The "golden ratio," represented by the Greek letter phi, has proven to be a seductive number. Leonardo Da Vinci used the ratio to compose "The Last Supper." Salvador Dali and George Seurat were also drawn to it. Architects cleave to the ratio, too, and it even shows up in natural forms—the spiral of leaves around a stem or seeds in the head of a sunflower.
Not all claims to have found an example of the golden ratio are true. But there's been another promising sighting: Pulsing stars dim and brighten in a pattern that is oh-so-close to the golden ratio.
The ratio’s attraction comes from how simple it is to construct: If one quantity (visualize a line or a rectangle) is divided such that the ratio between the smaller and larger parts is the same as the ratio between the larger part and the whole, then the quantities adhere to the golden ratio. The number is approximately 1.618. Another beautiful pattern—that of the Fibonacci sequence—is closely related because the ratio between the sequence’s consecutive numbers converges to phi.
The variable stars are of a class called RR Lyrae, often found in globular clusters. For Scientific American, Clara Moskowitz writes:
Unlike the sun, which shines at a near constant brightness (a good thing for life on Earth!), these stars brighten and dim as their atmospheres expand and contract due to periodic pressure changes. Each star pulses with a primary frequency and also shows smaller brightness fluctuations occurring on a secondary frequency.
The ratio of those frequencies can tell scientists about the structure of the stars and even their ages. A group of researchers noticed that four such stars oscillate between two frequencies that conform to the golden ratio. Teasing apart the pulses also revealed a pattern in variability of each part—fractals. And just as zooming into the convolutions of a coastline reveals more contortions at each smaller scale, the stars’ frequencies were fractal. "As we lower the threshold we see more and more frequencies," astronomer John Linder, of The College of Wooster in Ohio, told Scientific American.
The ratio in the pulse period of these stars may be meaningless, but the researchers have hope that it could provide information about the stars’ dynamics. They published their work online at arXiv.org. But other scientists are skeptical: "The fact that this period ratio (or its reciprocal) lies close to the golden ratio may be a coincidence," Robert Szabo of Konkoly Observatory in Hungary told New Scientist, "and in my opinion, more evidence is needed to demonstrate that it has a privileged role in the dynamics of [these] stars."
Still, the stars' flickering patterns do intrigue. Should this taste of the golden ratio whet your appetite, there are plenty of numerical sequences, naturally occurring and not, to keep you wondering.