Swiss physicist Auguste Piccard opened two new worlds in the 20th century. He was the first person to fly 10 miles above the earth and the first to travel 2 miles beneath the sea, using inventions that opened the doors to these new frontiers. In this week’s episode of the Futility Closet podcast we’ll follow Piccard on his historic journeys into the sky and the sea.
We’ll also admire some beekeeping serendipity and puzzle over a sudden need for locksmiths.
Shortly after conquering the Matterhorn on July 14, 1865, Edward Whymper watched four of his companions fall to their deaths down the mountain’s precipitous north face. Afterward he and his two Swiss guides, the Taugwalders, beheld a remarkable figure in the sky:
A mighty arch appeared, rising above the Lyskamm, high into the sky. Pale, colourless, and noiseless, but perfectly sharp and defined, except where it was lost in the clouds, this unearthly apparition seemed like a vision from another world; and, almost appalled, we watched with amazement the gradual development of two vast crosses, one on either side. If the Taugwalders had not been the first to perceive it, I should have doubted my senses. They thought it might have some connection with the accident, and I, after a while, that it might bear some relation to ourselves. But our movements had no effect on it. It was a fearful and wonderful sight; unique in my experience, and impressive beyond description, coming at such a moment.
What was this? Whymper later called it a fog bow, a bow that forms in fog rather than rain. Unfortunately, we have no photograph, only a sketch and a woodcut. In a 2002 simulation C.J. Hardwick tried to account for the features as Whymper had described them. “A fogbow and ice crystal arcs could have produced a circle and crosses in a direction consistent with the apparition,” he concluded in 2005. “However, while this simulation used a crystal type that can occur, it required an unusual alignment that would be very rare.”
(C.J. Hardwick, “Simulation of the Whymper Apparition,” Weather 57:12 [December 2002], 457-463; Cedric John Hardwick and Jason C. Knievel, “Speculations on the Possible Causes of the Whymper Apparition,” Applied Optics 44:27 [Sept. 20, 2005], 5637-5643.)
“Truel,” a mathematical romance by Tom Vaughan.
(This is based on a problem in game theory, but interestingly the hero is named Galois and one of his opponents is d’Herbinville — that’s the name of the man Alexandre Dumas identified as the opponent of Évariste Galois in his fatal duel of 1832.)
In 1917 Henry Dudeney asked: What’s the maximum number of lattice points that can be placed on an n × n grid so that no three points are collinear?
The answer can’t be more than 2n, since if we place one point more than this, we’re forced to put three into the same row or column. (The 10 × 10 grid above contains 20 points.)
For a grid of each size up to 52 × 52, it’s possible to place 2n points without making a triple. For larger grids it’s conjectured that fewer than 2n points are possible, but today, more than a century after Dudeney posed the question, a final answer has yet to be found.
In the 17th century the French mathematician Bernard Frénicle de Bessy described all 880 possible order-4 magic squares — that is, all the ways in which the numbers 1 to 16 can be arranged in a 4 × 4 array so that the long diagonals and all the rows and columns have the same sum.
These squares share a curious property: If we subtract 1 from each cell, to get a square of the numbers 0-15, then each of the rows and columns has a nim sum of 0. A nim sum is a binary sum in which 1 + 1 is evaluated as 0 rather than “0, carry 1.” For example, here’s one of Frénicle’s squares:
Translating each of these numbers into binary we get
And the binary sums of the four rows, evaluated without carry, are
0000 + 0101 + 1010 + 1111 = 0000
1110 + 1011 + 0100 + 0001 = 0000
1101 + 1000 + 0111 + 0010 = 0000
0011 + 0110 + 1001 + 1100 = 0000
The same is true of the columns. (The diagonals won’t necessarily sum to zero, but they will equal one another. And note that the property described above won’t necessarily work in a “submagic” square in which the diagonals don’t add to the magic constant … but it does work in all 880 of Frénicle’s “true” 4 × 4 squares.)
(John Conway, Simon Norton, and Alex Ryba, “Frenicle’s 880 Magic Squares,” in Jennifer Beineke and Jason Rosenhouse, eds., The Mathematics of Various Entertaining Subjects, Vol. 2, 2017.)
In 1804, when she was 5 years old, Mary Anning began to dig in the cliffs that flanked her English seaside town. What she found amazed the scientists of her time and challenged the established view of world history. In this week’s episode of the Futility Closet podcast we’ll tell the story of “the greatest fossilist the world ever knew.”
We’ll also try to identify a Norwegian commando and puzzle over some further string pulling.
A card curiosity via Martin Gardner: Deal 10 cards from an ordinary deck and hold this packet face down in your left hand. Turn the top two cards face up and then cut the packet anywhere you like. Again, turn the top two cards and cut. Continue doing this for as long as you like, turning over the top two cards and cutting the packet.
When you’ve finished, deal the cards in a row on the table and turn over the cards at even positions in the row: the second, fourth, sixth, eighth, and tenth cards.
This will always leave five cards face up.
(Martin Gardner, “Curious Counts,” Math Horizons 10:3 [February 2003], 20-22.)
This figure contains 15 lines and 15 points, with three points on each line and three lines through each point, yet no three points are connected by three lines to form a triangle.
It’s named after mathematicians Luigi Cremona and Herbert William Richmond, who studied it in the late 19th century.
As Washington State University anthropologist Grover Krantz was dying of pancreatic cancer, he told his colleague David Hunt of the Smithsonian:
“I’ve been a teacher all my life and I think I might as well be a teacher after I’m dead, so why don’t I just give you my body.”
When Hunt agreed, Krantz added, “But there’s one catch: You have to keep my dogs with me.”
Accordingly, in 2003, Krantz’s skeleton was laid to rest in a green cabinet at the National Museum of Natural History alongside the bones of his Irish wolfhounds Clyde, Icky, and Yahoo.
Krantz’s bones have been used to teach forensics and advanced osteology to students at George Washington University.
And in 2009 his skeleton was articulated and, along with Clyde’s, displayed in the exhibition “Written in Bone: Forensic Files of the 17th Century Chesapeake.”
From a letter of Charles Darwin to Charles Lyell, April 1860:
I must say one more word about our quasi-theological controversy about natural selection, and let me have your opinion when we meet in London. Do you consider that the successive variations in the size of the crop of the Pouter Pigeon, which man has accumulated to please his caprice, have been due to ‘the creative and sustaining powers of Brahma?’ In the sense that an omnipotent and omniscient Deity must order and know everything, this must be admitted; yet, in honest truth, I can hardly admit it. It seems preposterous that a maker of a universe should care about the crop of a pigeon solely to please man’s silly fancies. But if you agree with me in thinking such an interposition of the Deity uncalled for, I can see no reason whatever for believing in such interpositions in the case of natural beings, in which strange and admirable peculiarities have been naturally selected for the creature’s own benefit. Imagine a Pouter in a state of nature wading into the water and then, being buoyed up by its inflated crop, sailing about in search of food. What admiration this would have excited — adaptation to the laws of hydrostatic pressure, &c &c For the life of me I cannot see any difficulty in natural selection producing the most exquisite structure, if such structure can be arrived at by gradation, and I know from experience how hard it is to name any structure towards which at least some gradations are not known.
Ever yours,
C. Darwin.
