Some Palindromes

http://commons.wikimedia.org/wiki/File:Al_roverso_symfonie_47_Haydn.png

In the minuet in Haydn’s Symphony No. 47, the orchestra plays the same passage forward, then backward.

When Will Shortz challenged listeners to submit word-level palindromes to National Public Radio’s Weekend Edition Sunday in 1997, Roxanne Abrams offered the poignant Good little student does plan future, but future plan does student little good.

math palindromes

And Connecticut’s Oxoboxo River offers a four-way palindrome — it reads the same forward and backward both on the page and in a mirror placed horizontally above it.

| Art · Language · Science & Math

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

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

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

Insight

http://commons.wikimedia.org/wiki/File:Tree_of_knowledge.svg

Many a man floated in water before Archimedes; apples fell from trees as long ago as the Garden of Eden, and the onrush of steam against resistance could have been noted at any time since the discovery of fire and its use under a covered pot of water. In all these cases it was eons before the significance of these events was perceived. Obviously a chance discovery involves both the phenomenon to be observed and the appropriate, intelligent observer.

— Walter Cannon, The Way of an Investigator, 1945

| Science & Math

Brothers in Binary

A number is said to be perfect if it equals the sum of its divisors: 6 is divisible by 1, 2, and 3, and 1 + 2 + 3 = 6.

St. Augustine wrote, “Six is a number perfect in itself, and not because God created all things in six days; rather the converse is true; God created all things in six days because this number is perfect, and it would have been perfect even if the work of the six days did not exist.”

Perfect numbers are rare. No one knows whether an infinite quantity exist, and no one knows whether any of them are odd. The early Greeks knew the first four, and in the ensuing two millennia we’ve uncovered only 44 more. But they have one thing in common — they reveal a curious harmony when expressed in base 2:

brothers in binary

| Science & Math

Unquote

http://commons.wikimedia.org/wiki/File:Caspar_David_Friedrich_027.jpg

“We have not the reverent feeling for the rainbow that a savage has, because we know how it is made. We have lost as much as we gained by prying into that matter.” — Mark Twain

“At last I fell fast asleep on the grass & awoke with a chorus of birds singing around me, & squirrels running up the trees & some Woodpeckers laughing, & it was as pleasant a rural scene as ever I saw, & I did not care one penny how any of the beasts or birds had been formed.” — Charles Darwin, letter to his wife, April 28, 1858

| Quotations · Science & Math

Greetings

http://www.scribd.com/doc/48771623/Lageos-Press-Kit

Launched in 1976, NASA’s Laser Geodynamic Satellite needed a stable orbit to permit precise measurements of continental drift, so its designers gave it a high trajectory and a heart of solid brass. As a result, it’s not expected to return to Earth for 8 million years. That raised an interesting challenge: What message could we attach to the satellite that might be intelligible to our descendants or successors, who might recover it thousands of millennia in the future?

Tasked with that problem, Carl Sagan came up with the “greeting card” at left, which is affixed to LAGEOS on a small metal plaque. Using it, whoever comes upon the plaque can calculate roughly the time between his own epoch and ours. In Sagan’s words, the card says, “A few hundred million years ago the continents were all together, as in the top drawing. At the time LAGEOS was launched the map of the Earth looks as in the middle drawing. Eight million years from now, when LAGEOS should return to Earth, we figure the continents will appear as in the bottom drawing. Yours truly.”

| Science & Math

Balance

rectangle theorem

For any rectangle, the sum of the squares of the distances from any point P to two opposite corners is equal to the sum of the squares of the distances from that point to the two other corners (so, above, a2 + c2 = b2 + d2). This remains true whether the point is inside or outside the rectangle, on a side or a corner, or even outside the plane.

Pushkin wrote, “Inspiration is needed in geometry, just as much as in poetry.”

| Science & Math