or persecution by the Nazis. German artist Gunter Demnig began the project in 1992, installing the first plate before Cologne’s city hall to mark the 50th anniversary of Heinrich Himmler’s “Auschwitz decree” ordering the deportation of Sinti and Roma to extermination camps. In the ensuing 15 years he laid more than 13,000 stolpersteine in more than 280 cities, and last October the 70,000th stolperstein was installed in Frankfurt, Germany, for Willy Zimmerer, who was “euthanized” in 1944 at age 43.

Each plate is engraved with the victim’s name and dates of birth, deportation, and death, as well as the words Hier wohnte … (“Here lived …”) to emphasize the immediacy of the memorial, “tripping up” passersby (stolperstein means “stumbling stone”). “I wanted to bring back the names of the Jews who lived, loved, had children and a normal life, who lived in these houses,” Demnig has said. “It’s my life. We can’t allow this part of history to pass into oblivion.”

(Thanks, Hanno.)

Podcast Episode 245: Jeanne Baret

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The first woman to circumnavigate the world did so dressed as a man. In 1766, 26-year-old Jeanne Baret joined a French expedition hoping to conceal her identity for three years. In this week’s episode of the Futility Closet podcast we’ll tell the story of her historic journey around the globe.

We’ll also hear Mark Twain’s shark story and puzzle over a foiled con artist.

See full show notes …

Riddles

From a collection in Frank Mittler’s Little Book of Word Tricks (1958):

1. Pray tell me, listener, if you can,
Who is that highly-favored man
Who, though he marries many a wife,
May still stay single all his life?

2. I sit in fire, but not in the flame;
I follow the master, but not the dame;
I’m found in the church, but not in the steeple;
I belong to the monarch, but not the people.

3. Its light was mellow, soft and lazy;
One foot broke off — and it went crazy!

4. What is found in the very center of both America and Australia?

5. What divides by uniting and unites by dividing?

6. Why is a popular crooner like a doctor in an asylum?

Click for Answer

Podcast Episode 244: The Women’s Protest

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Image: Wikimedia Commons

In February 1943, hundreds of German women joined in a spontaneous protest in central Berlin. They were objecting to the roundup of some of the city’s last Jews — their husbands. In this week’s episode of the Futility Closet podcast we’ll describe the Rosenstrasse protest, a remarkable example of civil disobedience.

We’ll also ponder whether a computer can make art and puzzle over some unusual phone calls.

See full show notes …

Cover Story

A set of points has diameter 1 if no two points in the set are more than 1 unit apart. An example is an equilateral triangle whose side has length 1. What’s the smallest shape that can cover any such set? A circle of diameter 1 won’t cover our triangle; part of the triangle projects beyond the circle:

https://commons.wikimedia.org/wiki/File:Lebesgue-circle-triangle.svg

Of course a larger circle would work, but what’s the smallest shape will always do the job? Surprisingly, no one knows. When French mathematician Henri Lebesgue posed the problem to Gyula Pál in 1914, Pál suggested a modified hexagon (in black):

https://commons.wikimedia.org/wiki/File:P%C3%A1l%27s_solution_to_Lebesgue%27s_universal_covering_problem.svg

Here Pál’s shape manages to surround a circle (blue), a Reuleaux triangle (red), and a square (green), each of diameter 1, and in fact it will accommodate any such set. Its own area is 0.84529946. Will a smaller shape do the job? Well, yes, but the gains get increasingly fine: In 1936 Roland Sprague whittled Pál’s shape down to 0.844137708436, and in 1992 H.C. Hansen reduced it further to 0.844137708398. At this point observers Victor Klee and Stanley Wagon wrote, “[I]t does seem safe to guess that progress on [this problem], which has been painfully slow in the past, may be even more painfully slow in the future.” But in 2015 John Baez reached 0.8441153 with an exquisite adjustment to two regions in Hansen’s shape; the smaller of these would span only a few atoms if the shape were drawn on paper.

