Escalating Magic

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.)

Wayward Pigeons

In 1998, thousands of pigeons mysteriously went missing during two separate races in Virginia and Pennsylvania. More than 2,200 birds vanished, amounting to an 85 percent loss rate. The weather was calm, and it’s normal for a few birds to disappear, but the rate is usually closer to 5 percent.

It’s known that pigeons navigate by the sun and by sensing magnetic fields, but neither of those seems to be the culprit here. “Every year or so, you have one race like this where many disappear,” Cornell zoologist Charles Walcott told the Chicago Tribune. “But what is unusual is to lose so many birds from several races at the same time. What’s going on now is quite mysterious.”

Related: In 2010 a racing pigeon named Houdini disappeared during a 224-mile race in Britain and turned up five weeks later in Panama, 5,200 miles away.

“I was gobsmacked. I didn’t even know where Panama was,” owner Darren Cubberley told the Daily Mirror. “I’ve no idea how Houdini got there — I can only assume she hitched a lift on a ship across the Atlantic.”

The bird, reportedly in “perfect shape,” would have been too expensive to return, so she remained with Gustavo Ortiz, on whose roof she’d landed. At last report she was learning Spanish.

The Veiled Virgin
Image: Wikimedia Commons

Sculptor Giovanni Strazza probably completed this bust of a veiled Virgin Mary in the early 1850s. It was transported to Newfoundland and placed in the Episcopal Palace next to St. John’s Basilica.

“To say that this representation surpasses in perfection of art, any piece of sculpture we have ever seen, conveys but weakly our impression of its exquisite beauty,” wrote a local newspaper. “The possibility of such a triumph of the chisel had not before entered into our conception. Ordinary language must ever fail to do justice to a subject like this — to the rare artistic skill, and to the emotions it produces in the beholder.”

The Sea Island Problem

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 …
Image: Wikimedia Commons

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

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:

 \displaystyle \frac{CD \times DF}{FH - DG} + CD

and the distance between the first pole and the mountain:

 \displaystyle \frac{DG \times DF}{FH - DG}

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.”


Rejection letter sent by the University of Portland’s Portland Magazine, devised by editor Brian Doyle:

Thank you for your lovely and thoughtful submission to the magazine, which we are afraid we are going to have to decline, for all sorts of reasons. The weather is dreary, our backs hurt, we have seen too many cats today and as you know cats are why God invented handguns, there is a sweet incoherence and self-absorption in your piece that we find alluring but we have published far too many of same in recent years mostly authored by the undersigned, did we mention the moist melancholy of the weather, our marriages are unkempt and disgruntled, our children surly and crammed to the gills with a sense of entitlement that you wonder how they will ever make their way in the world, we spent far too much money recently on silly graphic design and now must slash the storytelling budget, our insurance bills have gone up precipitously, the women’s basketball team has no rebounders, an aunt of ours needs a seventh new hip, the shimmer of hope that was the national zeitgeist looks to be nursing a whopper of a black eye, and someone left the toilet roll thing empty again, without the slightest consideration for who pays for things like that. And there were wet towels on the floor. And the parakeet has a goiter. And the dog barfed up crayons. Please feel free to send us anything you think would fit these pages, and thank you for considering our magazine for your work. It’s an honor.

From Letters of Note.

Podcast Episode 234: The Dig Tree,_Wills_and_King,_1861.jpg

In 1860 a party of explorers set out to traverse the Australian continent, but bad management and a series of misfortunes sent it spiraling toward tragedy. In this week’s episode of the Futility Closet podcast we’ll tell the story of the Victorian Exploring Expedition and its dramatic climax at Cooper’s Creek.

We’ll also try to validate Archimedes and puzzle over an unlucky thief.

See full show notes …

Good and Ugly

Technical but interesting: Designer Iginio Lardani’s title sequence for The Good, the Bad and the Ugly contains only one error, and it’s in Lardani’s own credit (2:33 above).

“[A]n ‘error’ specific to optical printing caused by improper loading of the footage and mask being composited — Newton’s rings — appears in his title card alongside the text stating ‘TITLES | LARDANI,” notes film historian Michael Betancourt in Semiotics and Title Sequences (2017).

This appears to be an inside joke meant for other title designers. According to his son, Lardani had complete freedom in creating the design. Betancourt writes, “Recognizing this specific ‘error’ in the card stating ‘TITLES LARDANI’ depends on technical knowledge of the optical printing process. … given the technical perfection in the rest of the sequence, it is not just a ‘beginner’s mistake,’ but implies a conscious choice to include this compositing error in the design.”

“It is a joke only comprehensible (even recognizable) by an audience that recognizes the Newton’s rings and understands what they are — an error in the optical printing; this knowledgeable audience specifically includes title sequence designers rather than the general public. … Because it is specifically a specialized, technical mistake, its recognition will be severely limited to his peers — suggesting that they are the ones being addressed by it.”

Mood Indigo

In the 1870s Philadelphia’s Augustus Pleasonton convinced himself that blue light had almost miraculously beneficial properties: He and his followers insisted that it cured disease, promoted hair growth, banished deafness, even resolved insanity. In 1871 he patented a greenhouse (or bluehouse, I suppose) that would “accelerate the growth and maturity of plants and animals.”

It fell to Scientific American to point out that cobalt blue glass diminishes all rays across the visible spectrum; it just diminishes blue and violet light somewhat less than other wavelengths. A plant (or anything else) would receive more blue light simply standing in the sun than hidden in Pleasonton’s hut.

The fad faded, and by the inventor’s death in 1894 it had been forgotten. “It is amusing to see people making fools of themselves,” observed the Boston Globe, “but it soon grows wearisome.”

Set Dressing

William Wellman vowed that his 1927 dogfight movie Wings would contain no stock footage or studio fakery. But after assembling his own planes and film crew, he kept them resolutely on the ground for weeks. When Paramount asked what was missing, he gave them a bewildering answer: clouds.

“Motion on the screen is a relative thing,” he said. “A horse runs on the ground or leaps over fences or streams. We know he is going rapidly because of his relation to the immobile ground.” But a plane alone in the sky may produce no sense of motion at all unless there are clouds around it — and the skies above Wellman’s San Antonio shooting location remained stubbornly clear.

Producer Jesse Lasky later wrote, “There were days on end of perfect sunshine, and our $200-a-week director wouldn’t turn a camera, while overhead mounted at thousands of dollars a day. I confess that we were about ready to yank him off the picture and replace him with someone who would be more amenable.”

In the end they were glad they waited — when the clouds arrived, Wellman’s cameramen took off, and Wings won the first Academy Award for Best Picture.