A self-reproducing sentence by Lee Sallows — “Doing what it tells you to do yields a replica of itself”:

This reminds me of a short short story by Fredric Brown:

THE END

Professor Jones had been working on time theory for many years.

“And I have found the key equation,” he told his daughter one day. “Time is a field. This machine I have made can manipulate, even reverse, that field.”

Pushing a button as he spoke, he said, “This should make time run backward run time make should this,” said he, spoke he as button a pushing.

“Field that, reverse even, manipulate can made have I machine this. Field a is time.” Day one daughter his told he, “Equation key the found have I and.”

Years many for theory time on working been had Jones Professor.

END THE

In science it often happens that scientists say, ‘You know, that’s a really good argument; my position is mistaken,’ and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn’t happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion.

— Carl Sagan, in a 1987 address, quoted in Jon Fripp et al., Speaking of Science, 2000

On Jan. 9, 1793, two astonished farmers in Woodbury, N.J., watched a strange craft descend from the sky into their field. An excited Frenchman greeted them in broken English and gave them swigs of wine from a bottle. Unable to make himself understood, he finally presented a document:

The farmers helped the man fold his craft and load it onto a wagon for the trip back to Philadelphia. Before leaving, the Frenchman asked them to certify the time and place of his arrival. These details were important — he was Jean-Pierre Blanchard, and he had just completed the first balloon flight in North America.

One hundred sixty-nine years later, when John Glenn went into orbit aboard Friendship 7 in 1962, mission planners weren’t certain where he’d come down. The most likely sites were Australia, the Atlantic Ocean, and New Guinea, but it might be 72 hours before he could be picked up.

Glenn worried about spending three days among aborigines who had seen a silver man emerge from “a big parachute with a little capsule on the end,” so he took with him a short speech rendered phonetically in several languages. It read:

“I am a stranger. I come in peace. Take me to your leader, and there will be a massive reward for you in eternity.”

Theorem 1. A crocodile is longer than it is wide.

Proof. A crocodile is long on the top and bottom, but it is green only on the top; therefore a crocodile is longer than it is green. A crocodile is green along both its length and width, but it is wide only along its width; hence a crocodile is greener than it is wide. Therefore a crocodile is longer than it is wide.

Theorem 2. Napoleon was a poor general.

Proof. Most men have an even number of arms. Napoleon was warned that Wellington would meet him at Waterloo. To be forewarned is to be forearmed. But four arms is a very odd number of arms for a man. The only number that is both even and odd is infinity. Therefore, Napoleon had an infinite number of arms in his battle against Wellington. A general who loses a battle despite having an infinite number of arms is very poor general.

Theorem 3. If 1/0 = ∞, then 1/∞ = 0.

Proof. Given:

Rotate both sides 90° counterclockwise:

Subtract 8 from both sides:

Now reverse the rotation:

Only the brilliantly inventive Lee Sallows would think of this. The figure above combines Penrose triangles with Borromean rings: Each of the triangles is an impossible object, and they’re united in a perplexing way — although the three are linked together, no two are linked.

(Thanks, Lee.)

I’m not sure who came up with this — this simple diagram reflects all possible true trigonometric identities of the form x ÷ y = z or x × y = z, where x, y, and z are the basic trigonometric functions of the same angle t.

For any three neighboring functions on the perimeter of the star, the product of the ends always equals the middle (e.g., tan t × cos t = sin t) and the middle function divided by one of the end functions is equal to the other end function (e.g., sin t ÷ tan t = cos t and sin t ÷ cos t = tan t). If you memorize the diagram you can reel off a list of 18 simple relations.

I found it in Michael Stueben’s Twenty Years Before the Blackboard, 1998.

The index to the fourth edition of George Thomas’ Calculus and Analytic Geometry contains an entry for “Whales” on page 188. That page contains no reference to whales, but it does include the figure above.

German mathematician Erich Bessel-Hagen was often teased for his protruding ears.

In 1923 his colleague Béla Kerékjártó published a book, Vorlesungen Über Topologie, whose index lists a reference to Bessel-Hagen on page 151.

That page makes no mention of Bessel-Hagen, but it does contain this figure:

Is that libel?

From Martin Gardner, via Michael Stueben: Obtain a slab of gold measuring 10″ x 11″ x 1″. Divide it diagonally and then cut a triangular notch in two corners as shown. Remove these notches as profit, and slide the remaining halves together to produce a new 10″ x 11″ x 1″ slab. The process can be repeated to yield any amount of money you like!