Quick Thinking

In summer 1940, Germany demanded access to Swedish telephone cables to send encoded messages from occupied Norway back to the homeland. Sweden acceded but tapped the lines and discovered that a new cryptographic system was being used. The Geheimschreiber, with more than 800 quadrillion settings, was conveying top-secret information but seemed immune to a successful codebreaking attack.

The Swedish intelligence service assigned mathematician Arne Beurling to the task, giving him only a pile of coded messages and no knowledge of the mechanism that had been used to encode them. But after two weeks alone with a pencil and paper he announced that the G-schreiber contained 10 wheels, with a different number of positions on each wheel, and described how a complementary machine could be built to decode the messages.

Thanks to his work, Swedish officials learned in advance of the impending invasion of the Soviet Union. Unfortunately, Stalin’s staff disregarded their warnings.

“To this day no one knows exactly how Beurling reasoned during the two weeks he spent on the G-Schreiber,” writes Peter Jones in his foreword to The Codebreakers, Bengt Beckman’s account of the exploit. “In 1976 he was interviewed about his work by a group from the Swedish military, and became extremely irritated when pressed for an explanation. He finally responded, ‘A magician does not reveal his tricks.’ It seems the only clue Beurling ever offered was the remark, cryptic itself, that threes and fives were important.”

(Thanks, John.)



In 1962 mycologist R.W.G. Dennis reported a new species of fungus he had observed growing in Lancashire and East Africa. He called it Golfballia ambusta:

The unopened fruit body evidently closely resembles certain small, hard but elastic, spheres employed by the Caledonians in certain tribal rites, practised at all seasons of the year in enclosures of partially mown grass set apart for the purpose. The diameter of the volva is approximately 3 cm., its surface smooth or regularly furrowed, becoming much wrinkled after dehiscence, its texture extremely hard and tough. A gelatinous stratum, so characteristic of other phalloids, is wanting. The appearance and texture of the immature gleba is still unknown but at maturity it is extruded as a column, thickly set with short strap-like processes of an elastic consistency, each scarcely 1 cm. long and 1.5 mm. wide, abruptly truncated at the free end. As with other phalloids, there is a strong and distinctive odour, in this instance not unpleasant and identified independently by several observers as reminiscent of old or heated india-rubber. This is probably a reliable and important diagnostic character. Taste not recorded but probably mild; the fruit bodies are unlikely to be toxic but may well prove inedible from their texture. Spores have not been recovered and the means of reproduction therefore remains unknown.

It seems to be very prolific in America as well.

(R.W.G. Dennis. A remarkable new genus of phalloid in Lancashire and East Africa, Journ. Kew Guild. 8, 67 (1962): 181-182.)

Building Codes


Writing on “The Sagacity of the Bees” in fourth century, Pappus of Alexandria argued that bees had contrived the hexagonal shape of their honeycomb cells “with a certain geometrical forethought.” Irregularly shaped cells “would be displeasing to the bees,” he wrote, and only triangles, squares, or hexagons could fill the space regularly. “The bees in their wisdom chose for their work that which has the most angles, perceiving that it would hold more honey than either of the two others.”

In 1964, in a charming address titled “What the Bees Know and What They Do Not Know,” Hungarian mathematician László Tóth told the American Mathematical Society that he had found a slight improvement on the classic honeycomb design: Instead of closing the bottom of each cell with three rhombi, as bees do, it’s more efficient to use two hexagons and two rhombi.

But, he added immediately, “We must admit that all this has no practical consequence. By building such cells the bees would save per cell less than 0.35% of the area of an opening (and a much smaller percentage of the surface-area of a cell). On the other hand, the walls of the bee-cells have a non-negligible width which is, in addition, far from being uniform and even the openings of the bee-cells are far from being exactly regular. Under such conditions the above ‘saving’ is quite illusory. Besides, the building style of the bees is definitely simpler than that described above. So we would fail in shaking someone’s conviction that the bees have a deep geometrical intuition.”

(László Fejes Tóth, “What the Bees Know and What They Do Not Know,” Bulletin of the American Mathematical Society, 1964, 468-81.)

UPDATE: Wait — maybe they’re not as smart as we thought. (Thanks, Vic.)

