Living Math - Nature's Numbers

By Nicole Harms

Discover the fascinating world of math that exists right in your own backyard!

Did you know that math is everywhere? That’s right! And no, I am not looking for abstract references. Math is found in nature everywhere you look. The two most fundamental patterns found in nature are the number phi, and the Fibonacci number sequence.

The Fibonacci number sequence, which is fairly easy to teach to students of any age, begins as 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so forth. As you can see, the next number is always derived by adding the two preceding numbers. This sequence can continue indefinitely. It is fun for kids to see how far out they can take the sequence. What is fascinating as you take this sequence out is that the ratio between two sequential Fibonacci numbers is close to phi. Phi is a number, like pi, that is a non-repeating, non-ceasing decimal. It is rounded to 1.618 . . . The farther out you take Fibonacci numbers, the closer to phi you get when you take the ratios of two consecutive numbers.

So, as interesting as all of this is, what does it have to do with nature? Well, the phi ratio, which is often called the “Golden Ratio” is found everywhere in nature! Let’s start by looking at the human body. If you take the exact height of a person, and compare that in ratio form to the length from the top of the head to the fingertips, you will find it is a golden ratio! If you take the head-to-fingertip length and compare it to the length of the head to the navel, again you will find a golden ratio! The same applies to the head-to-navel length and the width of the shoulders! This is just in the human body.

Why is it that certain features on a person are considered beautiful, across cultural boundaries? It is because of phi. If you take a beautiful face, you will find fascinating instances of the golden ratio. Start by measuring the distance between the eyebrows. This, compared to the distance between the pupils of the eye, will create a golden ratio. Also, the distance from the corner of the mouth to the base of the chin (by drawing a straight line down) creates a golden ratio with the distance between the nostrils of the nose! These measurements continue many places throughout the human form, including the teeth! The ratio of the widths of the first tooth (the largest one) and the smaller tooth next to it is phi!

Moving on from the human body, the forms of animals are often related to Fibonacci numbers and phi. For example, the angelfish, one of the most beautiful freshwater aquarium fish, is an example of the golden ration. The distance between the top of the dorsal fin and the middle of the tail of the angelfish is a golden ratio to the length of the body.

Moving on to the plant world, plants that form spirals, whether it be pinecones or sunflowers, are created in Fibonacci numbers. A sunflower, for example, has seeds that go clockwise and counterclockwise. If these seeds are counted, the clockwise ones will be a Fibonacci number, and the counterclockwise ones will also be a Fibonacci number. Also, many plants branch in the Fibonacci sequence. Starting with the stem or truck, trace the plant up and you will find the next branching is 2, followed by 3, 5, 8, and so forth.

Do this experiment, if you can. Go to your garden and pick a flower. Carefully count the petals, as you pluck them. You will most likely find a Fibonacci number. Black-eyed susans, for example, have twenty-one petals. Or look at a rose. You will notice that in the spiral of the rose, there are clockwise, then counterclockwise petals. Just like the sunflower, these petals are Fibonacci numbers, usually consecutive numbers in the sequence!

Here’s another Fibonacci experiment. Take your favorite piece of fruit, and cut it in half side-to-side, so you create a cross section. Count the number of sections in the fruit, or the number of seeds, if you can. You will probably discover a Fibonacci number!

Now, there will be exceptions in nature. For example, a lily traditionally has six petals, which is not a Fibonacci number. However, if you look at the lily closely, you can still find the Fibonacci sequence. The petals of a lily are layered, and the top layer has three petals, and the bottom layer has three. Try counting the leaves and see what you find!

This is by no means an exhaustive discussion of the many places that math can be found in nature, but at the risk of being too technical, let’s think about what this can do for us as teachers. How can we use this information to motivate our students? The fact is that many children do not like math. They do not find it to be interesting or relevant to them. Fibonacci and his sequence can create some level of interest into the interesting world of math in nature.

Here is an activity you can do to show your children this relationship. Find a pinecone, and count the number of pines in the spiral going each direction. Record this, and then ask the child if they can find the pattern. Then introduce the Fibonacci sequence. They will be fascinated by the fact that they “discovered” the sequence in their own backyard! Move on to their own body. Have them measure some of the golden ratios of the body, such as height and head-to-fingertip length. Dividing this out, they will find that they get a number close to 1.618. Continue to other ratios, such as the distance between the hand and the forearm, or the distance between the joints on the finger. Each time, you will get a number close to phi. Challenge the child to find other comparisons on their own body, or the body of their pets, that exhibit the golden ratio.

Often in math, the work tends to be too many pencil to paper activities. Of course, these activities are extremely necessary to teach math, but at times, children find them boring. By creating more hands-on experiences for our children, we can create more excitement about the study of math. Using the mathematic patterns that already exist in nature, we can spur a life-long love of math! Or, at the very least, less hatred of the formidable subject!