# Look for mathematic sequences on your next nature hike

There's a reason I chose to be a naturalist. Calculus, physics, and statistics were never my friends. Fortunately, I was able to take a course titled Mathematics for Life Scientists. It was there I learned about Fibonacci's Sequence and the Golden Ratio. For those of you who are numerically inclined or like puzzles, this column is for you.

## False pretenses?

First off, Fibonacci was not an Italian sports carmaker. In fact, Fibonacci was just a nickname. His actual name was Leonardo Pisano Boglia, and he lived between 1170 and 1250.

Much as this Italian mathematician's name is not exactly real, his famous numerical progression is based on an inaccurate biological premise. He uses reproducing rabbits to explain his sequence in his 1202 volume titled "Liber Abaci."

Fibonacci's rabbits all live to keep reproducing. None are eaten by predators, die of disease, etc. The rabbits mate at the age of one month so that at the end of the second month a female can produce another pair, one male and one female. Hence, you'll have a new pair of bunnies every month from the second month on. Got it? So here it goes.

You start out with one pair of rabbits. After the first month you still have one pair of rabbits. After the second month, two pairs of breeding rabbits. After the third month, three pairs of rabbits.

Cutting to the chase, the Fibonacci series is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 ... Now look at this list. 1 + 1 = 2. 2 + 1 = 3. 3 + 2 = 5. See how the numbers progress? The next number is the sum of the previous two.

Is it any wonder that Nov. 23 is Fibonacci Day? Think about it, 11/23 are the first numbers of the series.

What does this have to do with nature? Hold on, one more disclaimer.

White trillium's three petals follow the Fibonacci Sequence. Courtesy of Jim Kaltsas

It wasn't even Fibonacci's idea. As he writes in the introduction to his book, he was the son of an official working in Bugia, Algeria. There, he met Indian merchants who introduced him to Indian mathematics. "In my book, I have published the doctrine of Mathematics completely according to the Method of Indians," he states.

Kudos to Fibonacci for giving credit where credit is due.

## Back to nature

If you are still with me, let's go for a mathematical nature walk. Depending on the season, here is a list of flowers you might see at Stillman followed by the number of petals in their blossoms: spiderwort (3), trillium (3), arrowhead (3), cardinal flower (5), phlox (5), columbine (5), and bloodroot (8).

Not all flowers have a fixed number of petals. Also, there are wildflowers, such as celandine poppy (4), whose blossoms deviate from Fibonacci's series. However, the pattern repeats itself enough that it caught the eyes of observers across continents and cultures.

## Now for some geometry

What are flowers for? To make seeds and fruit so that plants can reproduce.

Plants grow cells in spirals such as seeds in the center of a sunflower or the scales (seed holders) of a pine cone. If you stare at the sunflower seed head long enough, you will see geometric spirals that remind me of Op Art from the 1960s.

This is the most efficient way to pack seeds into a space. If seeds reproduced without twisting, they would be in straight lines and that would be a poor design.

These sunflower and pine cone spirals are an example of the Golden Ratio, or Spiral. The ratio brings us back to Fibonacci. Look at the diagram and note the curve's shape as it spirals through the proportionally-sized squares. The numbers on the squares correspond to Fibonacci's numbers.

Abstract enough for you? More math to come.

## Golden ratios and spirals abound!

Let's focus on the Golden Ratio number, which is 1.618. Well, that's kind of the number. It is an irrational number (expressed as an infinite decimal similar to pi) that never comes out exactly the same.

To get the Golden Ratio you look at the ratio between two successive Fibonacci numbers. The larger numbers you use, the more accurate the ratio. To wit: 3/5 = 1.666; 13/21 = 1.615; 144/233 = 1.618. Remember, the decimals would continue forever.

Besides seed heads and cones, Golden Spirals also appear everywhere, from the curve of nautilus and snail shells to the fiddleheads of newly emerging fern fronds.

You want bigger examples? Look at the spiral shapes of many galaxies or the pattern of clouds in hurricanes.

## Turning over a new leaf

Let's return to plants for a moment. Back in the 1830s, scientists found that each new leaf on a plant stem is positioned at a particular angle in relation to the previous leaf. The college word for "leaf arrangement" is phyllotaxis, and the angle they discovered was usually 137.5 degrees.

This Golden Angle, as you probably have guessed, relates back to Fibonacci's Sequence.

Don't worry, I'm not going to work this specific angle. I'll leave that to the Fibonnistas.

## Art and architecture

Leaving nature's designs for human ones, examples of the Golden Ratio can be found throughout the worlds of art and architecture. A partial list includes the U.N. building in New York, Great Pyramid at Giza, da Vinci's Last Supper, and Mozart's Sonata no. 1 in C major.

With all these instances of the Golden Ratio, it is not surprising to learn that there is a Fibonacci Quarterly. This publication's editorial board consists of math academics from North America, Europe, and Africa.

Wait a second, wouldn't the Fibonacci Quarterly come out four times a year? Four times?!

I don't see four in the Fibonacci Sequence. In order to practice what they preach, I suggest they publish on a thirdly basis. Just sayin.'

## Fascinating tendency

As I said at the outset, I am anything but a mathematical scholar. To be honest, I like to think that derivations and numerical formulas are few and far between in our living, natural world. Fibonacci's Sequence or the Golden Ratio is more of a rule than a law of nature (e.g. gravity).

Canadian mathematician Harold Coxeter said it best when he described the Golden Angle and phyllotaxis rules as "only a fascinating prevalent tendency."

• Mark Spreyer is executive director of Stillman Nature Center in Barrington. Email him at stillmangho@gmail.com.