Signposts

The “tombstone” (∎) used to denote the end of a proof was suggested by mathematician Paul Halmos. “The symbol is definitely not my invention,” he wrote. “It appeared in popular magazines (not mathematical ones) before I adopted it, but, once again, I seem to have introduced it into mathematics. It is the symbol that sometimes looks like ∎, and is used to indicate an end, usually the end of a proof. It is most frequently called the ‘tombstone’, but at least one generous author referred to it as the ‘halmos’.”

https://commons.wikimedia.org/wiki/File:Knuth%27s_dangerous_bend_symbol.svg
Image: Wikimedia Commons

The group of mathematicians who wrote under the name Nicolas Bourbaki would include a “dangerous bend” symbol in the margin next to tricky or difficult passages, “to forewarn the reader against serious errors, where he risks falling.” Other writers have adopted the symbol, including computer scientist Donald Knuth, who included American-style road signs in his Metafont and TeX typesetting systems.

https://commons.wikimedia.org/wiki/File:Krul.svg
Image: Wikimedia Commons

And teachers in the Netherlands use a distinctive “flourish of approval” when grading schoolwork to show that they have seen and agreed with a paragraph. The mark, which may have evolved from a hastily written g (for “good” [goed] or “seen” [gezien]), rarely appears outside the Netherlands and its former colonies.

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Turbulence

The roots of the word helicopter are not heli and copter but helico and pter, from the Greek “helix” (spiral) and “pteron” (wing).

G.L.M. de Ponton’s 1861 British patent says, “The required ascensional motion is given to my aerostatical apparatus (which I intend denominating aeronef or helicoptere,) by means of two or more superposed horizontal helixes combined together.”

| Language · Technology

Scattered Stars

When I heard the learn’d astronomer,
When the proofs, the figures, were ranged in columns before me,
When I was shown the charts and diagrams, to add, divide, and measure them,
When I sitting heard the astronomer where he lectured with much applause in the lecture-room,
How soon unaccountable I became tired and sick,
Till rising and gliding out I wander’d off by myself,
In the mystical moist night-air, and from time to time,
Look’d up in perfect silence at the stars.

That’s Walt Whitman. In 2000, mathematician Mike Keith noted a similar idea in Psalm 19:1-6:

The heavens declare the glory of God;
And the firmament sheweth his handywork.
Day unto day uttereth speech,
And night unto night sheweth knowledge.
There is no speech nor language,
Where their voice is not heard.
Their line is gone out through all the earth,
And their words to the end of the world.
In them hath he set a tabernacle for the sun,
Which is as a bridegroom coming out of his chamber,
And rejoiceth as a strong man to run a race.
His going forth is from the end of the heaven,
And his circuit unto the ends of it:
And there is nothing hid from the heat thereof.

So he married them by rearranging the psalm’s letters:

When I had listened to the erudite astronomer,
When his high thoughts were arranged and charted before me,
When I was shown the length and breadth and height of it,
The Earth, the horned Moon, the chariot of fire,
The hundredth flight of the shuttle through heavyish air,
How soon, mysteriously, I became sad and sick,
Had to wander out, ousted, charging through the forest,
Joining the sure chaos here in a foreign heath,
Having forgotten the vocation of the learned man,
And in the mystic clearing, once more looked up
In perfect silence at the sermon in the stars.

(Michael Keith, “Anagramming the Bible,” Word Ways 33:3 [August 2000], 180-185.)

| Language · Poems

Misc

https://pixabay.com/en/diamonds-eight-deck-playing-cards-884176/

  • The negative space in the eight of diamonds forms an 8.
  • William Brewster, leader of the Plymouth Colony, named his children Jonathan, Patience, Fear, Love, and Wrestling.
  • Wilfred Owen’s mother received the news of his death on Armistice Day.
  • SCHOOLMASTER = SMOTE SCHOLAR
  • “I never had any philosophic instruction, the first lecture on psychology I ever heard being the first I ever gave.” — William James
| History · Language · Literature · Oddities · Quotations

The Vowel Triangle

Chris McManus discovered this oddity. If W and Y are accepted as vowels, that gives us AEIOUWY. Starting with O, number these according to their positions on a circular alphabet without starting the count over for A (that is, O is the 15th letter of the alphabet, so it’s assigned number 15; beyond Z we’d reach A as the “27th” letter; and so on). Now write these numbers into a triangle, again starting with O:

   O                15
 U W Y           21 23 25
A  E  I         27  31  35

Each of the five lines in the figure gives a different arithmetic progression:

UWY: difference of 2
AEI: difference of 4
OUA: difference of 6
OWE: difference of 8
OYI: difference of 10

(David Morice, “Kickshaws,” Word Ways 34:4 [November 2001], 292-305.)

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Fortuitous Numbers

In American usage, 84,672 is said EIGHTY FOUR THOUSAND SIX HUNDRED SEVENTY TWO. Count the letters in each of those words, multiply the counts, and you get 6 × 4 × 8 × 3 × 7 × 7 × 3 = 84,672.

Brandeis University mathematician Michael Kleber calls such a number fortuitous. The next few are 1,852,200, 829,785,600, 20,910,597,120, and 92,215,733,299,200.

If you normally say “and” after “hundred” when speaking number names, then the first few fortuitous numbers are 333,396,000 (THREE HUNDRED AND THIRTY THREE MILLION, THREE HUNDRED AND NINETY SIX THOUSAND), 23,337,720,000, 19,516,557,312,000, 56,458,612,224,000, and 98,802,571,392,000.

And 54 works in both French and Russian.

(Michael Kleber, “Four, Twenty-Four, … ?,” Mathematical Intelligencer 24:2 [March 2002], 13-14.)

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Subscriber Trouble

A letter to the New York Times Book Review, March 22, 1998:

To the Editor:

I read the review of Nathan Miller’s ‘Star-Spangled Men’ (Feb. 22) by Douglas McGrath, who challenged the reader to produce a sentence with three prepositions in a row, after I had picked my copy of The New York Times up from under the front porch, thankful that I didn’t have to get it down from above the porch roof, and at the same time, knowing that the delivery boy usually threw it to within a foot of the door, leaving me a quick way back in out of the cold each morning, I decided not to yell at him, especially since an argument was not something I wanted to get into outside of the house at this time of the morning, but still thinking that this was a matter that should be taken up from inside of the house by writing a letter to the editor, being careful not to use up to over three or four prepositions in a row in any sentence.

George F. Werner
Edgewood, N.M.

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Ah

https://commons.wikimedia.org/wiki/File:Bundesarchiv_B_145_Bild-F026330-0024,_Paris,_Bankett_Einweihung_Deutsche_Botschaft.jpg
Image: Wikimedia Commons

Michael C. [Codron] told me this story about Lady Dorothy Macmillan saying to Mme. de Gaulle at the Élysée Palace, ‘Now that your husband has achieved so much, is there any particular wish, any desire you have for the future?’ and Madame replied, ‘Yes — a penis.’ Whereupon Gen. de Gaulle leaned over and said, ‘No, my dear, in English it is pronounced Happiness.’

— Kenneth Williams, diary, April 10, 1966

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