Fish Story


David Hume argued that reports of miracles can never be credited, because the weight of human experience must always favor a more natural explanation. “Nothing is esteemed a miracle, if it ever happen in the common course of nature. It is no miracle that a man, seemingly in good health, should die on a sudden: because such a kind of death, though more unusual than any other, has yet been frequently observed to happen. But it is a miracle, that a dead man should come to life; because that has never been observed in any age or country. There must, therefore, be a uniform experience against every miraculous event, otherwise the event would not merit that appellation.”

The sun is said to have danced in the sky in 1917. Well, which is more likely, that such an extraordinary event actually occurred, or that it was really a mass hallucination, an optical illusion, or any of a hundred more familiar explanations? A miracle, a suspension of natural law, is always the least likely possibility, so as rational creatures we must always reject it.

But Alfred Russel Wallace objected, “Such a simple fact as the existence of flying fish could never be proved, if Hume’s argument is a good one; for the first man who saw and described one, would have the universal experience against him that fish do not fly, or make any approach to flying, and his evidence being rejected, the same argument would apply to the second, and to every subsequent witness.”

Hume’s argument, he said, was “radically fallacious,” because if it were sound “no perfectly new fact could ever be proved, since the first and each succeeding witness would be assumed to have universal experience against him.” Who’s right?


I don’t know who came up with this — it’s been bouncing around science journals for 50 years:

hydromicrobiogeochemist: one who studies small underwater flora and their relationship to underlying rock strata by using chemical methods

microhydrobiogeochemist: one who studies flora in very small bodies of water and their relationship to underlying rock strata by using chemical methods

microbiohydrogeochemist: one who studies small flora and their relationship to underlying rock strata by using chemical methods and SCUBA equipment

biohydromicrogeochemist: a very small geochemist who studies the effect of plant life in hydrology

hydrobiomicrogeochemist: a very small geochemist who studies wet plants

biomicrohydrogeochemist: a very small, wet geochemist who likes lettuce

First Principles

This prudence of not attempting to give reasons before one is sure of facts, I learnt from one of your sex, who, as Selden tells us, being in company with some gentlemen that were viewing, and considering something which they called a Chinese shoe, and disputing earnestly about the manner of wearing it, and how it could possibly be put on; put in her word, and said modestly, Gentlemen, are you sure it is a shoe? — Should not that be settled first?

— Benjamin Franklin, letter to Mary Stevenson, Sept. 13, 1760

Crowd Control


In July 1968, ethologist John B. Calhoun built a “mouse utopia,” a metal enclosure 9 feet square with unlimited food, water, and nesting material. He introduced four pairs of mice, and within a year they had multiplied to 620. But after that the society began to fall apart — males became aggressive, females began neglecting their young, and the weaker mice were crowded to the center of the pen, where resources were scarce. After 600 days the females stopped reproducing and the males withdrew from them entirely, and by January 1973 the whole colony was dead. Even when the population had returned to its former levels, the mice’s behavior had remained permanently changed.

There were no predators in the mouse universe; the only adversity was confinement itself. Calhoun felt that his experiment held lessons as to the potential dangers of human overpopulation, and he urged his colleagues to study the effects of high population density on human behavior. “Our success in being human has so far derived from our honoring deviance more than tradition,” he said. “Now we must search diligently for those creative deviants from which, alone, will come the conceptualization of an evolutionary designing process. This can assure us an open-ended future toward whose realization we can participate.”

(Thanks, Pål.)

Club Science


From Sir Edward Victor Appleton’s speech at the 1947 Nobel Banquet:

Ladies and gentlemen, you should not … overrate scientific methods, as you will learn from the story of a man who started an investigation to find out why people get drunk. I believe this tale might interest you here in Sweden. This man offered some of his friends one evening a drink consisting of a certain amount of whisky and a certain amount of soda water and in due course observed the results. The next evening he gave the same friends another drink, of brandy and soda water in the same proportion as the previous night. And so it went on for two more days, but with rum and soda water, and gin and soda water. The results were always the same.

He then applied scientific methods, used his sense of logic and drew the only possible conclusion — that the cause of the intoxication must have been the common substance: namely the soda water!

That’s from Ronald Clark, Sir Edward Appleton, 1971. Clark adds, “Appleton was pleased but a little surprised at the huge success of the story. Only later did he learn that the Crown Prince drank only soda water — ‘one of those unexpected bonuses which even the undeserving get from Providence from time to time,’ as he put it.”