Opening a math textbook or sitting in a classroom are not the only ways to learn about mathematics. Mathematics can be discovered all around us in the natural world.

There can be so much fun in finding certain math aspects in nature, such as during a nature walk.

“Nature’s great book is written in mathematics. Nature is written in mathematical language.” —Galileo Galilei

## Fibonacci numbers

Fibonacci numbers arise from starting with the numbers zero and one, and then adding the previous two numbers together. So 0 + 1=1, and 1+1=2, and 2+1=3. Employing this reasoning Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, and so on.

It turns out that some of these numbers show up in nature, such as with the petals of flowers. For instance, Lillies have three petals, hibiscus has five petals, 8 petals are associated with delphiniums, 13 petals are associated with corn marigolds, black-eyed Susans tend to have 21 petals, 34 petals are associated with planting, and 55 and 89 petals are associated with certain daisies.

“Mathematics is the science of patterns, and nature exploits just about every pattern that this is.” —Ian Stewart

## Hexagons

Snowflakes form in hexagonal structures, the six-sided polygon. Certain minerals also form in hexagonal structures. If you’ve ever viewed part of a honeycomb, the bees build and put together the shape of a hexagon with the structure. Certain minerals have hexagonal structures as well as the eyes of some insects.

“What is mathematics? It is only a systematic effort of solving puzzles posed by nature.”—Shakuntala Devi

## Symmetry

Not only do certain shapes have significance in nature, but so does the manner in which they are reflected or transformed. For instance, most snowflakes are good examples of possessing six-fold radial symmetry, that also demonstrate the same patterns on their appendages. Certain flowers possess the same characteristics. Spider webs also demonstrate nearly perfect symmetry, containing equidistant radial support stemming from the middle.

“Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematics now, as in the past, are inspired by the art of mathematics rather than by any project of ultimate usefulness.” —E.T. Bell

## Circles and Spheres

Ripples in water mimic concentric circles. The full moon is celebrated in its complete and perfectly symmetrical manner. The rings of an onion are in circles. The cup of an acorn is a circle. The three-dimensional balls or spheres—related to a circle are also prevalent in nature. A sphere is sometimes referred to as a three-dimensional circle because all the points on the surface of a sphere are equidistance to its center.

Spheres in nature include bubbles, raindrops left upon a leaf, planets, various heirloom tomatoes and even certain flowers, like alliums.

“The beauty of mathematics only shows itself to more patient followers.” —Maryam Mirzaknani

## Spirals

Related to Fibonacci numbers is the Fibonacci spiral. The Fibonacci Spiral is based on the Golden Ratio. See heads of certain flowers like echinacea (cone flower) spiral in the Fibonacci Spiral. The Fibonacci spiral is formed by taking squares with the dimensions of the Fibonacci sequence, so it would start with a 1×1 square, next to another 1×1 square, then a 2×2 square, then a 3×3 square, next a 5×5 square, and so on (see the Fibonacci numbers explanation above).

These spirals can also be found in pine cones, pineapples, curls in hair, sea shells, galaxies, and much more.

“To those who do not know mathematics, it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature…If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.” —Richard P. Feynman

## Tessellations

The way that the hexagonal cells fit together in the honeycomb is an example of a tessellation pattern. Some scales on fish, snakes, and lizards also are tessellations because they form patterns. Other animals like turtles, dragonflies, and giraffes have kinds of tessellations as well. Ice crystals can make tessellation patterns as well.

“Mathematics are well and good but Nature keeps dragging us around by the nose.” —Albert Einstein

## Fractals

A fractal is a pattern that continues to generate forever. The Koch snowflake, for instance, begins as a three-sided equilateral triangle. Then new equilateral triangles are made on the initial three sides. And the replication of that creates the fractal. Some have analyzed trees and mimicked their growth in terms of fractals (Fractal Geometry) Others have analyzed coastlines as fractals and explored their self-similarity. Fractals are also analyzed in forests, rivers, mountains, plants, Romanesco broccoli and the human body.

“The tree is made by nature, mathematic by people. And combining the two is creating this beautiful alliance between humanity and nature. That’s why my forests are mathematical expansion systems, all of them.”—Agnes Denis

## Triangles

Triangles are common in nature, like circles and hexagons, leaves of plants, animal ears, and even human faces. Three is the basis of a triangle, which is also a Fibonacci number. The common clover makes up the trinity. They also appear in mountains and fault lines, cracks in the earth.

“The laws of Nature are written in the language of mathematics…the symbols are triangles, circles and other geometrical figures, without *show?* help it is impossible to comprehend a single word.” —Galileo Galilei

## Perspective

The idea of perspective is important in nature, math and art. It is made up of the horizon line, the vanishing line and the vanishing point. When your eye sees a path traversing forward, it is comprised of vanishing lines that go all the way to the horizon line where they meet at a vanishing point. This is why when looking outdoors, the further an object gets, the smaller it appears to us even though we know it is actually bigger. When we create 2-D representations of a scene in a photograph, painting, drawing, or other mathematical models, we show how it appears versus reality.

“Perspective is a most subtle discovery in mathematical studies, for by means of lines it causes to appear distant that which is near, and large that which is small.” Leonardo da Vinci

## Stars

Stars are another mathematical element that appears in nature. Most common are the five-pointed variety, starfish, certain leaves and fruits. In geometry, the star shape is defined as a particular kind on a non-convex polygon. In this case, all the angles inside the polygon are smaller than 180 degrees. The most recognizable star is probably the five-pointed star. But six-sided star, the hexagram is also found in nature, such as with snowflakes. Many cultures have assigned special significance to some shapes of stars as well.

“Mathematics directs the flow of the universe, lurks behind its shapes and curves, holds the reins of everything from tiny atoms to the biggest stars.”—Edward Frenkel

## Chaos and Chance

Some feel that there is more chaos and chance in the world than specific order inferred by specific shapes or special numbers. Certain flowers have totally random numbered petals. Birds fly in no desirable pattern. A storm system’s path is not 100% predictable. There is even a branch of mathematics called “chaos theory,” which suggests that seemingly orderly patterns also contain chaotic aspects and vice versa. It is also suggested that chaos is an unrecognizable system, whose order is not yet known. Or, also connected to nature, the butterfly effect, that a small new element, like a butterfly, can have a huge effect on an entire system.

Where do you see chaos in nature, and is it actually a different kind of order?

“Chaos Theory isn’t exactly about chaos. It’s about how a tiny change in a big system can affect everything.” —Hannah Baker