Here's Looking at Euclid: Chapters Six–Eight

Age 57, he has a trim white goatee and sideburns, and often grins effervescently.

Franklin’s square contains even more beguiling symmetries.

The branch of mathematics that involves the counting of combinations, such as all the 1,905 solutions to Sky TV’s faux Sudoku, is called combinatorics. It is the study of permutations and combinations of things, such as grids of numbers but also, famously, the schedules of traveling salesmen.

I love the tangram. Men, women and animals magically come to life. After a slight repositioning of just one piece the personality of the figure entirely changes. With their angular and often grotesque contours, the figures are wonderfully suggestive.

The tangram can serve as a warning against complacency, reminding you that the essence of objects may not always be what you first see.

“Neither the wrinkled front of age nor the cherubic brow of childhood is proof against the contagion,” exhorted the Boston Post.

“It has swept over the land from East to West with the violence of the sirocco, scorching men's brains as it passed, and apparently making them temporarily insane,” quivered the Chicago Tribune. According to the New York Times no pestilence “has ever visited this or any other country which has spread with [such] awful celerity.”

Puzzles are also a wonderfully concise way of conveying the wow factor of math.

One of the first, and most precocious, devisers of these problems was Sam Loyd. A New Yorker, Loyd was just 14 when he had his first conundrum printed in a local paper.

He moved on from chess to mathematically based puzzles, and by the end of the century Loyd was the world's first professional puzzle compiler and impresario. He published widely in the American media, and once claimed that his columns attracted 100,000 letters a day.

Loyd had a unique brilliance at turning mathematical problems into entertaining, distinctively illustrated puzzles. His most genial creation was invented for the Brooklyn Daily Eagle in 1896.

He became rich and famous, reveling in his reputation as the puzzle king of America.

If Loyd’s capitalist chutzpah and gift for self-publicity reflected the cut-and-thrust of turn-of-the-century New York, Dudeney embodied the more reserved English way of life.

He called this the Haberdasher’s Puzzle, since the shapes look like the leftover pieces of cloth that a haberdasher might have in his shop.

This was a period of great scientific advances—space travel, information technology and genetics—yet it was Gardner’s lively and lucid prose that really caught readers’ imaginations.

At the G4G it was impossible not to think that math wards off the onset of dementia. Many of the guests were over 70—some were in their 80s and even 90s.

Willowy and charmingly scruffy, with flowing white hair and a feathery beard, Smullyan was frequently entertaining guests on the hotel piano.

There have been only four international puzzle crazes with a mathematical slant: the tangram, the Fifteen puzzle, the Rubik’s Cube, and Sudoku. So far, the cube has been the most lucrative. More than 300 million have been sold since Ernö Rubik came up with the idea in 1974.

Apart from its commercial success, the gaudily colored cube is a popular-culture evergreen.

It is the nonpareil of puzzledom and, unsurprisingly, its presence was felt at the 2008 G4G.

To make sure that the starting position is sufficiently difficult in these competitions, the regulations stipulate that cubes must be scrambled by a random sequence of moves generated by a computer program.

He had a soft, open complexion, wisps of white hair, large tortoiseshell glasses and alert eyes. There was an ethereal aspect to him. He was slim and had excellent posture, possibly because he works every day standing up at his desk.

It quickly became clear that he was not entirely comfortable talking about his illustriousness among mathematicians.

Sloane’s list blossomed into the Online Encyclopedia of Integer Sequences, a compendium that now has more than 160,000 entries and expands by about 10,000 a year.

He is slight and bald and wears thick, square glasses. Yet he is also sinewy and tough, carrying himself with Zen-like poise—a benefit of his other passion, which is rock climbing.

In Sloane’s opinion, the similarity between studying sequences and rock climbing is that they both demand shrewd puzzle-solving skills.

We could carry on forever and each number could be winnowed down to a product of primes in only one possible way.

The music created by the Recamán sequence, however, is chilling. It is like the soundtrack of a horror movie. It is dissonant, but it does not sound random. You can hear noticeable patterns, as if there is a human hand mysteriously present behind the cacophony.

Like this one, all of Zeno’s paradoxes draw apparently absurd conclusions by dissecting continuous motion into discrete events.

According to lore, to refute this paradox Diogenes the Cynic silently stood up and walked from A to B, thereby demonstrating that such motion was possible. But Zeno's dichotomy paradox cannot be dismissed so easily.

The delights of the harmonic series are profuse, so let's have some more fun with it.

Though I had scribbled my name without any thought, I had adhered to the golden mean with uncanny precision.

The natural world has a predilection for Fibonacci numbers. If you look in a garden, you will discover that for most flowers the number of petals is a Fibonacci number.

In addition to its association with fruit and promiscuous rodents, the Fibonacci sequence has many absorbing mathematical properties.

Levin scoured through photographs and discovered that in the most attractive sets of teeth, the big top front tooth (the central incisor) was wider than the one next to it (the lateral incisor) by a factor of phi.

He continued to develop his ideas nonetheless, and, in 1978, he published an article expounding them in the Journal of Prosthetic Dentistry.

In the late nineteenth century the German Adolf Zeising most forcefully put forth the view that the golden proportion is beauty incarnate, describing the ratio as a universal law “which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.”

In order to discover if there was any empirical evidence that humans thought the golden rectangle more beautiful than any other sort of rectangle, Fechner devised a test in which subjects were shown a number of different rectangles and asked which they preferred.

“A guy in the Caribbean was using my matrix to trade in oil, a guy in China was using it to trade in currencies,” he said.

Meisner’s website has made him the go-to guy for every flavor of phi aficionado.