Science & Math

The Ames Window

Psychologist Adelbert Ames, inventor of the Ames room, also devised this illusion. In which direction is the window turning?

“The Chemist to His Love”

I love thee, Mary, and thou lovest me–
Our mutual flame is like th’ affinity
That doth exist between two simple bodies:
I am Potassium to thine Oxygen.
‘Tis little that the holy marriage vow
Shall shortly make us one. That unity
Is, after all, but metaphysical.
Oh, would that I, my Mary, were an acid,
A living acid; thou an alkali
Endow’d with human sense, that, brought together,
We both might coalesce into one salt,
One homogeneous crystal. Oh, that thou
Wert Carbon, and myself were Hydrogen;
We would unite to form olefiant gas,
Or common coal, or naphtha–would to heaven
That I were Phosphorus, and thou wert Lime!
And we of Lime composed a Phosphuret.
I’d be content to be Sulphuric Acid,
So that thou might be Soda. In that case
We should be Glauber’s Salt. Wert thou Magnesia
Instead we’d form the salt that’s named from Epsom.
Couldst thou Potassa be, I Aqua-fortis,
Our happy union should that compound form,
Nitrate of Potash–otherwise Saltpetre.
And thus our several natures sweetly blent,
We’d live and love together, until death
Should decompose the fleshly tertium quid,
Leaving our souls to all eternity
Amalgamated. Sweet, thy name is Briggs
And mine is Johnson. Wherefore should not we
Agree to form a Johnsonate of Briggs?

— “A Rochester druggist,” quoted in The Medical Age, Oct. 11, 1886

Math Notes

512 × 1953125 = 1000000000

262144 × 3814697265625 = 1000000000000000000

8589934592 × 116415321826934814453125 = 1000000000000000000000000000000000

Great Men

Dr. Franklin had a party to dine with him one day at Passy, of whom one-half were Americans, the other half French, and among the last was the Abbé Raynal. During the dinner he got on his favorite theory of the degeneracy of animals, and even of man in America, and urged it with his usual eloquence. The Doctor, at length, noticing the accidental stature and position of his guests at table, ‘Come,’ says he, ‘M. l’Abbé, let us try this question by the fact before us. We are here one-half Americans and one-half French, and it happens that the Americans have placed themselves on one side of the table, and our French friends are on the other. Let both parties rise, and we will see on which side nature has degenerated.’ It happened that his American guests were Carmichael, Harmer, Humphreys, and others of the finest stature and form; while those on the other side were remarkably diminutive, and the Abbé himself particularly, was a mere shrimp. He parried the appeal by a complimentary admission of exceptions, among which the Doctor himself was a conspicuous one.

— Thomas Jefferson, quoted in James Parton, Life and Times of Benjamin Franklin, 1864

Skyward

http://www.sxc.hu/photo/731929

In 1964, as the Apollo program prepared to land a man on the moon, it received unexpected news from Zambia. “I’ll have my first Zambian astronaut on the moon by 1965,” announced Edward Mukaka Nkoloso, a grade-school science teacher and director-general of the Zambian National Academy of Space Research.

“We are using our own system, derived from the catapult,” he explained. It would fire a 10-foot aluminum and copper rocket that would carry 10 Zambian astronauts ultimately to Mars.

“I’m getting them acclimatized to space travel by placing them in my space capsule every day. It’s a 40-gallon oil drum in which they sit, and I then roll them down a hill. This gives them the feeling of rushing through space. I also make them swing from the end of a long rope. When they reach the highest point, I cut the rope — this produces the feeling of free fall.”

Unfortunately, “I’ve had trouble with my spacemen and spacewomen,” Nkoloso complained. “They won’t concentrate on spaceflight; there’s too much lovemaking when they should be studying the moon. Matha Mwamba, the 17-year-old girl who has been chosen to be the first woman on Mars, has also to feed her 10 cats, who will be her companions on her long space flight.”

The U.N. denied the £700 million Nkoloso needed “to really get going,” but his enthusiasm remained undiminished. In 1968 he congratulated the returning Apollo 8 team but urged: “Let us make a Zambian rocket today. We shall never be content to remain behind other races. This is our heavenly destiny, our natural ambition and cultural hegemony.”

Rolling

http://commons.wikimedia.org/wiki/File:Tusi-couple.gif

Suppose we set a small circle rolling around the interior of a large circle of twice its diameter. If we follow a point on the small circle, what pattern will it draw?

