By George Johnson

New York Times
May 20, 2003

da Vinci painting Annunciation
Leonardo da Vinci. The Annunciation. c. 1472-1475. Oil and tempera on wood. Uffizi Gallery, Florence, Italy
This famous da Vinci painting uses the Golden Ratio
In Dan Brown's best-selling intellectual potboiler, "The Da Vinci Code," Sophie Neveu, fetching young cryptographer for the Paris police, and Robert Langdon, suave internationally acclaimed expert on religious symbology, are running through the deserted halls of the Louvre fleeing their would-be captors when Langdon, in one of the novel's typically awkward moments, pauses to reminisce. The subject: a lecture he recently gave at Harvard on the remarkable properties of the number phi.

Phi (not to be confused with pi) is often called the Divine Proportion, or the Golden Ratio, and describes the spiraling form of seashells and pine cones and other symmetries of nature. Said to be incorporated into masterpieces by Da Vinci and Michelangelo and expressed in the proportions of the Egyptian pyramids and the Greek Parthenon, the number (which, for the record, is 1.6180339) is reputed to be among the esoteric secret knowledge protected by generations of Freemasons, Rosicrucians, Illuminati and the other societies that secretly rule the world.

Most important, the dashing professor ("Harrison Ford in Harris tweeds") told his admiring students, the number is implicit in the pentacle, the five-pointed star representing the sacred feminine, the symbol of the goddess. And now phi has been woven into the conspiracy that Dr. Langdon and Ms. Neveu desperately must uncover: the plot to steal the secret of the Holy Grail.

It's probably not the math lesson that has propelled Mr. Brown's thriller onto the best-seller lists. But the mysterious aura that has long been attributed to the number adds to the playful illusion that readers are getting the inside scoop, an important clue to how the universe works.

In a world otherwise crippled by math anxiety, books about phi and other so-called constants of the universe are multiplying so quickly that "The Da Vinci Code" threatens to become part of a genre.

In "The Golden Ratio: The Story of Phi, the World's Most Astonishing Number," Dr. Mario Livio offers a more sober, and nonfictional, treatment of this seemingly ubiquitous parameter. "Gamma: Exploring Euler's Constant" by Dr. Julian Havil gives the starring role to another mysterious number, 0.5772156 . . ., which pops up in enough odd places to suggest to some that it may be an expression of the underlying beauty of mathematics.

Taking a broader sweep, Dr. John D. Barrow's new book, "The Constants of Nature: From Alpha to Omega - The Numbers That Encode the Deepest Secrets of the Universe," closely follows the work of Dr. Barrow's British colleague, Dr. Martin J. Rees, and his popular account, "Just Six Numbers: The Deep Forces That Shape the Universe."

Despite the mystical tone of some of the subtitles, these books are science, not science fiction, yet they tap into a mystery far more compelling than the shopworn Grail legend: why the world seems to operate according to mathematical laws.

Nowhere does the phenomenon emerge more forcefully than with the constants, quantities that seem to be encoded within the software of the universe. Not all of these numbers are inherently remarkable. The National Institute of Standards and Technology, keeper of the constants for the United States Government, defines the speed of light in a vacuum, called c, as 299,792,458 meters per second and the charge of the electron, or e, as 1.602176462 10-19 coulombs. (A coulomb is the quantity of electricity transported in one second by a current of one ampere.)

There is no point in trying to find some cabalistic significance to these constants any more than there is in the recently issued NIST peanut butter standard. Too much arbitrariness goes into the numbers' manufacture. If length is measured in something other than meters or time in something other than seconds, the numbers would be expressed by entirely different strings of digits. Nothing very magical there.

The weirdness comes when you combine c and e with another quantity called Planck's constant, the most important number in quantum theory. Twiddle with the units so that the meters and seconds and other parochial Earthling units cancel out and the result is a dimensionless, "pure" number, 7.297352533 10-3 or almost exactly one-137th. The number, called alpha, would be the same for alien races using measuring sticks based on the width of the fourth planet circling Betelgeuse or clocks with faces bearing 79 numerals.

Put together from values important to both special relativity and quantum mechanics, alpha is suspected by some physicists to be a key to the long-sought theory of everything. The secret of the universe is why alpha is one-137th and not, say, one-136th or one-138th. Calculations indicate that if it were slightly different, stars would not exist to produce elements and there would be no life.

Other constants appear to be just as finicky. If a number called omega, a measure of the density of matter in the universe, teetered one way, the Big Bang would have fizzled; if it teetered the other way, all matter would have rapidly diffused into a rarefied sterile gas. Some proponents of biblical creationism take these numerical balancing acts as proof of an intelligent being who fine-tuned the universe to be hospitable, others as a meaningless coincidence or a stroke of great luck.

Numbers probably emerged as a way of keeping track of stuff. Instead of laboriously counting a large pile of stones, divide them into several piles of 10 and then multiply. Or take one pile and make two smaller ones of equal size: 10 divided by 2 is 5.

History has lost the name of the joker who first asked, "But hey, what if you divide a pile of five stones into three equal piles . . ." Five divided by three is 1.6666666666 . . . the sixes repeating forever like an incantation of the Mark of the Beast.

It gets even weirder. Divide the circumference of a circle by its diameter and you get the never-ending expansion of pi. Centuries later people are still churning out pi's digits, billions so far, scrutinizing them for some kind of meaning, mathematics' own Holy Grail.

This kind of obsessive behavior started as far back as the sixth century B.C., when a cult of number worshipers called the Pythagoreans taught that fire was made of 24 right-angled triangles, surrounded by 4 equilaterals, which are made in turn of 6 right-angled triangles. "All is number," the brothers would intone.

Strip away the religious trappings and the notion that emerges is not so different from one embraced by numerous physicists and mathematicians today: God is a mathematician and the universe experienced by the senses is a shadow play of a deeper reality, the hidden realm of numbers.

Trying to get a grip on this, the physicist Eugene Wigner published an essay called "The Unreasonable Effectiveness of Mathematics in the Natural Sciences." After struggling through and rejecting some possible explanations for why the world seems to be ruled by numbers, he concluded that the phenomenon might always remain a mystery. It's "a wonderful gift," he wrote, "which we neither understand nor deserve."

Maybe the mystery could be cast into a thriller. Call it "The Einstein Code."

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