Somewhat like Eratosthenes, the Iranian polymath Al-Biruni (973-1048) was able to estimate the radius of the earth using just a few measurements and some clear thinking. If h is a mountain with a known height and the distance from the mountaintop A to the horizon C can be established accurately, then angle α is the same as angle AOC at the earth’s center and we have everything we need to calculate leg OC of right triangle AOC, which is the radius of the earth.

Biruni carried this out using a tall mountain near Nandana in present-day Pakistan. He estimated the earth’s radius at 6,339.9 km, which is only 16.8 km less than the current value of 6,356.7 km. This accuracy would not be obtained in the West until the 16th century.

06/22/2017 UPDATE: Wait, he didn’t even need the distance to the horizon, just the mountain height and the dip angle. Details here. (Thanks, Jacob.)

In a diary entry in 1843, Sir Oswald Brierly, manager of the whaling station at Twofold Bay in southeast Australia, noted a strange cooperative relationship that had grown up between killer whales and the local whalers:

They [the killer whales] attack the [humpback] whales in packs and seem to enter keenly into the sport, plunging about the [whaling] boat and generally preventing the whale from escaping by confusing and meeting him at every turn. … The whalemen of Twofold Bay are very favourably disposed towards the killers and regard it as a good sign when they see a whale ‘hove to’ by these animals because they regard it as an easy prey when assisted by their allies the killers.

By the early 20th century this curious custom had grown into a complex operation. The killer whales would herd a passing humpback into the bay and harass it there while others swam to the whaling station, breached, and thrashed their tails to alert the whalers. When the whalers arrived and harpooned the humpback, the killers would continue to leap onto its back and blowhole to tire it. In return, the whalers would anchor the dead whale to the bottom for a day or two so that the killers could feast on its lips and tongue.

The whalers came to know many of these killer whales by name: Hooky, Cooper, Typee, Jackson, and so on. The most famous, Old Tom, worked with the Twofold Bay whalers for almost four decades in the early 20th century — he grew famous for gripping the harpoon line with his teeth as each doomed humpback towed the whaleboat through the water. He died in 1930, and his skeleton, complete with grooves in the teeth, now resides in the Eden Killer Whale Museum in New South Wales.

(From Hal Whitehead and Luke Rendell, The Cultural Lives of Whales and Dolphins, 2014. See A Feathered Maître d’.)

In the Fibonacci sequence, each number is the sum of the two preceding ones:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, …

This produces two notable secondary patterns: Summing the squares of each pair of adjacent entries yields an even-numbered term in the sequence:

1² + 1² = 2
1² + 2² = 5
2² + 3² = 13
3² + 5² = 34
5² + 8² = 89
8² + 13² = 233
13² + 21² = 610

And the odd-numbered terms between these are the differences of squares of terms taken two by two, two places apart:

2² – 1² = 3
3² – 1² = 8
5² – 2² = 21
8² – 3² = 55
13² – 5² = 144
21² – 8² = 377
34² – 13² = 987

… and so on.

In 1988, workers found a 5-year-old bottlenose dolphin trapped in a canal lock in South Australia. She was taken to a commercial aquarium for medical treatment, where she was named Billie and housed with the trained dolphins who performed for oceanarium visitors there. One of the behaviors that the trained dolphins had learned was tailwalking, emerging vertically from the water and beating their tails to move backward as though walking on the water.

When Billie had recovered she was released back into the wild, but conservationist Mike Bossley continued to monitor her. He was surprised to see her tailwalking — she had never received any training during her recuperation and must have learned this spontaneously from the trained dolphins. “The behavior has no known utility in the wild and, as you can imagine, uses up a fair amount of energy, but Billie kept it up,” write biologists Hal Whitehead and Luke Rendell in The Cultural Lives of Whales and Dolphins (2014).

