Mathematics and the formation of coral reefs

SUMMARY: Mathematics helps describe and explain the formation of coral reefs.

Coral reefs are complex stony structures made of exoskeletons of coral polyps. Colonies of polyps form corals, with their stony parts consisting of calcium carbonate. All polyps in a single coral are genetically identical. Polyps get their energy from photosynthesis of their internal symbionts, one-cell algae living in the polyps. Some corals also have stinging tentacles for catching plankton, and can be painful for people to touch. The development and growth of coral reefs and atolls was fiercely debated in the nineteenth and early twentieth centuries. Charles Darwin argued in his 1842 publication Structure and Distribution of Coral Reefs, based on his personal observations, that the geometry of coral reefs resulted from the natural geological subsidence of oceanic islands.

In other words, coral reefs formed around islands, growing as the islands sank away. Darwin’s chief opponent in this debate was Alexander Agassiz, who advocated the theory that coral reefs were not wholly dependent on subsistence for their formation but rather arose from a variety of geological and biological factors. Agassiz collected data from nearly every coral reef on Earth before his death in 1910, but none of his research had been published at that time. Contemporaries of both Darwin and Agassiz were inhibited by the inability to collect data other than observations and relatively shallow rock samples. In the 1950s, geologist Harry Ladd conducted tests in conjunction with the U.S. War Department, including boring thousands of holes in the coral of Eniwetok Atoll. Ladd’s drill went to a depth of nearly 5,000 feet before finally passing completely through the coral into the soil below, confirming in many scientists’ minds that the atoll had been built up as the land had sunk away. Ladd purportedly erected a sign on Eniwetok that read, “Darwin was right!”

Measurements and Variables

The shape of a coral reef is determined by the sea floor and the historical changes in sea levels. Reef scientists recognize three main shape types: fringing reefs, barrier reefs, and atolls. Fringing reefs stay close to shores, and their shape is determined by the shore they circle. Barrier reefs start as fringing reefs, but as the water levels rise relative to the shore, there are deep, large lagoons separating the shore and the reef. When volcanic islands completely subside underwater, their fringing or barrier reefs can stay near the surface, forming a circular lagoon. Such reefs are called “atolls.”

In most places, sea levels rise over the land. The speed of reef growth depends on multiple variables, including temperature, water salinity, water clarity necessary for photosynthesis, and wave action. Reefs can grow up to 25 centimeters (about 10 inches) per year in height. Reefs cannot grow faster than sea levels rise, because the polyps can survive out of water only for a short time—for example, during the low tide. When the speed of reef growth matches the rise of the sea level, they are called “keep-up reefs.” When the speed of reef growth is slower than the rise of the sea level temporarily, reefs may become either “catch-up reefs” when the speeds eventually match, or “drowned out” reefs that die as they are submerged too deeply. Global warming threatens to increase the rate at which sea levels rise beyond the speed of reef growth.

Because reefs need clear waters for photosynthesis, they grow in the parts of the ocean that are relatively nutrient-poor. However, reefs themselves support rich and diverse ecosystems—the contradiction called “Darwin’s paradox.” Reefs underlie less than 1% of the world’s ocean beds but host about 25% of the marine species. They are called “underwater rainforests” because of their active biomass production, measured in weight per area per day.

Coral reefs have high fractal dimensions; in other words, their surface is rough, wrinkled, and uneven. This characteristic explains why corals thrive in moving waters. The fractal-like coral surfaces break the still water barrier surrounding them, with any agitation of water creating and amplifying turbulence. This turbulence means more water moves through the polyps, delivering nutrients to them and removing sediments that could prevent photosynthesis.

Mathematical Models

Coral reefs are vulnerable to storms, tsunamis, and other strong natural events. By modeling reef damage, it is possible to intervene, and to preserve some reefs that would otherwise be destroyed. Existing models include equations that measure the forces applied to reefs, and the forces reefs can withstand.

The ratio between the area of attachment of a reef and its total surface area plays a role in the models. The higher the surface area of the reef, the higher the pressure storms apply to it. On the other hand, the higher the area of attachment, the more force it takes to detach the reef. By modifying these variables, as well as the force of the storm, oceanologists can predict what happens to particular reefs. Moreover, with more computation power comes the opportunity to model detailed shapes of reefs, individual currents, and other local variables, making predictions more precise.

Dynamic systems of differential equations are the area of mathematics applicable to complex ecosystems such as coral reefs. More deterministic models such as algebraic or simple differential equations do not capture the reality as well.

Hyperbolic Crochet Coral Reef Project

The crocheted coral reef is a collaborative project with hundreds of contributors and several exhibits worldwide, and is coordinated by the Institute for Figuring. It demonstrates hyperbolic geometry, which is a non-Euclidian geometry discovered about 200 years ago and found in nature—including corals. “Hyperbolic crocheting,” the process for modeling corals, was first described in the late 1990s. It involves a simple repeating algorithm with introduced “mutations” that produce varied forms.

The models explore mathematical entities that can be found in coral reefs, such as the hyperbolic radius of curvature, pseudospheres, hyperbolic planes, and geodesics.

Bibliography

Dobbs, David. Reef Madness: Charles Darwin, Alexander Agassiz, and the Meaning of Coral. New York: Pantheon Books, 2005.

Institute for Figuring. “Hyperbolic Crochet Coral Reef.” http://crochetcoralreef.org.

Sale, Peter. Coral Reef Fishes: Dynamics and Diversity in a Complex Ecosystem. San Diego, CA: Academic Press, 2002.