Reckoning

Mercyhurst College mathematician Charles Redmond used to get frantic phone calls at the end of each term from colleagues in other departments who’d written syllabi saying “Quiz 1 is 10% of your final grade” and now couldn’t figure out how to do the necessary calculation.

“There’s a good reason,” Redmond wrote. “They said something they didn’t mean to say, and they had only a vague notion of what they meant to say in the first place.”

He suggested that students might turn this confusion to their advantage. Suppose Quiz 1 is worth 10 or more points and your score is 9. If x is your final grade and “Quiz 1 is 10% of your final grade,” then 0.10x = 9 and suddenly your final grade is 90, an A.

“Not bad for about a week’s worth of work. Take the rest of the semester off.”

(Ed Barbeau, “Fallacies, Flaws, and Flimflam,” College Mathematics Journal 33:2 [March 2002], 137-139.)

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Another Way

Carnegie-Mellon mathematician Po-Shen Loh has offered a new, simple proof of the quadratic formula that provides a natural, intuitive algorithm for solving general quadratic equations.

More here. “May this story encourage the reader to think afresh about old things; seeing as how new progress was made on this 4,000 year old topic, more surprises certainly await the light of discovery.”

(Thanks, Jason.)

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First Friends

In 1978, archaeologists excavating a late Paleolithic tomb in northern Israel uncovered the skeletons of an elderly human and a 5-month-old puppy. They had lain there together for 12,000 years.

“The most striking thing about these remains was the fact that whoever presided over the original burial had carefully arranged the dead person’s left hand so that it rested, in a timeless and eloquent gesture of attachment, on the puppy’s shoulder,” writes James Serpell in In the Company of Animals (1996).

“The contents of this tomb not only provide us with some of the earliest solid evidence of animal domestication, they also strongly imply that man’s primordial relationship with this particular species was an affectionate one. In other words, prehistoric man may have loved his dogs and his other domestic animals as pets long before he made use of them for any other purpose.”

(Simon J.M. Davis and François R. Valla, “Evidence for Domestication of the Dog 12,000 Years Ago in the Natufian of Israel,” Nature 276:5688 [1978], 608.)

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The Friendly Floatees

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Image: Wikimedia Commons

During a storm in January 1992, a container was swept overboard from a ship in the North Pacific. As it happened, it contained 28,800 children’s bath toys, and oceanographer Curtis Ebbesmeyer realized they offered the basis for a serendipitous study of surface currents. Working with his colleague James Ingraham, Ebbesmeyer began to track the toys as they drifted around the globe, accumulating reports from beachcombers, coastal workers, and local residents as they began to wash up on beaches. Using computer models, they were able to predict correctly that toys would make landfall in Washington state, Japan, and Alaska, and even become trapped in pack ice and spend years creeping across the top of the world before making an eventual reappearance in the North Atlantic. “Ultimately,” Ebbesmeyer wrote, “the toys will turn to dust, joining the scum of plastic powder which rides the global ocean.”

For some reason, media accounts of the story always carried the image of a solitary rubber duck, though the toys had also included beavers, turtles, and frogs. “Maybe it’s a kind of racism,” Ebbesmeyer speculated to journalist Donovan Hohn in 2007. “Speciesism.”

See Ambassador.

| Oddities · Science & Math

One of a Kind

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Image: Wikimedia Commons

The Yemeni island Socotra, off the tip of the Horn of Africa in the Arabian Sea, is so isolated that nearly 700 of its species are found nowhere else on Earth. The island’s bitter aloe has valuable pharmaceutical and medicinal properties, and the red sap of Dracaena cinnabari, above, was once thought to be the blood of dragons.

And this is only what remains after two millennia of human settlement; the island once featured wetlands and pastures that were home to crocodiles and water buffaloes. Tanzanian zoologist Jonathan Kingdon says, “The animals and plants that remain represent a degraded fraction of what once existed.”

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A Circle Theorem

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Image: Wikimedia Commons

If two chords of a circle intersect in a particular angle at S, then the sum of the opposite arc lengths (say, AB + CD) remains the same regardless of the position of S within the circle.

(Nick Lord and David Wells, “A Circular Tour of Some Circle Theorems,” Mathematical Gazette 73:465 [1989], 188-191.)

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What Needs More Words?

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Ecologists often have to estimate the number of unseen species in an ecosystem: If I count x species of butterfly during my time on an island, how many species probably live there that I did not see? In 1975, Stanford statisticians Bradley Efron and Ronald Thisted applied the same question to the works of William Shakespeare: If we take the Bard’s existing works as a sample, what can we infer about the size of his total vocabulary?

Shakespeare’s known works comprise 884,647 words, which fall into 31,534 “types,” or distinguishable arrangements of letters. Efron and Thisted applied two approaches and found that they produced the same estimate: If a new cache of the playwright’s works were discovered today, equal in size to the old, it would likely contain about 11,460 new word types, with an expected error of less than 150.

So how many word types altogether did Shakespeare know? No upper bound is possible, but they established a lower bound of 35,000 beyond the 31,534 already used — in other words, to write the works that we know of, he likely used less than half his total vocabulary.

(Bradley Efron and Ronald Thisted, “Estimating the Number of Unseen Species: How Many Words Did Shakespeare Know?”, Biometrika 63:3 [1976], 435-447.) (Thanks, Brent.)

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Mercury Pendulums

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This is just a detail that I found interesting — early regulator clocks tended to slow down as their pendulums lengthened in warm conditions. One solution, offered by George Graham in 1721, was to attach two vials of mercury to the pendulum — as the pendulum warmed and expanded, so did the mercury, creeping upward in its vials and, at least in theory, preserving the pendulum’s center of mass.

One difficulty is that the mercury tends to warm more slowly than the pendulum itself, but the system worked well enough to persist into the 20th century.

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All Together Now

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Image: Wikimedia Commons

A neat astronomy fact: At the equinox, the sun rises due east and sets due west at every latitude on Earth (except at the poles, where east and west are undefined).

The celestial equator is the great circle writ on the sky above our own equator. For any point on Earth (except the poles), due east and due west mark the intersection of that circle with the horizon. At the equinox the sun is on the celestial equator, so it rises due east and sets due west, not just on our equator but everywhere.

(Thanks, Sanford.)

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