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.”

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.)

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:

XVIII = IV × II + X

LXXXIII = IX^X×X/L + II

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

LXXXIV = LX / X × XIV

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

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:

Another example, for Row 10:

The same can be done for any desired row.

(Thanks, Alex.)