Dark Doings

There is a legend that after Buddha died, his shadow lingered in a cave. It actually is possible for a shadow to persist without any sustaining object. Light travels at 299,792,458 meters per second in a vacuum. The moon is about 384,400,000 meters away from Earth. Hence, if the moon were instantly obliterated during a solar eclipse, its shadow would linger for more than a second on the surface of Earth. If the moon were farther away, its shadow could last several minutes. We can extrapolate to posthumous shadows that postdate their objects by millions of years. We can also speculate about an infinite past in which a shadow is sustained by a beginningless sequence of objects. As one object is destroyed, an object of the same shape and size seamlessly replaces it. This shadow antedates any object in the sequence and so refutes the principle that every shadow is caused by an object. Shadows are not dedicated dependents. Although slaves to some object or other, they can switch masters.

— Roy Sorensen, Seeing Dark Things, 2008

Franklin the Wizard


But the sage was not too grave to play a joke on his friends. One day, when they were walking in the park at Wycombe, he said that he could quiet the waves on a small stream which was being whipped by the wind. He went two hundred paces above where the others stood, made some magic passes over the water, and waved his bamboo cane three times in the air. The waves gradually sank and the stream became as smooth as a mirror. After they had marvelled Franklin explained. He carried oil in the hollow joint of his cane, and a few drops of it spreading on the water had caused the miracle.

— Carl Van Doren, Benjamin Franklin, 1938

Chasing Leo


In 1938, the American Mathematical Monthly published an unlikely paper: “A Contribution to the Mathematical Theory of Big Game Hunting.” In it, Ralph Boas and Frank Smithies presented 16 ways to catch a lion using techniques inspired by modern math and physics. Examples:

  • “The Method of Inversive Geometry. We place a spherical cage in the desert, enter it, and lock it. We perform an inversion with respect to the cage. The lion is then in the interior of the cage, and we are outside.”
  • “A Topological Method. We observe that a lion has at least the connectivity of the torus. We transport the desert into four-space. It is then possible to carry out such a deformation that the lion can be returned to three-space in a knotted condition. He is then helpless.”
  • “The Dirac Method. We observe that wild lions are, ipso facto, not observable in the Sahara Desert. Consequently, if there are any lions in the Sahara, they are tame. The capture of a tame lion may be left as an exercise for the reader.”
  • “A Relativistic Method. We distribute about the desert lion bait containing large portions of the Companion of Sirius. When enough bait has been taken, we project a beam of light across the desert. This will bend right round the lion, who will then become so dizzy that he can be approached with impunity.”

The article has inspired a tradition of updates by other mathematicians over the years:

  • “Let Q be the operator that encloses a word in quotation marks. Its square Q2 encloses a word in double quotes. The operator clearly satisfies the law of indices, QmQn = Qm + n. Write down the word ‘lion,’ without quotation marks. Apply to it the operator Q-1. Then a lion will appear on the page. It is advisable to enclose the page in a cage before applying the operator.” (I.J. Good, 1965)
  • “Game Theoretic Method. A lion is big game. Thus, a fortiori, he is a game. Therefore there exists an optimal strategy. Follow it.” (“Otto Morphy,” 1968)
  • “Method of Analytics Mechanics. Since the lion has nonzero mass it has moments of inertia. Grab it during one of them.” (Patricia Dudley et al., 1968)
  • “Method of Natural Functions. The lion, having spent his life under the Sahara sun, will surely have a tan. Induce him to lie on his back; he can then, by virtue of his reciprocal tan, be cot.” (Dudley)
  • “Nonstandard Analysis. In a nonstandard universe (namely, the land of Oz), lions are cowardly and may be caught easily. By the transfer principle, this likewise holds in our (standard) universe.” (Houston Euler, et al., 1985)

Dudley also suggested a “method of moral philosophy”: “Construct a corral in the Sahara and wait until autumn. At that time the corral will contain a large number of lions, for it is well known that a pride cometh before the fall.”

“Another Paradox”


If a cork ball about an inch in diameter be tied at the end of a thread about a foot in length, and then swung so that it enters a smooth stream of water flowing from a tap at about three inches from the mouth of the latter, it will be found that the ball will remain in the water, and that the thread will make an angle of about thirty degrees with a vertical line passing through the ball. The latter, it should be added, must be thoroughly wetted before this result is produced.

Strand, September 1908

Math Notes

20864448472975628947226005981267194447042584001 = (2 + 0 + 8 + 6 + 4 + 4 + 4 + 8 + 4 + 7 + 2 + 9 + 7 + 5 + 6 + 2 + 8 + 9 + 4 + 7 + 2 + 2 + 6 + 0 + 0 + 5 + 9 + 8 + 1 + 2 + 6 + 7 + 1 + 9 + 4 + 4 + 4 + 7 + 0 + 4 + 2 + 5 + 8 + 4 + 0 + 0 + 1)20

Moving Without Motion?

Achilles-weed is prostrate and grows along the ground at the amazing rate of 10 cm per hour. An exceeding slow tortoise munches one end of the Achilles-weed at the same rate as it grows at the other end. So the tortoise appears to chase the Achilles-weed round the garden. But, strictly speaking, the Achilles-weed does not move at all, it grows and is eaten. Yet its location changes, and it is made up of parts whose location changes (the left and right-hand halves of the Achilles-weed). Hence being made up of parts whose location changes is not sufficient for motion.

— Peter Forrest, “Is Motion Change of Location?”, Analysis, 1984