borwein integrals

These are called Borwein integrals, after David and Jonathan Borwein, the father-and-son mathematicians who first presented them in 2001.

Engineer Hanspeter Schmid writes, “[W]hen this fact was recently verified by a researcher using a computer algebra package, he concluded that there must be a ‘bug’ in the software. It is not a bug, though; this series of integrals really only results in π/2 up to a certain point, and then breaks down. This astonishes most mathematically educated readers, as especially those readers mentally extrapolate the sequence shown above and find it surprising that something fundamental should change when the factor sinc(x/15) is introduced.” He gives a graphic explanation of what’s happening.

(David Borwein and Jonathan M. Borwein, “Some Remarkable Properties of Sinc and Related Integrals,” Ramanujan Journal 5:1 [March 2001], 73–89.) (Thanks, Dan.)

Sanity and Simpson

In his 2008 book Impossible?, Julian Havil presents an argument offered in Massachusetts in 1854 contending that foreigners were more likely to be insane than native-born Americans. These figures were offered:

Whole Population
Insane Not Insane Totals
Foreign-Born 625 229375 230000
Native-Born 2007 892669 894676
Totals 2632 1122044 1124676

The probability that a foreign-born person was deemed insane was 625/230000 = 2.7 × 10-3, and for a native-born person the probability was 2007/894676 = 2.2 × 10-3, which seems to support the claim.

But we get a different story when we divide the data by social hierarchy, into what were called the pauper and independent classes:

Pauper Class
Insane Not Insane Totals
Foreign-Born 182 9090 9272
Native-Born 250 12513 12763
Totals 432 21603 22035
Independent Class
Insane Not Insane Totals
Foreign-Born 443 220285 220728
Native-Born 1757 880156 881913
Totals 2200 1100441 1102641

In the pauper class the probability of a foreign-born person being deemed insane is 182/9272 = 0.02, which is the same as that for a native-born person (250/12763 = 0.02). And the same is true in the independent class, where both probabilities are 2.0 × 10-3. Havil writes, “So, if an adjustment is made for the status of the individuals we see that there is no relationship at all between sanity and origin” (an example of Simpson’s paradox).

Current Events
Image: Wikimedia Commons

“Squaring the square” refers to tiling a square with other squares, each with sides of integer length.

In a “perfect” squared square, like the one above, each smaller square is of a different size. The Cambridge University team that first sought perfect squares found a novel way to go about it — they transformed the square tiling into an electrical circuit in which each square is a resistor that connects to its neighbors above and below, and then applied Kirchhoff’s circuit laws to that circuit.

The example below isn’t perfect, but the technique did succeed — the smallest perfect square they found is 69 units on a side.
Image: Wikimedia Commons

(Rowland Leonard Brooks, et al., “The Dissection of Rectangles into Squares,” Duke Mathematical Journal 7:1 [1940], 312-340.)

Curioser and Curioser

“I’m sure I’m not Ada, for her hair goes in such long ringlets, and mine doesn’t go in ringlets at all; and I’m sure I can’t be Mabel, for I know all sorts of things, and she, oh! she knows such a very little! Besides, she’s she, and I’m I, and — oh dear, how puzzling it all is! I’ll try if I know all the things I used to know. Let me see: four times five is twelve, and four times six is thirteen, and four times seven is — oh dear! I shall never get to twenty at that rate!”

So frets Alice outside the garden in Wonderland. Is the math only nonsense? In his book The White Knight, Alexander L. Taylor finds one way to make sense of it: In base 18, 4 times 5 actually is 12. And in base 21, 4 times 6 is 13. Continuing this pattern:

4 times 7 is 14 in base 24
4 times 8 is 15 in base 27
4 times 9 is 16 in base 30
4 times 10 is 17 in base 33
4 times 11 is 18 in base 36
4 times 12 is 19 in base 39

And here it becomes clear why Alice will never get to 20: 4 times 13 doesn’t equal 20 in base 42, as the pattern at first seems to suggest, but rather 1X, where X is whatever symbol is adopted for 10.

Did Lewis Carroll have this in mind when he contrived the story for Alice? Perhaps? In The Magic of Lewis Carroll, John Fisher writes, “It is hard … not to accept Taylor’s theory that Carroll was anxious to make the most of two worlds; the problem as interpreted by Taylor interested him, and although it wouldn’t interest Alice there was no reason why he shouldn’t use it to entertain her on the level of nonsense. It is even more difficult to suppose that he, a mathematical don, inserted the puzzle in the book without realising it.”

In the Winter 1971 edition of Jabberwocky, the quarterly publication of the Lewis Carroll Society, Taylor wrote, “If you find a watch in the Sahara Desert you don’t think it grew there. In this case we know who put it there, so if he didn’t record it that tells us something about his recording habits. It doesn’t tell us that the mathematical puzzle isn’t a mathematical puzzle.”

