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
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.”
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.
Discovered by J.A.H. Hunter.
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.)
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.
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.
Properly speaking, can the top half of an apple exist without the whole?
What is it the top half of?
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.
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?