More Magic

Albrecht Dürer’s 1514 engraving Melencolia I includes this famous magic square: The magic sum of 34 can be reached by adding the numbers in any row, column, diagonal, or quadrant; the four center squares; the four corner squares; the four numbers clockwise from the corners; or the four counterclockwise.

In Power Play (1997), University of Toronto mathematician Ed Barbeau points out that there’s even more magic when we consider squares and cubes. Take the numbers in the first two and the last two rows:

16 + 3 + 2 + 13 + 5 + 10 + 11 + 8 = 9 + 6 + 7 + 12 + 4 + 15 + 14 + 1

162 + 32 + 22 + 132 + 52 + 102 + 112 + 82 = 92 + 62 + 72 + 122 + 42 + 152 + 142 + 12

Or alternate columns:

16 + 5 + 9 + 4 + 2 + 11 + 7 + 14 = 3 + 10 + 6 + 15 + 13 + 8 + 12 + 1

162 + 52 + 92 + 42 + 22 + 112 + 72 + 142 = 32 + 102 + 62 + 152 + 132 + 82 + 122 + 12

Most amazingly, if you compare the numbers on and off the diagonals, this works with both squares and cubes:

16 + 10 + 7 + 1 + 13 + 11 + 6 + 4 = 2 + 3 + 5 + 8 + 9 + 12 + 14 + 15

162 + 102 + 72 + 12 + 132 + 112 + 62 + 42 = 22 + 32 + 52 + 82 + 92 + 122 + 142 + 152

163 + 103 + 73 + 13 + 133 + 113 + 63 + 43 = 23 + 33 + 53 + 83 + 93 + 123 + 143 + 153

Image: Wikimedia Commons

Colombian activist César López came up with a striking new peace symbol in 2003 — the escopetarra, a guitar fashioned from a gun.

The word combines the Spanish escopeta (shotgun) and guitarra (guitar). López made the first from a Winchester rifle and a Stratocaster; he’s since built four more and given them to various Latin American artists and cities and to the United Nations, which displayed it at a disarmament conference.

López told the BBC that he got the idea when he saw a soldier carrying his weapon like a guitar. “From there sprang the idea of joining the worst invention of mankind can be joined with the most beautiful,” he said. “Some of the AK-47s have the barrels marked with each of the victims. So we mark the barrels with the songs we play.”


The Treaty of Versailles contained an odd provision: It established a standard pitch to which orchestras could tune. For hundreds of years the agreed pitch might vary widely from one region to the next. When it became clear that the average pitch was rising over time, a French commission settled that the standard pitch for the A above middle C would be 435 hertz, and an 1885 convention of European nations adopted that as an international standard. The Treaty of Versailles in 1919 ratified that decision.

It would change again with time and technology. In the 20th century American musicians came to prefer 440 hertz, and that came to be adopted as the new standard. Today the standard tuning frequency is set by the International Organization for Standardization: ISO 16 “specifies the frequency for the note A in the treble stave and shall be 440 hertz.”

Note Worthy,_Charles)

Charles Ives set himself an impossible problem [in his 114 Songs of 1922]. He wanted to use pitch distance to represent the fact that God is infinitely close to man. But what is an infinitesimally close pitch distance? In the end Ives gave up and left it to the singer to decide. Maybe what Ives wanted was a smallest perceptible pitch difference. There is no standard notation for this.

— Wilfrid Hodges, “The Geometry of Music,” in John Fauvel, ed., Music and Mathematics, 2006

A Late Night,_mosa%C3%AFque.jpg

As a joke, Sosus of Pergamon created an “unswept floor” on which the refuse of a dinner table lies scattered.

Sosus laid the original floor at the royal palace in Pergamon in the second century B.C. It quickly became a favorite — wealthy Romans copied it, and it’s the only mosaic mentioned in Pliny’s history of art.


Ballerina Anna Pavlova was famous for creating The Dying Swan, a four-minute solo ballet depicting the last moments in the life of a swan, after the cello solo “Le Cygne” in Camille Saint-Saëns’ Le Carnaval des animaux. She performed the role some 4,000 times; American critic Carl Van Vechten called it “the most exquisite specimen of [Pavlova’s] art which she has yet given to the public.”

Two days after Pavlova’s death in 1931, the orchestra at London’s Apollo Theater paused between selections and began to play “The Death of the Swan.” Dance writer Philip J.S. Richardson recorded what came next:

The curtain went up and disclosed an empty, darkened stage draped in grey hangings, with the spotlight playing on someone who was not there. The large audience rose to its feet and stood in silence while the tune which will forever be associated with Anna Pavlova was played.