A Helping Fin

In January 2009, marine ecologists Robert Pitman and John Durban were watching a group of Antarctic killer whales wash a seal off an ice floe when, surprisingly, a pair of humpback whales intervened:

Exposed to lethal attack in the open water, the seal swam frantically toward the humpbacks, seeming to seek shelter, perhaps not even aware that they were living animals. (We have known fur seals in the North Pacific to use our vessel as a refuge against attacking killer whales.)

Just as the seal got to the closest humpback, the huge animal rolled over on its back — and the 400-pound seal was swept up onto the humpback’s chest between its massive flippers. Then, as the killer whales moved in closer, the humpback arched its chest, lifting the seal out of the water. The water rushing off that safe platform started to wash the seal back into the sea, but then the humpback gave the seal a gentle nudge with its flipper, back to the middle of its chest. Moments later the seal scrambled off and swam to the safety of a nearby ice floe.

“I was shocked,” Pitman said later. “It looked like they were trying to protect the seal.”

Perhaps they were. Humpbacks organize to protect their own calves, of course, but they also protect other species — indeed, this happened in nearly 90 percent of attacks where the killer whales’ prey could be identified.

Is this evidence of a moral sense among the whales? Donald Broom, an emeritus professor of animal welfare at Cambridge University, thinks so — if right means meeting individuals’ basic needs and maintaining their rights, and wrong means impeding these things, then there’s considerable evidence for a sense of morality among nonhumans. In The Cultural Lives of Whales and Dolphins, Hal Whitehead and Luke Rendell write, “When whales and dolphins go out of their way to help other creatures with their needs that looks pretty moral, at least on the surface.”

Surf’s Up

La Ola, or the “Mexican wave,” seems like the ultimate in spontaneous behavior, but biological physicist Illés Farkas of Eötvös University found that stadium waves can be studied quite effectively using the methods of statistical physics. Examining videos of waves in stadia holding more than 50,000 people, Farkas and his colleagues found that a crowd behaves like an excitable medium — the first group to stand acts as a “perturbation,” and in less than a second the wave begins, dying out on one side and continuing on the other (3 out of 4 Mexican waves travel clockwise around the stadium). And the speed of waves is surprisingly consistent — 22±3 seats per second, with an average width of 15 seats.

Farkas defined three states that a spectator can take: inactive (sitting, ready to stand), active (standing up), and refractory (sitting again afterward). Shizuoka University mechanical engineer Takashi Nagatani found that the local behavior of the spectators can be interpreted in terms of a chemically excitable medium with the following reaction set:

Passive + Activated ↦ 2 · Activated
Activated ↦ Refractory
Refractory ↦ Passive

Farkas wrote, “For a physicist, the interesting specific feature of this spectacular phenomenon is that it represents perhaps the simplest spontaneous and reproducible behaviour of a huge crowd with a surprisingly high degree of coherence and level of cooperation. In addition, La Ola raises the exciting question of the ways by which a crowd can be stimulated to execute a particular pattern of behaviour.”

(From Andrew Adamatzky, Dynamics of Crowd-Minds, 2005.)

Et Tu?

A Friedman number, named after Stetson University mathematician Erich Friedman, is a number that can be calculated using its own digits, such as 736 = 36 + 7 or 3281 = (38 + 1) / 2.

A “nice” Friedman number is one in which the digits are used in order, such as 3685 = (36 + 8) × 5 or 3972 = 3 + (9 × 7)2.

Might this be done in other number systems? In a sense all Roman numerals are automatically Friedman numbers, but there are some interesting nontrivial examples as well:



And it turns out that “nice” examples are possible here too, in which a number’s letters are used in order:

LXXVI = L / X × XV + I


Is there more? Friedman has begun looking for examples in Mayan numerals — see his website.

A Triangle Calculator
Image: Wikimedia Commons

Edric Cane came up with a simple way to establish any row in Pascal’s triangle, creating a simple sequence of fractions that, when multiplied successively, will produce the numbers in any desired row. Here’s an example for Row 7, giving the coefficients for (a + b)7 = a7 + 7a6b + 21a5b2 + 35a4b3 + 35a3b4 + 21a2b5 + 7ab6 + b7:

triangle calculator - row 7

Another example, for Row 10:

triangle calculator - row 10

The same can be done for any desired row.

(Thanks, Alex.)

An Odd Fact

Mentioned in James Tanton’s Mathematics Galore!:

In 1740 the French mathematician Philippe Naudé sent a letter to Leonhard Euler asking how many ways a positive integer could be written as a sum of distinct positive integers (regardless of their order). In considering the problem Euler found something remarkable.

Let D(n) be the number of ways to write n as a sum of distinct positive integers. So, for example, D(6) is 4 because there are four ways to do this for 6: 6, 5 + 1, 4 + 2, and 3 + 2 + 1.

And let O(n) be the number of ways to write n as a sum of odd integers. So O(6) is 4 because 6 can be written as 5 + 1, 3 + 3, 3 + 1 + 1 + 1, or 1 + 1 + 1 + 1 + 1 + 1.

Euler showed that O(N) always equals D(N).