Wonderful, mysterious, beautiful 1.61803 ...

When plunging in for the kill, a peregrine falcon swoops toward its prey at close to 200 mph. But the real miracle of this dive is the path the bird takes through the air: With almost computer-like precision, the falcon follows a mathematical curve known as a logarithmic spiral. The shortest path, of course, would be a straight line, but since a falcon’s eyes are opposed on either side of its head, straight flight would require the bird to cock its head to keep the prey in sight. Wind tunnel tests reveal that the drag produced by a tilted head seriously reduces airspeed. By flying in a logarithmic spiral, falcons take advantage of the curve’s unique properties to maintain a fixed gaze on their prey while retaining body alignment. Avian hunger, evolution and aerodynamics thus conspire to sculpt in the sky a form that has entranced mathematicians for centuries.

Nature loves the logarithmic spiral. From sunflowers, seashells and whirlpools to hurricanes and spiral galaxies, we discover this extraordinary form at all scales from the microscopic to the cosmological. It has been a staple of life for millions of years, manifesting, for example, in the minuscule fossils of foraminifera whose shape has remained unchanged through eons. In “The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number,” Mario Livio calls this curve, with unabashed admiration, nature’s “favorite ornament.”

Encoded within the logarithmic spiral is perhaps the most mystical of all numbers, phi, whose value is approximately 1.61803. Like pi, phi is one of the irrationals and cannot be expressed as any kind of fraction; its decimal expansion goes on forever without repeating. Paradoxically, this algebraic disorder serves as the foundation for a staggering abundance of geometric order, and phi is at the heart of many of geometry’s most enchanting constructions.

Today, its most famous manifestation of Phi is in the golden rectangle, whose sides are in the proportions of phi to 1, the so-called “perfect proportions,” which are said to have determined the architecture of the Parthenon and the composition of the Mona Lisa. The logarithmic spiral itself is derived from a nested sequence of golden rectangles, though it can also be constructed from a nested sequence of its lesser known cousin, the golden triangle.

Throughout history, no number has been imbued with more mystical or outright religious portent. During the Renaissance, the golden ratio was known as “the divine proportion.” Luca Pacioli, the 16th century mathematician whose book on geometry helped to propagate awareness of phi among Renaissance painters such as his friend Leonardo da Vinci, compared the mystery of phi to the ineffability of God. Phi’s association with the divine preceded Christianity, for the Greeks also read in this number the signature of transcendence. Livio tells us that the Greeks’ obsession with phi did not revolve around the golden rectangle but around the pentagon and the pentagram, both of which are geometric homages to phi. To draw either form accurately, the geometer must constantly apply the golden ratio. A nested sequence of pentagons and pentagrams can be seen indeed as a mathematical fugue in phi major, and for 2 1/2 millenniums that diagram was regarded as a symbol -- almost a guarantee -- of the true iconic reality beyond the material world.

Further fuel for phi’s mystical status was provided by the discovery of the five Platonic solids: the cube, tetrahedron, octahedron, dodecahedron and icosahedron. These are the only solid forms that can be constructed from wholly regular sides: The cube is made from six squares, the tetrahedron from four equilateral triangles, and so on. Plato, the great iconolater, associated four of these solids with the four basic elements: earth, air, fire and water. The fifth, the dodecahedron, he associated with the universe itself, declaring it to be the form “which the god used for embroidering the whole heaven.” The only one of the Platonic solids constructed from pentagons, the dodecahedron was regarded as the most divine of geometry’s most perfect forms.

As with the pentagon, the dodecahedron must be understood in terms of phi. The golden ratio also plays a crucial role in the dimensions and symmetries of other Platonic solids and in the relationships between these figures, with each new discovery further cementing phi’s status as a truly special number.

Nor, it seems, have we exhausted the repertoire of this remarkable mathematical wraith. Livio relates the story of how in the early 1970s cosmologist Roger Penrose made an extraordinary discovery about phi in a branch of mathematics known as tiling theory, a discovery that turns out to have profound consequences in the physical world. Tiling theory deals with the ways in which it is possible to fill an area with some set of repeated shapes, a process mathematicians poetically call “tessellating” the plane. As every bricklayer knows, you can fill an area with square bricks, or rectangular ones. You can also do it with hexagonal bricks. But you cannot do it with pentagonal bricks: There will always be gaps.

For hundreds of years, mathematicians believed it was not possible to tessellate a plane with fivefold symmetry. Then in 1974, Penrose discovered a simple set of two interlocking shapes derived from a pentagon that do just that, leaving no gaps. Both shapes are defined by the golden ratio. Penrose’s discovery sent waves of excitement through the mathematics world; then, in 1984 came the news that this unlikely pattern appeared to be present in nature. Israeli materials scientist Danny Schectman found that crystals of an aluminum manganese alloy exhibited long-range fivefold symmetry. Dubbed “quasi-crystals,” these compounds and others like them are now believed to form from atoms arranged in a Penrose-like pattern.

In recent years a slew of numbers -- notably pi, e and zero -- have been the subjects of extensive biographies. Phi, perhaps the most magnetic personality of them all, is long overdue. Sadly, Livio seems overwhelmed by his subject, and his arrangement of the material is so random that at times I felt as if I was watching a load of washing tumbling in a dryer. “The Golden Ratio” is at its best when Livio is simply presenting the mathematics of phi, whose crystalline beauty is enough to make a Keplerian out of any reader. But when dealing with the surrounding cultural context, “The Golden Ratio” reads like a first draft. I mention this not to malign Livio, whose intentions are clearly of the most honorable kind -- is there any more difficult task than trying to explain mathematics to nonmathematicians? -- but to make a comment on the state of science publishing in general. Too many science books (and I include math here) are shoved out with less than rigorous oversight. Rather than being honed with rewrites, good ideas are pumped through the publishing mill and evacuated into book stores like so much raw sludge. In “The Golden Ratio,” there is the germ of a truly splendid text. Another two drafts, and it could have become the book it ought to have been.