Science & Math

Skyline Trouble

In June 1978, a Princeton engineering student called structural engineer William LeMessurier with some worrying calculations. LeMessurier’s new Citicorp Tower, which had opened the previous year, was vulnerable to quartering winds — winds that blew from a 45-degree angle. On investigating, LeMessurier found also that the welded joints he had specified had been replaced with weaker bolted joints during construction. This meant that a strong wind could shear the bolts and topple a 59-story building into midtown Manhattan.

With hurricane season approaching, welders worked from 8 p.m. to 4 a.m. every night, reinforcing the building’s joints, and the Red Cross worked out an evacuation plan for the surrounding neighborhood. Because of a press strike at the time, many of these details came to light only 20 years later.

That year’s Hurricane Ella actually bore down on New York as the workers were finishing the job, but the storm veered out to sea before reaching the city. The welding was completed in October, and it’s now estimated that a storm strong enough to rock the tower will occur only once every 700 years.


How fast does time pass? We have no way to measure this. We can reply, helplessly, that it passes at one second per second, but this is not a rate of change — 1 second divided by 1 second is 1. Not 1 of anything, just 1.

“‘One’ can be an answer, right or wrong, to the questions ‘How many children had Lady Macbeth?’, ‘How many Gods are there?’, and ‘How many minutes do sixty seconds make?’,” writes Notre Dame philosopher Peter van Inwagen. “‘One’ can never be an answer, not even a wrong one, to any other sort of question — including those questions that ask ‘how fast?’ or ‘at what rate?’ Therefore, if time is moving, it is not moving at any rate or speed.”

Ceramic Geometry

pythagorean tiling

This tiling pattern is sometimes referred to as Pythagorean because it can be construed to prove the Pythagorean theorem.

The red area is a right triangle. The square of its shorter side is equivalent to a green square, and the square of its longer side is equivalent to a yellow square.

One green and one yellow square can be cut up and reassembled to fit into one of the canted white squares, which is equivalent to the square of the red triangle’s hypotenuse. Hence a2 + b2 = c2.

Keeping Things Interesting

Each year, when the last flight of the summer field season departs the U.S. research station at the South Pole, the remaining staff gather to watch The Thing.

The next flight won’t arrive for eight months.

Time and Again

Imagine that I first walk through Divinity Avenue, and then imagine that the powers governing the universe annihilate ten minutes of time with all that it contained, and set me back at the door of this hall just as I was before the choice was made. Imagine then that, everything else being the same, I now make a different choice and traverse Oxford Street. You, as passive spectators, look on and see the two alternative universes,–one of them with me walking through Divinity Avenue in it, the other with the same me walking through Oxford Street. Now, if you are determinists you believe one of these universes to have been from eternity impossible: you believe it to have been impossible because of the intrinsic irrationality or accidentality somewhere involved in it. But looking outwardly at these universes, can you say which is the impossible and accidental one, and which the rational and necessary one? I doubt if the most ironclad determinist among you could have the slightest glimmer of light on this point. In other words, either universe after the fact and once there would, to our means of observation and understanding, appear just as rational as the other.

— William James, “The Will to Believe,” 1896

Healthcare Reform

The statement “You will recover from this illness” is either true or false. If it’s true, then it has been true for all eternity, and you’ll recover whether you summon a doctor or not.

If the statement is false, then it has always been false, and you will not recover even with a doctor’s aid.

So there is no point in calling a doctor.

(From Cicero’s De Fato.)

A Good Man

Should other species be regarded as human? In 1779 Lord Monboddo proposed that orangutans should: They walk upright, use weapons, form societies, build shelters, and behave with “dignity and composure.” “If … such an Animal be not a Man, I should desire to know in what the essence of a Man consists, and what it is that distinguishes a Natural Man from the Man of Art?”

