Whether they're from humans, whales, or elephants, the brains of many mammals are covered with elaborate folds. Now, a new study shows that the degree of this folding follows a simple mathematical relationship—called a scaling law—that also explains the crumpling of paper. That observation suggests that the myriad forms of mammalian brains arise not from subtle developmental processes that vary from species to species, but rather from the same simple physical process.

In biology, it rare to find a mathematical relationship that so tightly fits all the data, say Georg Striedter, a neuroscientist at the University of California, Irvine. "They've captured something," he says. Still, Striedter argues that the scaling law describes a pattern among fully developed brains and doesn't explain how the folding in a developing brain happens.

The folding in the mammalian brain serves to increase the total area of the cortex, the outer layer of gray matter where the neurons reside. Not all mammals have folded cortices. For example, mice and rats have smooth-surfaced brains and are "lissencephalic." In contrast, primates, whales, dogs, and cats have folded brains and are "gyrencephalic."

For decades, scientists have struggled to relate the amount of folding in a species' brain to some other characteristic. For example, although animals with tiny brains tend to have smooth ones, there is no clean relationship between the amount of folding—measured by the ratio of the total area of the cortex to the exposed outer surface of the brain—and brain mass. Make a plot of folding versus brain mass for various species and the data points fall all over and not on a unified curve. Similarly, there is no clean relationship between the amount of folding and the number of neurons, the total area of the cortex, or the thickness of the cortex.

But now, Suzana Herculano-Houzel and Bruno Mota—a neuroscientist and physicist, respectively, at the Federal University of Rio de Janeiro in Brazil—have found a mathematical relation for folding in mammals' brains that appears to be universal. Using data for 62 different species, the duo plotted the area of cortex times the square root of its thickness versus of the exposed area of the brain. All the data points fell on a single universal curve—for both lissencephalic and gyrencephalic species—as the researchers report online today in Science. And the curve showed that the combination of total area and thickness grew with the exposed outer area raised to the power 1.25—just as the area of a circle grows with its radius raised to the power 2.

It may sound complicated, but that universal relationship is the same one that describes crumpled wads of paper—as Herculano-Houzel showed by scrunching up sheets of paper of different sizes and thickness at her dining room table and measuring their surface areas. The relationship comes about because the bent-up paper settles into the configuration that minimizes its energy. So presumably, in folding, the cortex also simply settles into the configuration of least mechanical energy.

But Striedter argues that the analogy with crumpling paper is not completely reliable. The paper is exposed to external forces applied by the hands, he notes, whereas the forces in the cortex presumably arise internally. Scientists have yet to determine how those forces arise and the folding occurs, says Striedter, who co-authored a commentary on the paper that also appears in Science. For example, he says, some models suggest that the growing cortex folds as its outer layer grows faster than its inner layer. "The scaling relationship is one thing," Strieder says. "The mechanism of how you get there is another."

However, Herculano-Houzel sees it differently. At every stage of development, the growing cortex is indeed subject to the external force of the cranium, she says. So as the cortex grows in the confined space, it must crumple, and at every stage of development the scaling relationship should hold, she argues. That prediction can be tested in, say, developing pigs, Herculano-Houzel says, which is what she intends to do next. If the scaling relationship holds at all stages of development, she says, then there is no need for another mechanism to explain the folding.