For all its flaws, The Da Vinci Code, by Dan Brown, has done more than any other book to publicize the so-called "divine proportion," "golden ratio," or in mathematical terminology, simply phi. It pops up all over the place, in art, architecture, nature, and music. And now it's cropped up in the world of subatomic particles, according to a new paper that appeared in Science on January 8.
Its commonly cited value is 1.618, although in reality, phi (pronounced "fee") is an irrational number with infinite decimal places, like pi. Phi was first mentioned in Euclid's Elements around 300 BC. Geometric shapes are said to be in "divine proportion" if the ratios of their various sides closely resemble phi.
The most common such shapes are the golden rectangle, the golden triangle, and the algorithmic spiral, which can be seen in any chambered nautilus shell.
Phi is related to the Fibonacci Sequence: 1, 2, 3, 5, 8, 13, 21.... After the first two terms, each number is equal to the sum of the two previous numbers, to wit: 1+1=2; 1+2=3; 2+3=5; 3+5=8, and so on, into infinity. But the fun doesn't end there. If you take each number in the sequence and divide it into the one that follows, dividing one number into the next, the answer will come closer and closer to the value for phi, without ever actually reaching. Try it: 5 divided by 3 is 1.666; 13 divided by 8 is 1.625; 21 divided by 13 is 1.615; and so forth. Plot the Fibonacci Sequence graphically, and you get pretty logorithmic spirals, like this:
The same sequence of numbers occurs frequently in nature; so does the golden ratio. For example, the total number of petals in most flowers is a Fibonacci number. An iris has three petals, a buttercup five, and an aster has 21 petals. In a nautilus shell, the ratio of the first spiral to the next is roughly 1.618. The golden ratio can also be found in certain crystal structures, and in the swooping, spiral flight path of a falcon attacking its prey And now the quantum realm turns out to have a similar kind of hidden symmetry at the nanoscale, if experiments performed by a team of German and British scientists are any indication. Those with a solid passing knowledge of physics know that in the quantum world, particles can exist in a superposition of states (dubbed a quantum critical or "Schroedinger cat state"). The scientists induced this kind of state in experiments with cobalt niobate using a judiciously applied magnetic field. (Cobalt niobate is comprised of magnetic atoms linked into a long chain, similar to an atom-thin bar magnet.)
The scientists were able to "tune" the cobalt niobate system, such that the chain of atoms behaved like "a nanoscale guitar string," according to principal author Radu Coldea of Oxford University, and used neutron scattering to "see" the actual vibrations at the atomic scale. "Here the tension comes from the interaction between spins causing them to magnetically resonate," Coldea explains. "For these interactions we found a series (scale) of resonant notes. The first two notes show a perfect relationship with each other. Their frequencies (pitch) are in the ratio of 1.618... which is the golden ratio famous from art and architecture."
Coincidence? Coldea doesn't think so. "It reflects a beautiful property of the quantum system - a hidden symmetry," he maintains. And this experiment is the first observation of such a symmetry in a material. I can't wait to see what Dan Brown makes of it all in his next bestselling potboiler.