John A. Adam says the beauty of nature is revealed by mathematics and the beauty of mathematics is revealed in nature.

As professor of mathematics at Old Dominion University in Norfolk, Mr. Adam teaches a class called “Mathematics in Nature.” He has written a book by the same name.

“I want to encourage students to think about mathematical and physical principles underlying many natural phenomena,” says Mr. Adam, who holds a doctorate in theoretical astrophysics. “I want to point students to patterns around them.”

Many mathematicians and scientists consider mathematics to be the science of patterns. It is a way for the invisible world to become visible.

Coat patterns on animals such as leopards, cheetahs, tigers and giraffes, and wing markings on insects such as butterflies and moths can be analyzed through mathematics, Mr. Adam says.

Although scientists debate the causes of these patterns, many animal markings are believed to occur in the biochemistry of the embryo. The chemicals, also known as morphogens, produce spots, stripes or other patterns, he says.

“How did the leopard get its spots?” Mr. Adam asks. “That’s what science is. You see a pattern and try to explain it.”

The rainbow is one of Mr. Adam’s favorite mathematical patterns, characterized by the distribution of light in the sky.

“The subtle features of rainbows can only be explained in terms of advanced physics and mathematics,” Mr. Adam says.

Waves in puddles, ponds, lakes or oceans are directed by the mathematical correlations among the speed, wavelength and the depth of the water,he says.

The wakes created by ships and ducks produce comparable patterns, depending on their size. Generally, the triangular pattern trailing the ship or duck often has the same angle, about 38 degrees, he says.

Sand dunes are another example of wave patterns in nature, he says.

Cracks that form in drying mud, tree bark or rock usually have distinctive mathematical patterns based on polygonal shapes, frequently hexagonal.

Another prevalent pattern is the sixfold symmetry of snowflakes.

Also, large bends in a river, or “meanders,” define the likely form of a river. Despite its size, a river is never straight for more than 10 times its average width, Mr. Adam says.

Nature seems to abide by minimization principles, says Ilhan M. Izmirli, assistant professor of mathematics at American University. He holds a doctorate in mathematics.

For example, in the construction of a beehive, the bees use hexagonal shapes. The kernels on a corncob and soap bubbles also usually each touch six other spheres.

Circles aren’t usually an efficient way to build something, he says. Triangles, squares and hexagons are a better use of space. Hexagons, which are closest in shape to circles, cover the largest area for a given perimeter.

“The bees are doing something extraordinarily economical,” Mr. Izmirli says. “It’s the most optimal way of building something in nature.”

Even the “bee dance,” in the form of a figure eight, gives direction to the food source. The length of the routine gives the exact distance of the food source from the hive, he says.

The Fibonacci numerical sequence is a mathematical code in nature, says Dean L. Overman, author of “A Case Against Accident and Self-Organization.” Mr. Overman is a retired Washington lawyer and chairman of the advisory board of First Trust Portfolios, an asset management company in Lisle, Ill.

In the Fibonacci sequence, “0, 1, 1, 2, 3, 5, 8, 13 …,” each number is the sum of the preceding two numbers.

The numbers show up in the construction of items such as pine and fir cones, leaves on palm trees, seashells, and seeds in sunflowers, Mr. Overman says. Vegetables, such as artichokes and broccoli, and flowers, such as oxeye daisies, trilliums, wild roses, cosmoses and irises, also exhibit the pattern in that the original number of petals on a flower is often a Fibonacci number.

Further, the ratio of successive pairs of Fibonacci numbers approximates the “golden section” or the “golden ratio,” a proportion that is considered to be aesthetically attractive. The ratio is roughly 1.61803, and its reciprocal is about 0.61803, Mr. Overman says.

When the ratio is used to create a rectangle, it often is referred to as a “golden rectangle.” Artist Leonardo da Vinci supposedly matched Mona Lisa’s face with a golden rectangle, he says. Well-known architects, such as Le Corbusier, have based famous works on the shape, Mr. Overman says.

There’s even an argument that Austrian composer Wolfgang Amadeus Mozart used Fibonacci numbers and the “golden section” when writing his music, he says.

The concept of taking an abstract mathematical idea, such as the curvature of space in relationship to Albert Einstein’s theory of relativity, and matching it with the structure of the physical universe indicates that the universe is inherently mathematical, Mr. Overman says.

“In the most effective scientific theories, we do not create and then impose our mathematics on the universe, but rather discover the mathematics which are already present in nature,” Mr. Overman says. “It appears that there is a cosmic mathematician behind all this.”

How a mathematical description can describe the physical world is a fundamental mystery, says Robert Kaita, a physicist at Princeton University’s plasma physics laboratory in Princeton, N.J. He holds a doctorate in physics.

For instance, the universal force of gravity discovered by Isaac Newton illustrates the identical mathematical description that describes the force between any two objects that have mass, such as an apple and the Earth, the Earth and the sun, the solar system and other masses in the galaxy, he says.

A world with different mathematical descriptions would make it difficult for scientists to create advancements in technology, Mr. Kaita says.

Without a mathematical basis for the universe, he wonders if the universe could exist at all.

“If the laws of gravity changed, maybe the Earth couldn’t hold together, and we wouldn’t have an Earth,” Mr. Kaita says.

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