Is that the end of the story? No: Last October Philip Gibbs claimed a further reduction to 0.8440935944, and the search goes on. In 2005 Peter Brass and Mehrbod Sharifi showed that the universal cover must have an area of at least 0.832, so there’s room, at least in theory, for still further improvements.

(Thanks, Jacob.)

His and Hers

In the Ubang language of Nigeria, men and women speak different languages. They understand each other perfectly, but “It’s almost like two different lexicons,” says anthropologist Chi Chi Undie. “There are a lot of words that men and women share in common, then there are others which are totally different depending on your sex. They don’t sound alike, they don’t have the same letters, they are completely different words”:

English Male Female
yam itong irui
clothing nki ariga
dog abu okwakwe
tree kitchi okweng
water bamuie amu
cup nko ogbala
bush bibiang déyirè
goat ibue obi

Raised by their mothers and other women, boys grow up speaking the female language, but at age 10 they’re expected to switch, unbidden, to the male. “There is a stage the male will reach and he discovers he is not using his rightful language,” says Chief Oliver Ibang. “Nobody will tell him he should change to the male language. … When he starts speaking the men language, you know the maturity is coming into him.”

“God created Adam and Eve and they were Ubang people,” he says. He had planned to give two languages to each ethnic group, but after the giving two to the Ubang he realized there were not enough languages to continue. “So he stopped. That’s why Ubang has the benefit of two languages — we are different from other people in the world.”

Grice’s Maxims

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What rules underlie natural conversation? In a lecture at Harvard in 1967, British philosopher H.P. Grice set out to specify them using a mathematical approach, as Euclid had done in plane geometry. First, he said, the participants in a conversation follow a Cooperative Principle:

Make your conversational contribution such as is required, at the stage at which it occurs, by the accepted purpose or direction of the talk exchange in which you are engaged.

Then he derived more specific principles under four headings:

  • Quantity
    1. Make your contribution as informative as is required.
    2. Do not make your contribution more informative than is required.
  • Quality
    1. Try to make your contribution one that is true.
    2. Do not say what you believe to be false.
    3. Do not say that for which you lack adequate evidence.
  • Relation
    1. Be relevant.
  • Manner
    1. Be perspicuous.
    2. Avoid obscurity of expression.
    3. Avoid ambiguity.
    4. Be brief.
    5. Be orderly.

These are useful, but they’re not axioms. “[I]t is possible to engage in a genuine and meaningful conversation and yet fail to observe one or more of the maxims Grice listed,” writes Stanford mathematician Keith Devlin. “The maxims seem more a matter of an obligation of some kind.” In Grice’s own words, “I would like to be able to think of the standard type of conversational practice not merely as something which all or most do in fact follow, but as something which it is reasonable for us to follow, which we should not abandon.”

(Keith Devlin, “What Will Count as Mathematics in 2100?”, in Bonnie Gold and Roger A. Simons, eds., Proof & Other Dilemmas: Mathematics and Philosophy, 2008.)

Secret Admirer

In 1952, strange love letters began to appear on the notice board of Manchester University’s computer department:

HONEY DEAR
YOU ARE MY FERVENT CHARM. MY AVID HEART ARDENTLY IS WEDDED TO YOUR DEVOTED LIKING. MY DEVOTED LOVE PANTS FOR YOUR HUNGER. MY HUNGER CHERISHES YOUR IMPATIENT CHARM. MY FONDNESS DEVOTEDLY PANTS FOR YOUR ADORABLE PASSION.
YOURS KEENLY
M.U.C.

DARLING SWEETHEART
YOU ARE MY AVID FELLOW FEELING. MY AFFECTION CURIOUSLY CLINGS TO YOUR PASSIONATE WISH. MY LIKING YEARNS FOR YOUR HEART. YOU ARE MY WISTFUL SYMPATHY: MY TENDER LIKING.
YOURS BEAUTIFULLY
M.U.C.

M.U.C. was the Manchester University Computer; professor Christopher Strachey was testing its ability to select information randomly by asking it to string romantic words into impromptu billets-doux. You can see the word lists, and generate your own love letter, here.