Click for Answer

Misc

  • Dorothy Parker named her dog Cliche.
  • 27639 = 27 × 63 – 9
  • Tikitiki cures beriberi.
  • Can an object move itself?
  • “The best way out is always through.” — Robert Frost

Math Notes

By Royal V. Heath:

12 + 43 + 65 + 78 = 87 + 56 + 34 + 21

That’s not terrifically impressive on its face. But:

  • Each side uses the digits 1-8.
  • The whole equation is a palindrome.
  • It remains valid if you square each term.

Teamwork

http://en.wikipedia.org/wiki/File:Polly_%26_Annie.jpg

At the start of her career, NIH immunologist Polly Matzinger disliked writing in the passive voice and felt too insecure to adopt the first person. So she listed her dog, Galadriel Mirkwood, as a coauthor and wrote as “we.”

Their paper was published in 1978 in the Journal of Experimental Medicine. When the editor learned Galadriel’s species, he barred Matzinger from his pages for the rest of his life.

Sibling Rivalry

Do men have more sisters than women do? Intuitively it seems they must. In a family with two children, a boy and a girl, the boy has a sister but the girl doesn’t. In a family with four children, two boys and two girls, each boy has two sisters but each girl has one. It seems inevitable that, on average, men must have more sisters than women.

But it isn’t true. There are four possible two-child families, all equally likely: BB, BG, GB, GG. Half of the children in these families have a sibling of the same sex, and half have a sibling of the opposite sex. This observation can be extended to larger families. So men have the same number of sisters as women.

Cut the Knot has a good discussion of the statistics, including a javascript simulator.

A Population Puzzle

http://commons.wikimedia.org/wiki/File:Chodowiecki_Basedow_Tafel_6_c.jpg

If I have two children, there’s a 50 percent chance that I’ll have a boy and a girl.

But if I have four children, the chance that I’ll have an equal number of boys and girls drops to 6 in 16, or 37.5 percent.

This trend continues — as the number of offspring rises, the chance of having precisely the same number of boys and girls drops:

2 children: 50 percent
4 children: 37.5 percent
8 children: 27.34375 percent
16 children: 19.6380615 percent
32 children: 13.9949934 percent
64 children: 9.9346754 percent
128 children: 7.0386092 percent

This is worrying. Does it mean that in a large population Jacks might drastically outnumber Jills?

No. As the population grows, the distribution assumes the shape of a normal bell curve concentrated near 1/2. So while the chance of precise parity drops, the chance that a large population will have approximately equal numbers of girls and boys actually increases, as we’d expect.

“Constantly Mean”

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

The golden mean is quite absurd;
It’s not your ordinary surd.
If you invert it (this is fun!),
You’ll get itself, reduced by one;
But if increased by unity,
This yields its square, take it from me.

— Paul S. Bruckman, The Fibonacci Quarterly, 1977

Both Sides Now

Bach’s “crab canon” rendered as a Möbius strip:

Bach and Handel were both blinded by the same oculist, John Taylor, “the poster child for 18th-century quackery,” according to University of Wisconsin ophthalmologist Daniel Albert. Bach probably died of a post-operative infection; Handel wrote the lyrics to Samson (“Total eclipse! No sun, no moon! / All dark amidst the blaze of noon!”) after Taylor’s botched cataract surgery.

Random Möbius anecdote: In 1957, B.F. Goodrich patented a half-twisted conveyor belt for carrying hot material such as cinders and foundry sand, “thereby permitting each face of the belt to cool during one half of the operating period.”

Going Down

http://books.google.com/books?id=CpgkAQAAIAAJ&printsec=frontcover#v=onepage&q&f=false

What would happen if you jumped into a tunnel that passed through the center of the earth? If you encountered no air resistance, dinosaurs, Mad Hatters, or Morlocks, you’d accelerate until you passed through the center at 18,000 mph, then slow as you ascended through the opposite hemisphere. At the far end you’d have just time to tip your hat to the surprised antipodeans before you fell home again, and you’d continue oscillating like this forever.

“If this shaft had its starting-point on one of the mountain plateaux of South America at an elevation of seven thousand feet,” wrote Camille Flammarion in 1909, “and if it issued at the sea-level at the other side, a man who had fallen into the shaft would arrive at the antipodes still travelling at such a speed that the spectators would see this strange projectile shot to a height of seven thousand feet into the air.”