Thomas Love Peacock mocked this view in his 1817 novel Melincourt, in which a civilized orangutan (“Sir Oran Hout-ton”) is elected to Parliament. And an anonymous wag objected even to the satire:

The author of a novel lately written,
Entitled “Melincourt,”
(‘Tis very sweet and short),
Seems indeed by some wondrous madness bitten,
Thinking it good
To take his hero from the wood:
And though I own there’s nothing treasonable
In making ouran-outangs reasonable,
I really do not think he should
Go quite the length that he has done,
Whether for satire or for fun,
To make this creature an M.P.
As if mankind no wiser were than he.
However, those who’ve read it
Must give the author credit
For skill and ingenuity,
Although it have this monstrous incongruity.

But today Monboddo’s view is on the ascendancy. Harvard legal scholar Steven M. Wise argues that orangutans — as well as chimpanzees, bonobos, elephants, parrots, dolphins, and gorillas — deserve legal personhood. “Ancient philosophers claimed that all nonhuman animals had been designed and placed on this earth just for human beings,” he writes. “Ancient jurists declared that law had been created just for human beings. Although philosophy and science have long since recanted, the law has not.”

Heavenly Form

The U.S. Customs Service received an odd bit of paperwork on July 24, 1969:

apollo 11 customs form

All three Apollo 11 astronauts signed the form, which was filed at the Honolulu airport on the day they splashed down.

“Yes, it’s authentic,” NASA spokesperson John Yembrick told “It was a little joke at the time.”

See Business Trip and Space Bills.

Math Notes

math notes

(Thanks, Pablo.)

Surface Matters

If you touch a gold ball, you touch its surface and you touch gold. It seems reasonable to conclude that the surface is made of gold. But University of Exeter computer scientist Antony Galton points out that the surface is two-dimensional; it can’t contain any quantity of gold.

What then is it? We can’t say it’s the outermost layer of gold atoms, for that’s a film with two surfaces. And we can’t say it’s an abstract boundary with no physical existence, for we can see it and touch it. So what is it?

J.L. Austin asked, “Where and what exactly is the surface of a cat?”

Sure Thing

[Lewis Carroll] told me of a simple, too simple, rule by which, he thought, one could be almost sure of making something at a horse-race. He had on various occasions noted down the fractions which represented the supposed chances of the competing horses, and had observed that the sum of these chances amounted to more than unity. Hence he inferred that, even in the case of such hard-headed men as the backers, the wish is often father to the thought; so that they are apt to overrate the chances of their favourites. His plan, therefore, was to bet against all the horses, keeping his own stake the same in each case. He did not pretend to know much about horse-racing, and I probably know even less; but I understand that it would be impossible to adjust the hedging with sufficient exactitude — in fact, to get bets of the right amount taken by the backers.

— Lionel Arthur Tollemache, Old and Odd Memories, 1908

Pi Coincidences

The number 360 is centered across the 360th decimal place of π:

pi coincidences - 360

6998970 = 36 + 19 + 49 + 18 + 59 + 97 + 20

6998971 = 36 + 19 + 49 + 18 + 59 + 97 + 21

(Thanks, Pablo.)

pi coincidences - approximations

The 22nd, 7th, 355th, 113th, and 52163rd digits of π (counting from the 3) are 2s.

The 16604th digit, alas, is a 1 — but it’s flanked by 2s.

After You

At the end of your visit to an elderly, infirm relative who lives alone, the relative says, ‘I’m sorry but my arthritis won’t let me get up from this chair today. You’ll have to show yourself out.’ How can you show yourself out of someone’s house? If you know the way out, you can act as a guide to someone else. But how can you act as your own guide?

— T.S. Champlin, Reflexive Paradoxes, 1988

Johnson Circles

johnson circles

If three hula hoops cover a common point, then a fourth hoop will cover their remaining intersections.

Safe Passage

Mathematician G.H. Hardy had an ongoing feud with God. Once, after spending a summer vacation in Denmark with Harald Bohr, he found he’d have to take a small boat across the tempestuous North Sea to return to England. Before boarding, he sent Bohr a postcard that said “I have proved the Riemann hypothesis. — G.H. Hardy.”

