UA Mathematicians Predict Patterns in Fingerprints, Cacti

Patterns in nature can be seen every day, yet in many cases, little is understood about how and why they form. Now University of Arizona mathematicians have found a way to predict natural patterns, including fingerprints and the spirals seen in cacti.

UA graduate student Michael Kuecken developed a mathematical model that can reproduce fingerprint patterns, while UA graduate student Patrick Shipman created a mathematical model to explain the arrangement of repeated units in various plants. Shipman’s report on his work will be published in an upcoming issue of Physical Review Letters.

Even though the use of fingerprints for identification began more than 2000 years ago in China and they have been studied experimentally for over two hundred years, there is no widely accepted explanation for their occurrence. Likewise, the reasons behind nature’s choice of patterns in plants have been difficult for mathematicians to explain, despite these patterns having been identified centuries ago.

“What I like about this research is the interplay between math and biology. It is actually quite difficult, because the disciplines require a somewhat different mindset and biology is notoriously bewildering and full of detail,” Kuecken said. “In a way, dealing with this problem was like putting together a jigsaw puzzle of facts. I had to try out different things and could use math, and sometimes common sense, to see if the pieces actually fit.”

Human skin has multiple layers, including the outermost epidermis and the inner dermis. The outer and inner layers are separated by the basal layer, which is composed of cells that constantly divide. Growth occurs in a similar fashion in plants, which have areas of continuous cell growth, such as the tip of a cactus, that allow the plant to grow larger.

The basal layer in human skin and the equivalent layer in plant skin grow at a faster rate than either the surface layers or the thick dermis layer. As the basal layer continues to grow, pressure increases. In both plants and fingertips, the growing layer buckles inward toward the softer inner layer of tissue, relieving the stress. As a result, ridges are formed on the surface.

The undulations from the buckling form fingerprints and various patterns in plants, from the ridges in saguaro cacti to the hexagons in pineapples. The way a pattern is formed, regardless whether it is a fingerprint or a plant, is related to the forces imposed during ridge formation.

The basic properties responsible for the mechanism of buckling in plants and fingerprints happen in other materials as well. Kuecken and Shipman’s graduate advisor, UA professor of mathematics Alan Newell, said, “In material science, high-temperature superconductors seem to be connected with stresses that compress to build the structures in various high-temperature materials. Indeed, the idea that buckling and surface stresses would have something to do with the patterns you see in plants is fairly recent.”

In fingerprints, ridge formation is influenced by discrete elevations of the skin on the fingertips, called volar pads, which first appear in human embryos at about six and a half weeks. The volar pads’ location is where the epidermal ridges for fingerprints will arise later in development.

Kuecken explained that as the volar pads shrink, it places stress on the skin layers. The ridges then form perpendicular to this stress. There are three basic patterns of fingerprints known as arches, loops and whorls that form in response to the different directions of stress caused by shrinking of the volar pads. Other research on ridge formation has already shown that if a person has a high, rounded volar pad, they will end up with a whorl pattern. Kuecken’s mathematical model was able to reproduce these large patterns, as well as the little intricacies that make an individual fingerprint unique.

Shipman’s model, like Kuecken’s, also took into account stresses that influenced ridge formation. In plants, forces acting in multiple directions result in complex patterns. For example, when buckling occurs in three different directions, all three ridges will appear together and form a hexagonal pattern.

“I’ve looked at cacti all my life, I really like them, and I’d really like to understand them,” Shipman said. To study these patterns, Shipman looked at the stickers on a cactus or florets on a flower.

When a line is drawn from sticker to sticker on a cactus in a clockwise or in a counterclockwise direction, the line ends up spiraling around the plant. This occurs in many plants, including pineapples and cauliflower. When these spirals are counted, it results in numbers that belong to the Fibonacci sequence, a series of numbers that appears frequently when scientists and mathematicians analyze natural patterns.
Shipman found that cactus stickers predicatably align in spiral patterns.

From his model, Shipman found that the initial curvature of a plant near its growth tip influences whether it will form ridges or hexagons. He found that plants with a flat top, or less curved top, such as saguaro cacti, will always form ridges and tend not to have Fibonacci sequences. Plants that have a high degree of curvature will produce hexagonal configurations, such as those in pinecones, and the number of spirals will always be numbers in the Fibonacci sequence.

Newell says that Shipman’s mathematical model demonstrates that the shapes chosen by nature are those that take the least energy to make. “Of all possible shapes you can have, what nature picked minimizes the energy in the plant.”