Swiss artist Max Bill conceived the Möbius strip independently of August Möbius, who discovered it in 1858. Bill called his figure Eindeloze Kronkel (“Endless Ribbon”), after the symbol of infinity, ∞, and began to exhibit it in various sculptures in the 1930s. He recalled in a 1972 interview:

I was fascinated by a new discovery of mine, a loop with only one edge and one surface. I soon had a chance to make use of it myself. In the winter of 1935-36, I was assembling the Swiss contribution to the Milan Triennale, and there was able to set up three sculptures to characterize and accentuate the individuality of the three sections of the exhibit. One of these was the Endless Ribbon, which I thought I had invented myself. It was not long before someone congratulated me on my fresh and original reinterpretation of the Egyptian symbol of infinity and of the Möbius ribbon.

He pursued mathematical inspirations actively in his later work. He wrote, “The mystery enveloping all mathematical problems … [including] space that can stagger us by beginning on one side and ending in a completely changed aspect on the other, which somehow manages to remain that selfsame side … can yet be fraught with the greatest moment.”

When chemists at the University of California at Berkeley discovered elements 97 and 98, they named them berkelium and californium. The New Yorker suggested that the school showed “a surprising lack of public-relations foresight”: “Now it has lost forever the chance of immortalizing itself in the atomic tables with some such sequence as universitium (97), ofium (98), californium (99), berkelium (100).”

The discoverers sent back a reply: “By using these names first, we have forestalled the appalling possibility that after naming 97 and 98 ‘universitium’ and ‘ofium’, some New Yorker might follow with the discovery of 99 and 100 and apply the names ‘newium’ and ‘yorkium’.”

The magazine answered, “We are already at work in our office laboratories on ‘newium’ and ‘yorkium’. So far we just have the names.”

Mathematician Paul Erdős had no home, no job, and no hobbies. Instead, for 60 years he wandered the world, staying with each of hundreds of collaborators just long enough to finish a project, and then moving on. In this week’s episode of the Futility Closet podcast we’ll meet the “magician of Budapest,” whose restless brilliance made him the most prolific mathematician of the 20th century.

We’ll also ponder Japanese cannibalism in World War II and puzzle over a senseless stabbing.

Ronald L. Graham, Jaroslav Nesetril, Steve Butler, eds., The Mathematics of Paul Erdős, 2013.

Rodrigo De Castro and Jerrold W. Grossman, “Famous Trails to Paul Erdős,” Mathematical Intelligencer 21:3 (January 1999), 51–53.

Bruce Torrence and Ron Graham, “The 100th Birthday of Paul Erdős/Remembering Erdős,” Math Horizons 20:4 (April 2013), 10-12.

Krishnaswami Alladi et al., “Reflections on Paul Erdős on His Birth Centenary,” Parts I and II, Notices of the American Mathematical Society 62:2 and 62:3 (February and March 2015).

Béla Bollobás, “To Prove and Conjecture: Paul Erdős and His Mathematics,” American Mathematical Monthly 105:3 (March 1998), 209-237.

N Is a Number: A Portrait of Paul Erdős, Kanopy Streaming, 2014.

“Paul Erdős,” MacTutor History of Mathematics Archive (accessed June 10, 2017).

Above: Erdős teaching 10-year-old Terence Tao in 1985. Tao is now recognized as one of the world’s finest mathematicians; he received the Fields Medal in 2006.

This week’s lateral thinking puzzle was contributed by listener Waldo van der Waal, who sent this corroborating link (warning — this spoils the puzzle).

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You can also make a one-time donation on the Support Us page of the Futility Closet website or buy merchandise in our store.

Many thanks to Doug Ross for the music in this episode.

If you have any questions or comments you can reach us at podcast@futilitycloset.com. Thanks for listening!

If we make a solid wooden frame with the sentence written on its front, and the time-scale on one of its sides, if we spread flatly a sheet of India rubber over its top, on which rectangular co-ordinates are painted, and slide a smooth ball under the rubber in the direction from 0 to ‘yesterday,’ the bulging of the membrane along this diagonal at successive moments will symbolize the changing of the thought’s content in a way plain enough, after what has been said, to call for no more explanation. Or to express it in cerebral terms, it will show the relative intensities, at successive moments, of the several nerve-processes to which the various parts of the thought-object correspond.

He was grappling with the stream of consciousness, the notion that thought is a flowing stream rather than a distinct chain of ideas, and with the realization that studying this by introspection is ultimately futile: “The rush of thought is so headlong that it almost always brings us up at the conclusion before we can arrest it. Or if our purpose is nimble enough and we do arrest it, it ceases forthwith to be itself. … The attempt at introspective analysis in these cases is in fact like seizing a spinning top to catch the motion, or trying to turn up the gas quickly enough to see how the darkness looks.”

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.)

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.”

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.

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).

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.)