On the other hand, if our straight tunnel connected two points that were not precise antipodes, then we could install a train powered by gravity — it would roll “downhill” on the first part of its journey, and momentum would carry it through the second (again neglecting air resistance and friction). Curiously, in all these cases the total trip would take the same length of time — about 42 minutes.

Math Notes

j.a.h. hunter math notes

Discovered by J.A.H. Hunter.

Piece Work

Mathematician Henri Picciotto was visiting a local McDonald’s with his son when he noticed that Chicken McNuggets are served in boxes of 6, 9, and 20 nuggets. This means that you can easily buy, say, exactly 15 nuggets, but not 16. What’s the largest “non-McNugget” number?

He worked it out on a napkin: It’s 43. If you want any number of Chicken McNuggets larger than 43, you can get them by buying some combination of 6-, 9-, and 20-nugget boxes. But if you want exactly 43, you’re out of luck.

(That was in the 1980s. Today you can buy 4 McNuggets in a Happy Meal, which simplifies the problem. Perhaps McDonald’s was listening.)

Misc

  • Newton was born the year that Galileo died.
  • Cole Porter’s summer home was called No Trespassing.
  • 66339 = (6 × 6)3 + 39
  • Could you have had different parents?
  • “A good conscience is a continual Christmas.” — Ben Franklin

UPDATE: The first item here is incorrect. The dates coincide only if one uses the Gregorian calendar to date Galileo’s death and the Julian to date Newton’s birth. The two events occurred 361 days apart, which puts them in separate years on both calendars. Apparently this is a very common error. (Thanks, Igor.)

Snappy New Year

In Insurmountable Simplicities (2006), Roberto Casati points out that a traveler flying east may miss midnight — by entering a new time zone, he may jump from the 11:00 hour into the 12:00 hour without passing through midnight:

Mathematics and geography tell us that during the flight it will happen more than once that we reach the stroke of the hour without leaving the time zone we are in. If our flight lasts eight hours and the time difference between New York and Paris is six hours, this will happen at least twice, and at most eight times. But there is no guarantee that the stroke of midnight will be among these cases.

Thus, if it’s New Year’s Eve, an eastbound traveler may get no champagne.

Frozen Fire

Lightning can fuse sand into curious rootlike tubes up to 5 meters long, called fulgurites. Because their shape records the path of the strike as it passes into the ground, they’re sometimes known as petrified lightning.

Lightning had a ruinous history before the introduction of Ben Franklin’s lightning rod. The campanile of St. Mark in Venice was destroyed three times over. In 1769, a bolt struck the tower of St. Nazaire in Brescia, whose magazine contained 100 tons of gunpowder. One-sixth of the town was destroyed, and 3,000 people died.

Compounding the harm was the disastrous belief that ringing bells during thunderstorms would allay lightning. In one 33-year period, lightning struck 386 church towers and killed 103 bell ringers.

Modern strikes are less dire. In 1919, Cleveland Indians pitcher Ray Caldwell was struck by lightning during a game against the Philadelphia Athletics. “It felt like a sandbag hit me,” he said. He refused to leave the game and pitched to Joe Dugan for the final out. The Indians won, 2-1.

Bites and Pieces

http://www.sxc.hu/photo/893214

Properly speaking, can the top half of an apple exist without the whole?

What is it the top half of?

Crowded

Any set of 10 positive integers smaller than 100 will always contains two subsets with the same sum.

In any such group, the number of possible subsets (excluding the empty set) is 210 – 1, or 1023. And the largest possible sum of any subset is 90 + 91 + … + 99 = 945. Hence, no matter which numbers are chosen, there will always be more subsets than possible sums, and some subsets (dozens, actually) must yield the same sum.

Squeeze Play

A rail one mile long is lying on the ground. If you push its ends closer together by a single foot, so that the distance between them is 5279 feet rather than 5280, how high an arc will the rail make?

Click for Answer

The Pizza Theorem

http://commons.wikimedia.org/wiki/File:Pizza_theorem_1.svg

If you’re sharing a pizza with another person, there’s no need to cut it into precisely equal slices.

Make four cuts at equal angles through an arbitrary point and take alternate slices. You’ll both get the same amount of pizza.

Also: If a pizza has thickness a and radius z, then its volume is pi z z a.

Misc

  • Dorothy Parker named Alexander Woollcott’s apartment “Wit’s End.”
  • Can you look at something and imagine it at the same time?
  • 36850 = (36 + 8) × 50
  • AGNOSTIC is an anagram of COASTING.
  • “The errors of a man are what make him really lovable.” — Goethe