When Bohr excitedly asked about this later, “Oh, that!” Hardy said. “That was just insurance. God would never let me drown if it meant I’d get undue credit.”


  • Connecticut didn’t ratify the Bill of Rights until 1939.
  • Can one pity a fictional character?
  • 64550 = (64 – 5) × 50
  • BILLOWY is in alphabetical order, WRONGED in reverse.
  • “The essence of chess is thinking about the essence of chess.” — David Bronstein

Presto Chango

jones reversible magic square

From Samuel Isaac Jones, Mathematical Wrinkles (1929), a magic square with a twist:

“It will be observed that this square when turned upside down is still magic.”

Math Notes

hunter figure curio

Discovered by J.A.H. Hunter.

Half and Half

half and half 1

A line that bisects the right angle in a right triangle also bisects a square erected on the hypotenuse.


half and half 2

Time and Motion

If a second is defined by reference to the rotation of the earth on its axis, i.e. as 1/60 of 1/60 of 1/24 of the time between 2 identical positions of the Greenwich meridian relatively to the fixed stars, then, if the earth rotated 10 times more slowly than it does now, it would be possible to run 10 yds. in a second, instead of only a yard as now, and a second would be 10 times longer than it is now; but if cinema machines still moved as fast as they do now, it would still be quite impossible for any one to see a succession of static pictures instead of a moving one. Don’t we mean by a second the length of time which is now 1/60 of 1/60 of 1/24 of the time between etc.?

— G.E. Moore, Commonplace Book, 1962

Compound Interest

On Jan. 18, 1897, California farmer George Jones bought a quantity of livestock feed from Henry B. Stuart of San Jose. As security he signed a $100 promissory note that bore 10 percent interest per month, compounded monthly.

They had agreed that Jones would pay the debt in three months, but the note had run for almost 25 years when Stuart got tired of waiting and told his lawyer to sue. Judge J.R. Welch of the Superior Court of Santa Clara entered this judgment on March 6, 1922:

“Wherefore, by virtue of the law and the facts, it is Ordered, Adjudged and Decreed that said Plaintiff have and recover from said Defendant the sum of $304,840,332,912,685.16 with interest thereon at the rate of 7% per annum until paid, together with the further sum of $50.00 Plaintiff’s attorney’s fees herein with interest thereon at the rate of 7% per annum until paid.”

That’s $304 trillion, “more money than there is in the world, outside of Russia,” the New York Tribune reported drily. Jones paid $19.69 and filed for bankruptcy.

Divide and Conquer

In 1980, Colorado math teacher William J. O’Donnell was explaining that

divide and conquer 1

when a student noted that

divide and conquer 2

“My immediate reaction was that this student had stumbled onto a special case where this algorithm worked,” O’Donnell wrote in a letter to Mathematics Teacher. “Later, a couple of minutes of work revealed that this technique works for all fractions. Let a, b, c, and d be integers. Then

divide and conquer 3

“Whereas this method can be conveniently applied on occasion, it does not offer the student much advantage when c does not divide a and d does not divide b.”


Darth Vader is piloting a barge to Salt Lake City to give a workshop on evildoing. Suddenly he finds himself approaching a crumbling brick aqueduct, at the foot of which is a basket of adorable kittens. He struggles to stop the barge, but it’s too late. The terrified kittens mew piteously, but they’re too weak to escape. Inexorably, implacably, the barge floats out directly over the basket. What happens?

Nothing happens. The barge displaces its weight in water, so there’s no additional load on the aqueduct.

The workshop is a great success.


  • No bishop appears in Through the Looking-Glass.
  • Can a law compel us to obey the law?
  • 98415 = 98-4 × 15
  • Why does the ghost haunt Hamlet rather than Claudius?
  • “Put me down as an anti-climb Max.” — Max Beerbohm, declining to hike to the top of a Swiss Alp