“It was an unforgettable moment,” he wrote.

To the Life

Perhaps inspired by his interest in natural science, American painter Charles Willson Peale set out to make his 1795 Staircase Group as realistic as possible. Not only did he paint his two sons in perfect detail on a full-length canvas, but he installed the finished painting in a door frame in his studio, with a real step at the bottom that seemed to merge into the staircase he’d painted.

Rembrandt Peale, another son, recalled that George Washington tipped his hat to the young men as he walked by.

A Company of Soldiers

During World War II, British thespian Maurice Evans toured the Central Pacific with a stripped-down version of Hamlet in which all the male actors were G.I.s. He cut down and sped up the text so that the play ran in 2 hours and 45 minutes, so that the men could return to their quarters on time.

“As I remember, on opening night, Hamlet seemed like a flop,” recalled one participant. “There was barely a murmur of response from the house and a fair amount of gloom backstage. We were later told that the audience had been reminded that this was not some cheap skin show but a classic, and that they were to show respect for Shakespeare and for Hamlet, who happened to be, by God, an officer in the United States Army. With this misunderstanding cleared up, future audiences were much more enthusiastic.”

In fact the “G.I. Hamlet” proved so successful that when the war was over Evans moved it to New York and then toured the country. He didn’t mean to be typed as a Shakespearean actor, he said; he just wanted audiences to enjoy the plays as “rattling good shows.”

The Telharmonium,_Holyoke,_Massachusetts.jpg

Inventor Thaddeus Cahill offered a startling advance in 1895: An electronic keyboard instrument that could distribute music over the nation’s telephone networks. By combining sine waves according to Hermann Helmholtz’s new theories, the device could approximate the tone of any given instrument using electrical dynamos.

After hearing a demonstration at the Hotel Hamilton, Ray Stannard Baker wrote in McClure’s, “The first impression the music makes upon the listener is its singular difference from any music ever heard before: in the fullness, roundness, completeness, of its tones.”

Unfortunately, the device required an enormous amount of electricity, it disrupted the New York telephone network, and it was rapidly overtaken by other inventions in an immensely fruitful period. Cahill had hoped to fund it through subscriptions, and this quickly became impossible. But it had its adherents — Mark Twain’s friend Albert Bigelow Paine recalled a social gathering at the Clemens home at which the author demonstrated the instrument:

“Clemens was filled with enthusiasm over the idea. He made a speech a little before midnight, in which he told how he had generally been enthusiastic about inventions which had turned out more or less well in about equal proportions. He did not dwell on the failures, but he told how he had been the first to use a typewriter for manuscript work; how he had been one of the earliest users of the fountain-pen; how he had installed the first telephone ever used in a private house, and how the audience now would have a demonstration of the first telharmonium music so employed. It was just about the stroke of midnight when he finished, and a moment later the horns began to play chimes and ‘Auld Lang Syne’ and ‘America.'”

The Lamppost Trick

In 1916, when Norman Rockwell began his career painting covers for the Saturday Evening Post, he faced a “recurring crisis” in coming up with new ideas. “I’d feel all washed out, blank, nothing in my head but a low buzzing noise,” he wrote in his 1960 autobiography, My Adventures as an Illustrator. “One day, after I’d been aimlessly sketching and crumpling up sheets of paper for hours, I said to myself, This has got to stop; I can’t sit here and muse all day. So I figured out a system and used it for 20 years or so.”

When I had run out of ideas, I’d eat a light meal, sharpen 20 pencils, and lay out a dozen pads of paper on the dining room table. Then I’d draw a lamppost (after a while I got to be the best lamppost artist in America). Then I’d draw a drunken sailor leaning on the lamppost. I’d think about the sailor. Did his girl marry someone else while he was at sea? He’s stranded in a foreign port without money? No. I’d think of the sailor patching his clothes on shipboard. That would remind me of a mother darning her little boy’s pants. Well, what did she find in the pocket? A top. A knife handle. A turtle — I’d sketch a turtle slouching slowly along to —

He would spend three or four hours following this random train of thought while drawings piled up on the floor; then he’d go to bed miserable and desperate. The next morning, still desperate, “I’d kick my trash bucket and suddenly, as it rolled bumpety-bump across the floor, an idea would come to me like a flash of lightning.” He’d follow up this new idea, and once he understood enough about the scene, he removed the lamppost.

“I’d given my brain such a beating the night before that it was in a sensitive state,” he explained. “Pretty soon I’d have a Post cover.”