The SNARC Effect

In 1993, cognitive neuroscientist Stanislas Dehaene asked respondents to classify a number as larger or smaller than 65, using response keys held in their hands. Interestingly, the subjects who held the “smaller” key in their left hand and the “larger” key in their right responded more quickly and with fewer errors than those in the opposite group. This suggests that we carry around a mental number line in our heads, implicitly associating left with “small” and right with “large”; the subjects in the slower group may have been fighting against this prejudice. Dehaene calls this the SNARC effect, for “spatial-numerical association of response codes.”

The effect was borne out in later studies. When subjects were asked to cross their arms, the group whose “smaller” button lay to their left were still faster than their counterparts. And the effect still obtains regardless of the range of numbers used, and even in tasks where the size of the number is irrelevant: In another experiment subjects were asked to report whether a given number was odd or even; here again, responses to numbers in the upper half of the test range were quicker when the appropriate response key was on the right, and likewise for small numbers on the left.

Interestingly, Iranian students living in France who had initially learned to read from right to left showed a reverse SNARC effect (associating small numbers with the right and large numbers with the left) if they’d recently immigrated, but those who had lived in France for some time showed the same SNARC effect as native French students.

“Very probably, then, this number-space association is learned, not innate,” writes M. Giaquinto in Visual Thinking in Mathematics. “But there may very well be an innate propensity in operation here. A left-right association has been found for familiar ordered sets of non-numerical items, namely, months and letters. This suggests that we have a tendency to form a linear spatial representation of any set of things whose customary presentation is well ordered (in the mathematical sense).”

(S. Dehaene, S. Bossini, and P. Giraux, “The Mental Representation of Parity and Numerical Magnitude,” Journal of Experimental Psychology: General 122, 371-396. See Number Forms.)

Figure and Ground

Typographer John Langdon designed this ambigram for the Department of English & Philosophy at his institution, Drexel University.

“This illusion was a particularly difficult challenge,” he told Brad Honeycutt for The Art of Deception (2014). “My attempts to create more ‘conventional’ (rotational, mirror-image, etc.) ambigrams for these two words were unsuccessful. But my personal investments in both philosophy and language seem to inspire me to some of my best work. This ‘perception shift’ ambigram was very difficult to develop, but my stubborn persistence finally paid off. The two words ‘philosophy’ and ‘English’ can be difficult to discern, but with a little patience and a voluntary perception shift, finding them is particularly satisfying.”

There’s much more at Langdon’s site.

Tatlin’s Tower

tatlin's tower

After the Bolshevik Revolution, architect Vladimir Tatlin proposed this enormous monument to house Communist headquarters in Petrograd. Two large helixes would spiral 400 meters into the air, surpassing the Eiffel Tower as the world’s foremost symbol of modernity. The helixes would point to Polaris, so that the star and the tower would remain motionless relative to each other. Suspended from the framework would be three office buildings of glass and steel, each moving in harmony with the cosmos: A is a cylindrical auditorium that rotates once a year, B is a cone-shaped office structure that rotates once a month, C is a cubical information center that rotates once a day, and on top is an open-air screen on which messages could be projected. (During overcast weather they planned to project the news onto clouds.)

In the end it was never built — even if Russia could have produced the steel, it’s not clear that it would have stood up.

The Banana Bat

This would have livened things up: In 1890 inventor Emile Kinst promoted an “improved ball-bat” that he said would set baseballs spinning: “The object of my invention is to provide a ball-bat which shall produce a rotary or spinning motion of the ball in its flight to a higher degree than is possible with any present known form of ball-bat, and thus to make it more difficult to catch the ball, or if caught, to hold it.” It would also enable hitters to drive the ball more easily to every part of the field.

“Owing to the peculiar form of my bat, the game becomes more difficult to play, and therefore much more interesting and exciting, because the innings will not be so easily attained, and consequently the time of the game will also be shortened.” The Major League Rules Committee said no.

BTW, in recent weeks I’ve come across two sources that say that Ted Williams once returned a set of bats to the manufacturer with a note saying, “Grip doesn’t feel just right.” The bats were found to be 0.005″ thinner than he had ordered. I don’t know whether it’s true. The sources are Spike Carlsen’s A Splintered History of Wood and Dan Gutman’s Banana Bats & Ding-Dong Balls: A Century of Unique Baseball Inventions (where I found the bat above).

World View

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 the early 20th century, inspired by the scientific hope that the mind could evolve to ever-higher levels of consciousness, Russian poets tried to paint this higher reality with paradoxical statements that defied common sense. This movement reached its apotheosis in 1913 when Aleksei Kruchenykh wrote “Dyr bul shchyl,” an untranslatable arrangement of letters on a page. Kruchenykh added the legend “3 poems written in my own language different from others: its words do not have a definite meaning.” Kruchenykh named this new language zaum, Russian for transrational, because it transcends common sense and logic.

“For whom was Kruchenykh writing these poems?” asks Lynn Gamwell in Mathematics + Art. “He wrote for other avant-garde poets — he was a poet’s poet — and, according to his late-nineteenth-century biological worldview, the poets in his audience possessed expanded (more highly evolved) minds. Kruchenykh did not communicate in the ordinary sense — he deliberately chose obfuscation, made-up words. He composed his poems to reach a small, elite art audience whose brains, to his way of thinking, had evolved enough to perceive an actual infinity — the Absolute. In other words, the subject matter of his verse was neither nonsense nor an occult realm but rather an alleged higher ‘transrational’ level of reasoning.”


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 1899 Winston Churchill was covering the Boer War as a correspondent when he was captured and put in a Pretoria prison. He climbed a wall and set out to flee 300 miles to neutral Portuguese East Africa while the Afrikaners raised the alarm and circulated a rather unflattering description:

Escaped prisoner-of-war Winston Spencer Churchill Englishman 25 years old about 5 foot 8 inches tall medium build walks with a slight stoop. Pale features. Reddish-brown hair almost invisible small moustache. Speaks through his nose and cannot pronounce the letter S. Had last a brown suit on and cannot speak one word of Dutch.

Churchill fled on foot for two days, hid in a coal mine for three, and finally boarded a freight train, where he hid under bales of wool to evade a Boer search party. When he reached safety, publicity of his adventure set him on the path toward a career in government.