Not only that, but other wild dolphins began tailwalking themselves. Bossley reported, “Another female dolphin called Wave began performing the same behavior, but does so with much greater regularity than Billie. Four adult female dolphins have also been seen tailwalking,” as have several calves. Just last month conservationist Jenni Wyrsta saw a wild dolphin named Bianca do 33 tailwalks in a row in the Port River, in sets of two and three. “I’ve never seen a dolphin do double, let alone triple tailwalks,” she told the Portside Messenger.

Whitehead and Rendell write, “So, a behavior learned in captivity, with no obvious function beside play, has apparently gone on to become something of a hit in the wild and persists to this day, twenty-five years following Billie’s release and after Billie’s own death in 2009.”

Allan William Johnson discovered this “magic square of squares” in 1990:

 302   2462   1722    452

 932   1162    662   2582

1262   1382   2372    442

2602     32    542   1502

Each row, column, and long diagonal totals 93025, or 305².

In the humble tetrahedron, each face shares an edge with each other face. Surprisingly, there’s only one other known polyhedron in which this is true — the Szilassi polyhedron, discovered in 1977 by Hungarian mathematician Lajos Szilassi:

If there’s a third such creature it would have 44 vertices and 66 edges, and no one knows whether such a shape could even be contrived. It remains an unsolved problem.

Arrange the first n² odd numbers in a square (here n = 6):

 1  3  5  7  9 11
13 15 17 19 21 23
25 27 29 31 33 35
37 39 41 43 45 47
49 51 53 55 57 59
61 63 65 67 69 71

Now, no matter n, the sum of the first row is n², the sum of either long diagonal is n³, and the sum of the whole array is n⁴.

(From Edward Barbeau’s Power Play, 1997.)

In Circularity, Ron Aharoni mentions a story by Raymond Smullyan. On a certain island there are two kinds of people, those who always lie and those who always tell the truth. One day an islander is arrested on suspicion of murder. At his trial he says, “The murderer is a liar.”

Smullyan argues that this piece of evidence alone should acquit him. If the man is honest, then what he says is true, the murderer is a liar, and since he himself is a truth-teller he cannot be the guilty party. On the other hand, if he’s a liar then his testimony is false, which means that the murderer is in fact not a liar, and once again he cannot be guilty. Either way, he proves his innocence by showing that the murderer and himself belong to two different tribes.

Aharoni adds, “The problem is that the man was found beside the corpse with a bloody knife in his hand and a wide smile on his face. He is obviously the murderer, which means that he managed to prove an obvious fallacy. It seems that using his method, he can prove anything. And indeed he can. See what he is claiming when stating that the murderer is a liar: ‘If I am the murderer, then I am a liar’, which means ‘if I am the murderer then this is a lie’. In other words — ‘If I am the murderer then L is true’. And … this proves that ‘I am not the murderer.'”

Write down a 0:

0

Now mentally “flip” this string in binary, exchanging 0s and 1s, and append this new string to the existing one:

0 1

Keep this up and you’ll get a growing string of 0s and 1s:

0 1 1 0
0 1 1 0 1 0 0 1
0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
…

This is the Thue–Morse sequence, named after two of its discoverers, Axel Thue and Marston Morse. One interesting property of the sequence is that, no matter how far it’s extended, it contains no “cubes,” no instances in which some nonempty string occurs three times in a row. For example, the last line above contains both 11 and 00 but no instance of 111 or 000. (It also contains 1001 twice in a row, but not three times.)

Max Euwe, the Dutch mathematician who was world chess champion from 1935 to 1937, used this principle to show that chess was not a finite game. Under the rules at the time, a chess game would end in a draw if a sequence of moves (with all pieces in the same positions) were played three times in a row. Euwe used the Thue-Morse sequence to show that this need never happen: If 0 represents one set of moves, and 1 represents another, and each set leaves the board position unchanged, then the Thue-Morse sequence shows that two players might step through these routines forever without ever playing one three times in a row.

Modern chess rules have dropped the threefold sequence provision. Instead a draw results when the same board position occurs three times, or when 50 successive moves occur without a capture or a pawn move. Both of these rules limit a game to a finite length (although one player must actually claim the draw).

(Thanks, Pål.)