Avid summer readers are immersed in thrillers, pot-boilers, and fantasies whose plots swirl around sleuthing and solving mysterious codes. But look beyond the printed page, and you will quickly discover evidence of secret numbers hidden in the elegant patterns and intricate designs of nature itself.
This past March, I was flying to Michigan when I looked out the window of the plane at the expanse of Lake Erie. Excited by what I saw, I wanted to stand and yell, “Look, everybody, isn’t that amazing?!” Down below, the springtime ice-out had begun. Cracks had created enormous plates of ice hundreds of feet wide, forming a giant jigsaw puzzle whose pieces were in the shape of hexagons and pentagons.
Search for this pattern in nature, and you will find it everywhere, from the hexagonal cells of honeycomb and the angles in which soap bubbles fit together in a mass of foam, to the shapes of a giraffe’s spots and the scales on the backs of turtles, lizards, snakes, and armadillos. Look for hexagons amid corn kernels on the cob, in the vertical columns of basalt that cooled from volcanic lava, and where newly divided living cells lend a spherical start to life.
Closely packed objects in nature often form six sides because hexagons create the least amount of surface compared to area when objects are crowded together, while still leaving no wasted space in between. But the objects that are in close relationship, such as kernels and scales, must form simultaneously in order to create hexagons because all the angles that create the hexagons need to be 120 degrees. Things that form haphazardly over time, such as ice floes and the cracked plates on a dried-up mud flat, result in an imperfect mix of hexagons, pentagons and the odd square
Unlike hexagons, pentagons do not meet evenly in space and so are not found in minerals, crystals, and other rigid molecular structures. You will never see a five-armed snowflake. But living things can grow more fluidly, so five-petaled flowers are common. Cut an apple across the equator and you will unveil the pentacle-shaped seed chamber, the result of pollination in a five- petaled flower typical of the rose family.
Spirals are some of the most intriguing and universal patterns in nature. The arrangements of scales on conifer cones form equi-angular spirals, in which each line drawn out from the center crosses the lines of the spiral at the same angle. Similar designs can be found among the seedheads of sunflowers and daisies, flower-bud scales, flower-petal arrangements, artichoke leaves, and the growth patterns of palm tree leaves and pineapples. The same numbers that underlie these geometric patterns are also behind the shape of the chambered nautilus and the curvature of horns in wild sheep.
Pick up a pinecone, or the seedhead of a sunflower, and solve the mystery. Scales and seeds form in spiral patterns that curve out from the center and contain a particular number in each spiral. Look closely and you will notice that the spirals run both clockwise and counterclockwise.
Another spiral pattern is found in the arrangements of leaves and thorns on stems. Start with a particular leaf and move up the stem. Count the number of passes you make around the stem and also the number of leaves you go by before you find a leaf whose stem lies almost exactly over the one where you began. Create a fraction with the number of passes on top and the number of leaves on the bottom. Among apples and oaks, for example, you will pass twice around the stem and count five leaves, so this fraction is 2/5. In beech and hazelnut, it is 1/3.
What is the secret key? All of the numbers contained in these spirals are in the Fibonacci sequence, discovered by a mathematician in Pisa, Italy, 800 years ago. Fibonaccis are calculated by starting with the number 1 and arriving at each subsequent number by adding the value of the last number in the sequence to the one that precedes it: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on.
Pine needles always occur in bundles of 2, 3, or 5 ( all Fibonaccis). The pentagon is a common Fibonacci in living things, appearing in the number of arms on a sea star, the patterns found on a sand dollar, in many flowers and fruits, and in the patterns formed by branches. Fibonacci numbers also show up in art, architecture, poetry, music, science, and technology.
Living things do not follow the code of Fibonaccis by design or calculation: they simply form patterns that create the most efficient spatial relationships. Fortunately for us, the geometry of Fibonaccis creates a world of natural beauty and visual poetry.