In Visual Thinking in Mathematics, M. Giaquinto writes, “Calculus grew out of attempts to deal with quantitative physical problems which could not be solved by means of geometry and arithmetic alone. Many of these problems concern situations which are easy to visualize. In fact visual representations are so useful that most books on calculus are peppered with diagrams.” But there’s an intriguing footnote: “Moshé Machover brought to my attention a notable exception: Landau (1934). It has no diagram, and no geometrical application.”

That’s Differential and Integral Calculus, by Edmund Landau, a professor of mathematics at Gottingen University. Machover is right — the 366-page volume contains not a single diagram. Landau writes, “I have not included any geometric applications in this text. The reason therefor is not that I am not a geometer; I am familiar, to be sure, with the geometry involved. But the exposition of the axioms and of the elements of geometry — I know them well and like to give courses on them — requires a separate volume which would have to precede the present one. In my lecture courses on the calculus, the geometric applications do, of course, make up a considerable portion of the material that is covered. But I do not wish to wait any longer to make generally available an account, rigorous and complete in every particular, of that which I have considered in my courses to be the most suitable method of treating the differential and integral calculus.”

The book was quite successful — the first English edition appeared in 1950, and subsequent editions have continued right up through 2001.

Which of these faces is male, and which female? In fact both photos show the same androgynous face; the only difference is the amount of contrast in the image. But most people see the face on the left as female and the one on the right as male.

Gettysburg College psychologist Richard Russell says, “Though people are not consciously aware of the sex difference in contrast, they unconsciously use contrast as a cue to tell what sex a face is. We also use the amount of contrast in a face to judge how masculine or feminine the face is, which is related to how attractive we think it is.”

Cosmetics may serve to make a female face more attractive by heightening this contrast. “Cosmetics are typically used in precisely the correct way to exaggerate this difference,” Russell says. “Making the eyes and lips darker without changing the surrounding skin increases the facial contrast. Femininity and attractiveness are highly correlated, so making a face more feminine also makes it more attractive.”

(Richard Russell, “A Sex Difference in Facial Pigmentation and Its Exaggeration by Cosmetics,” Perception 38:8 [August 2009], 1211-1219.)

The digits 1-9 can be arranged into a 3 × 3 magic square in essentially one way (not counting rotations or reflections) — the so-called lo shu square:

4    3    8

9    5    1

2    7    6

As in any magic square, each row, column, and diagonal produces the same total. But surprisingly (to me), the sum of the row products also equals the sum of the column products:

4 × 3 × 8 + 9 × 5 × 1 + 2 × 7 × 6 = 96 + 45 + 84 = 225

4 × 9 × 2 + 3 × 5 × 7 + 8 × 1 × 6 = 72 + 105 + 48 = 225

Even more surprisingly, the same is true of the Fibonacci sequence, if we arrange its first nine terms into a square array in the same pattern:

 3    2   21

34    5    1

 1   13    8

3 × 2 × 21 + 34 × 5 × 1 + 1 × 13 × 8 = 126 + 170 + 104 = 400

3 × 34 × 1 + 2 × 5 × 13 + 21 × 1 × 8 = 102 + 130 + 168 = 400

It turns out that this is true of any second-order linear recursion. (The sums won’t always be squares, though.)

From Edward J. Barbeau’s Power Play, 1997.

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.”

A low-tech model of human cognition, from William James’ The Principles of Psychology, 1890:

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.)

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 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’.)