By Gene Mirabelli
By Mario Livio
Broadway Books 294 pages $24.95
Books about science are popular these days. The physicist
Brian Greene wrote a best-seller untangling string theory
and last year the cosmologist Stephen Hawkings gave us a lavishly
illustrated coffee-table tome, compressing the universe into
a nutshell. Books about numbers are not that popular. But
those who like to play with calculations and geometric diagrams
will want to look at Mario Livio’s entertaining book about
a very special numeral.
In The Golden Ratio Livio tells the story of Phi, “the
world’s most astonishing number.” There are some numbers that
turn up on special occasions and have such curious characteristics
that mathematicians eventually name them or, more precisely,
symbolize them by a letter, usually from the Greek alphabet.
Probably the best-known numeral to be honored this way is
the one you come up with when you divide the circumference
of a circle by its diameter: the endless 3.1415 . . . , the
number called Pi.
The Golden Ratio is 1.61803 . . . and is symbolized by the
Greek letter Phi. Phi is not so well-known as Pi, but it’s
considerably more interesting and has been called the Golden
Number, the Golden Ratio, the Golden Section, perhaps because
of its esthetic value.
The first clear definition of Phi was given around 300 B.C.
by Euclid, the Alexandrian who first formalized geometry.
He pointed out that you can divide a straight line into two
unequal parts, and if you divide it at precisely the right
place the whole line will be to the larger segment as the
larger segment is to the smaller segment. Or, to say it another
way, the ratio of the whole line to the larger line segment
is the same as the ratio of the larger segment to the smaller
So what? Well, it turns out that the ratio of those line segments
occurs in nature all the time and in remarkably beautiful
forms. The lovely widening spiral of a chambered nautilus
shell, the pattern of seeds in the face of a sunflower, the
common arrangement of shoots on a plant stem and petals on
a rose, all follow this number. Even a rectangular room whose
sides are in the Golden Ratio makes a particularly harmonious
volume. This handsome book has a lucid diagram or picture
to illustrate such points on just about every other page.
Mario Livio’s easygoing narrative takes him to some strange
places in the history of mathematics. Among the more interesting
people we meet is the Greek philosopher and mathematician
Pythagoras. The old professor wrote little or nothing, but
happily his students formed a society, the Pythagoreans, to
preserve and carry on his work. Pythagoras, who was said to
have a golden thigh, taught the doctrine of metempsychosis,
that the soul is immortal and is reborn in human and animal
incarnations. As you might guess, Pythagoreans were vegetarians.
They were also crazy about geometry, and probably were the
first to discover that the Golden Ratio is an unending problem
in division, what mathematicians call an irrational number.
Livio’s book is not a history of geometry or mathematics—definitely
not a heavy text—but a remarkably lucid exploration of just
about everything even remotely associated with Phi. It’s an
exceptionally simple read. To be blunt, if there’s a weakness
in The Golden Ratio it’s the book’s utter simplicity.
People who are tempted to open a book on Phi do not need to
be told that “Plato (428/427 B.C.-348/347 B.C.) was one of
the most influential minds of ancient Greece and of western
civilization in general.” And it’s plain silly to digress
on such trivia as the first mention of Fibonacci’s name. (It
occurred in a footnote in Histoire des Sciences Mathematique
en Italie. Does that matter?)
Books about numbers tend to stay in print. So if you’re interested,
or if you’re searching for a gift for that nerdy-but-nice
friend who has numerical interests, take a look at books from
earlier years, too. Probably the numeral with the most intriguing
history is zero, a relative newcomer in mathematical calculations.
Robert Kaplan’s The Nothing That Is: A Natural History
of Zero provides an entertaining chronicle of the great
naught, as does Charles Seife’s rather supercharged narrative
Zero: The Biography of a Dangerous Idea.
You may have come across e, or 2.71828 . . . , while playing
around with logarithms or perhaps you accidentally computed
it while trying to calculate compound interest on a savings
account. Eli Maor’s e: The Story of a Number
is probably the best popular book on e. And, of course, there’s
always Pi. Petr (or Peter) Beckmann’s short, concise A
History of Pi is still available. Beckman is cranky author—mathematics
attracts a special kind of crank—but his book is uncluttered
and to the point. And David Blatner’s The Joy of Pi
is a sane, wholesome and breezy work. But the hot little number
for this year is Phi, irrational and beautiful, in Mario Livio’s
The Golden Ratio.