Heart and Soul

pedal triangle theorem

From a point P, drop perpendiculars to the sides of a surrounding triangle. This defines three points; connect those to make a new triangle and drop perpendiculars to its sides. If you continue in this way, the fourth triangle will be similar to the original one.

In 1947, Mary Pedoe memorialized this fact with a poem:

Begin with any point called P
(That all-too-common name for points),
Whence, on three-sided ABC
We drop, to make right-angled joints,
Three several plumb-lines, whence ’tis clear
A new triangle should appear.

A ghostly Phoenix on its nest
Brooding a chick among the ashes,
ABC bears within its breast
A younger ABC (with dashes):
A figure destined, not to burn,
But to be dropped on in its turn.

By going through these motions thrice
We fashion two triangles more,
And call them ABC (dashed twice)
And thrice bedashed, but now we score
A chick indeed! Cry gully, gully!
(One moment! I’ll explain more fully.)

The fourth triangle ABC,
Though decadently small in size,
Presents a form that perfectly
Resembles, e’en to casual eyes
Its first progenitor. They are
In strict proportion similar.

The property generalizes: Not only is the third “pedal triangle” of a triangle similar to the original triangle, but the nth “pedal n-gon” of an n-gon is similar to the original n-gon.

| Poems · Science & Math

Abuse of Power

http://commons.wikimedia.org/wiki/File:JGCannon.jpg

One hot summer day in 1904, Speaker of the House Joe Cannon of Illinois visited the House dining room and asked for a bowl of bean soup. He was told that, in view of the sultry weather, it had been omitted from the menu.

“Thunderation!” Cannon roared. “I had my mouth set for bean soup! From now on, hot or cold, rain, snow, or shine, I want it on the menu every day.”

And so it has been, ever since. The recipe was published on the menu in 1955:

2 lb. No. 1 white Michigan beans.
Cover with water and soak overnight.
Drain and re-cover with water.
Add a smoked ham hock and simmer slowly for about 4 hours until beans are cooked tender. Then add salt and pepper to suit taste.
Just before serving, bruise beans with large spoon ladle, enough to cloud. (Serves about six persons)

| History · Trivia

Who’s Counting?

In the 14th century, an unnamed Kabbalistic scholar declared that the universe contains 301,655,722 angels.

In 1939, English astrophysicist Sir Arthur Eddington calculated that it contains 15,747,724, 136,275,002,577,605,653,961,181,555,468,044,717,914,527,116,709,366,231,425,076,
185,631,031,296 electrons.

“Some like to understand what they believe in,” wrote Stanislaw Lec. “Others like to believe in what they understand.”

| Religion · Science & Math

As Time Goes By

http://www.sxc.hu/photo/1188156

A piano keyboard can be used as a calendar mnemonic: If the notes in the chromatic scale from F to E are assigned to the calendar months from January to December, then the white keys correspond to months with 31 days, the black keys to those with 30 days or fewer.

| Trivia

Higher Learning

Acknowledgment from an anonymous doctoral dissertation in the University Microforms International database:

If I had a dime for every time my wife threatened to divorce me during the past three years, I would be wealthy and not have to take a postdoctoral position which will only make me a little less poor and will keep me away from home and in the lab even more than graduate school and all because my committee read this manuscript and said that the only alternative to signing the approval to this dissertation was to give me a job mowing the grass on campus but the Physical Plant would not hire me on account of they said I was over-educated and needed to improve my dexterity skills like picking my nose while driving a tractor-mower over poor defenseless squirrels that were eating the nuts they stole from the medical students’ lunches on Tuesday afternoon following the Biochemistry quiz which they all did not pass and blamed on me because they said a tutor was supposed to come with a 30-day money-back guarantee and I am supposed to thank someone for all this?!!

(From a UMI press release, quoted in The Whole Library Handbook 2, 1995)

| Society

Good Behavior

prisoner magic square

Back in 2010 I posted a prime magic square created by a prison inmate and published anonymously in the Journal of Recreational Mathematics. The same prisoner composed the 7×7 square above, which has some remarkable properties of its own:

  • Here again every cell is prime.
  • The numbers in each row, column, and the two main diagonals add to the magic constant of 27627.
  • That same constant, 27627, is the sum of each broken diagonal (that is, each pair of parallel diagonals that include seven numbers, for example 3881 + 827 + 9257 + 5471 + 1741 + 29 + 6421).
  • If the units digit is removed from each number (changing 9341 to 934, 6367 to 636, etc.), then it remains a pandiagonal magic square, with all the properties mentioned above for the primes.

Both squares appeared in the October 1961 issue of Recreational Mathematics Magazine — editor Joseph S. Madachy noted that they had been “sent to Francis L. Miksa of Aurora, Illinois from an inmate in prison who, obviously, must remain nameless.”

It’s not clear to me why the prisoner shouldn’t get credit for this work, whatever his crime — presumably he created both squares while working alone and without tools or references, a remarkable achievement. If I learn any more I’ll post it here.

| Science & Math

Paper Fight

Most mutilated journals in the library of the University of Nebraska, Omaha, September 1982-May 1983:

  1. Personnel Psychology
  2. Journal of Conflict Resolution
  3. Journal of Politics
  4. Judicature
  5. Education and Urban Society
  6. ASCE Journal of Hydraulics
  7. Phylon
  8. Journal of Humanistic Philosophy
  9. Journal of Marriage and the Family
  10. Journal of Experimental Psychology

(“Saving and Securing Library Materials,” American Libraries, November 1983, p. 651.)

| Literature

Parting Shot

http://commons.wikimedia.org/wiki/File:Samuel_Wilberforce_(1805%E2%80%931873),_Bishop_of_Oxford,_by_George_Richmond_1868.jpg

Bishop Samuel Wilberforce was fond of riddles. After his death in 1873, this one was found among his literary papers:

I’m the sweetest of sounds in Orchestra heard,
Yet in Orchestra never was seen.
I’m a bird of gay plumage, yet less like a bird,
Nothing ever in Nature was seen.
Touch the earth I expire, in water I die,
In air I lose breath, yet can swim and can fly;
Darkness destroys me, and light is my death,
And I only keep going by holding my breath.
If my name can’t be guessed by a boy or a man,
By a woman or girl it certainly can.

No one knows the answer.

07/05/2013 UPDATE: A great many readers have sent me proposed answers since I posted this item. The overwhelming favorite is “a whale” (or “orca”); others include “a woman’s voice” and “a soap bubble.” The latter was favored by Henry Dudeney (in his 300 Best Word Puzzles) — he, like everyone, is confident of his solution:

“We have no doubt that the correct answer is that we gave (apparently for the first time in print) in the Guardian for 6th February, 1920. This answer is the word BUBBLE. It is an old name for Bagpipes, the word exactly answers every line of the enigma, though the final couplet may be perplexing. The explanation is that ‘Bubble’ is an old name for breast.”

| Puzzles

Noted

http://commons.wikimedia.org/wiki/File:TurdusCrossleyiKeulemans.jpg

Letter to the Times, June 15, 1962:

Sir,

All thrushes (not only those in this neck of the Glyndebourne woods) sooner or later sing the tune of the first subject of Mozart’s G minor Symphony (K. 550) — and, what’s more, phrase it a sight better than most conductors. The tempo is always dead right and there is no suggestion of an unauthorized accent on the ninth note of the phrase.

Yours, &c.,

Spike Hughes

See Bird Songs.

| Oddities