In 1958 psychologist Robert Plutchik suggested that there are eight primary emotions: joy, sadness, anger, fear, trust, disgust, surprise, and anticipation. Each of the eight exists because it serves an adaptive role that gives it survival value — for example, fear inspires the fight-or-flight response.

He arranged them on a wheel to show their relationships, with similar emotions close together and opposites 180 degrees apart. Like colors, emotions can vary in intensity (joy might vary from serenity to ecstasy), and they can mix to form secondary emotions (submission is fear combined with trust, and awe is fear combined with surprise).

When all these combinations are included, the system catalogs 56 emotions at 1 intensity level. And in his final “structural model” of emotions, the petals are folded up in a third dimension to form a cone.

In 1833, to show that vultures found their prey by sight rather than smell, naturalist John Bachman made “a coarse painting representing a sheep skinned and cut open”:

This proved very amusing — no sooner was this picture placed on the ground than the Vultures observed it, alighted near, walked over it, and some of them commenced tugging at the painting. They seemed much disappointed and surprised, and after having satisfied their curiosity, flew away. This experiment was repeated more than fifty times, with the same result.

He confirmed the result by setting the painting within two feet of a heap of camouflaged offal in his garden. “They came as usual, walked around it, but in no instance evinced the slightest symptoms of their having scented the offal which was so near them.” He concluded that, while vultures may have a sense of smell, they don’t use it to find food.

See Vulture Picnic.

In 1956 Harvard psychologist George Miller pointed out a pattern he’d observed. If a person is trained to respond to a given pitch with a corresponding response, she’ll respond nearly perfectly when up to six pitches are involved, but beyond that her performance declines. Humans seem to have an “information channel capacity” of 2-3 bits of information: We can distinguish among 4-8 alternatives and respond appropriately, but beyond that number we start to founder.

A similar limit appears in studies of memory span. One psychologist read aloud lists of random items at a rate of one per second and then asked subjects to repeat what they’d heard. No matter what items had been read — words, letters, or numbers — people could store a maximum of about seven unrelated items at a time in their immediate memory.

It’s probably only a coincidence that these tasks have similar limits, but it’s still a useful rule of thumb: The number of objects an average person can hold in working memory is about seven.

(George A. Miller, “The Magical Number Seven, Plus or Minus Two: Some Limits on Our Capacity for Processing Information,” Psychological Review 63:2 [1956], 81-97.)

In 2003 mathematician Gregory Galperin of Eastern Illinois University offered a remarkable way to calculate π: Launch two masses toward an elastic wall, count the resulting collisions, and you can generate π to any precision, at least in principle.

“On the one hand, our method is purely mathematical and, most likely, will never be used as a practical way for finding approximations of π. On the other hand, this method is the simplest one among all the known methods (beginning from the ancient Greeks!).”

The video above, by 3Blue1Brown, gives the setup; the continuation is below. Via MetaFilter.

(Gregory Galperin, “Playing Pool With π (The Number π From a Billiard Point of View),” Regular and Chaotic Dynamics 8:4 [2003], 375-394.)

Each number in this pandiagonal order-4 magic square is a three-digit prime:

277 197 631 431
661 401 307 167
137 337 491 571
461 601 107 367

Add 30 to each cell and you get a new magic square, also made up of 16 three-digit primes:

307 227 661 461
691 431 337 197
167 367 521 601
491 631 137 397

Add 1092 to each cell in that one and you get a magic square of four-digit primes:

1399 1319 1753 1553
1783 1523 1429 1289
1259 1459 1613 1693
1583 1723 1229 1489

(Allan William Johnson Jr., “Related Magic Squares,” Journal of Recreational Mathematics 20:1 [January 1988], 26-27, via Clifford A. Pickover, The Zen of Magic Squares, Circles, and Stars, 2011.)

The Chinese mathematician Liu Hui offered this technique in a text composed about 500 years after Euclid. We’re on the mainland, and we want to find the height of a mountain on a distant island without crossing the sea.

Liu Hui showed that this can be accomplished by setting up two poles of a known height in a line with the mountain …

… and by appealing to a principle of complementary rectangles — here the red and the blue rectangles have the same area:

By using that principle it’s possible to recast the problem in terms of values that we can measure: the height of the poles (CD), the “offset” from which the top of the mountain can just be sighted from ground level over the top of each pole (DG and FH), and the distance between the poles (DF). Putting all that together we can find both the height of the mountain:

and the distance between the first pole and the mountain:

without ever leaving the mainland. Penn State University mathematician Frank Swetz concluded that “in the endeavours of mathematical surveying, China’s accomplishments exceeded those realized in the West by about one thousand years.”