It was in the 13th century that an Italian mathematician named Fibonacci asked a question about the multiplication of rabbits. The solution revealed a sequence of numbers which, two centuries later, seemed to have been embedded into nature itself. The Fibonacci series begins with 0 and 1. Any number is always the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc. The seemingly mystical sequence of numbers is anything but an anomaly. It is occurring with brain-shattering regularity in the form of plants, animals, and flowers, not as a law of logic but as the emergent signature of a successful, beautiful hidden process of growth. The prevalence of the Fibonacci sequence in the universe is an overwhelmingly strong indication of becoming acquainted with the hidden mathematical language of life itself.
The most blindly self-evident and ubiquitous duplication of this formula is in the formula of seeds, leaves, and flowers. Look at a sunflower. Its center is not merely some arbitrary group of seeds; it is a splendid, orderly progression of spirals that move in opposite directions. If you count spirals on one side in counter-clockwise and on the other in clockwise, you will have most frequently two neighboring numbers of the Fibonacci sequence, typically 34 and 55 or 55 and 89. The same with the pattern on the scales of pinecones, pineapple tubercles, and cauliflower florets. This is phyllotaxis, the clever answer of the plant to the simple problem: how to have most parts (seeds, leaves) optimally.
To look upon this as the optimal arrangement, imagine a new leaf bulging out at the tip of a growing stem. If the plant is to thrive, each leaf will receive the maximum amount of light and rain, without a leaf or leaves above it chopping it off. If each new leaf pushed the previous one to the right, it would push it. The plant will need to half-turn past the stem that directs the new leaf into the most full space obtainable. The best angles seem to use the so-called "golden ratio," about 1.618, which is an extremely close correspondence to the Fibonacci series. The golden angle, slightly greater than 137.5 degrees, is the angle of a circle with this ratio. The plant then uses this angle afterward in leaf deployment so that every leaf arrives on time to prune off the leaves below it from as much light as there is. With each recirculation, the number of spiraling faces in sight is an approximation of a Fibonacci number because of an intrinsic bias. The plant is not doing the math; it is using a blind, nondescript rule of growth that creates a complex but efficient mathematical form.
The same packing efficiency tool is demonstrated to yield Fibonacci numbers in the number of flower petals. A number of the most common flowers have a Fibonacci number of petals. A lily or an iris contains 3. Wild roses and buttercups 5. Delphiniums 8. Marigolds 13. Asters 21, and most daisies 34, 55, or 89. No iron law—nature is nothing if not whimsical—but it is a statistical knockout pattern. The flowers themselves are re-shaped leaves that grew outward from some point of focus, the meristem. When the flower bud is developing, the primordia (the tiny developing petals) will emerge at the golden angle, bent inwards towards the center. The number that emerges by doing so is a Fibonacci number. It is the safest and most secure way so that there won't be any big gaps or excess when the flower opens.
Mathematics of the Fibonacci sequence can be discovered in plant and even animal life. The nautilus shell's spiral is the classic instance. The nautilus, in growing, piles on new, progressively larger but always related chambers. Its shell is a logarithmic spiral, or a proportionate increase curve by the golden section. It is this helix that permits the creature to grow in magnitude without modification of shape, a gnomonic growth. Acquired is a proportionately well-built, strong, light-filled home that increases in size with its resident. Rooms, sliced in cross-section, replicate the same helical shapes in the sunflower head, from the same mathematical forces acting on successful growth.
Even honeybee reproduction is Fibonacci, as in the original problem's rabbits' reproduction. One queen reproducing eggs exists in the bee colony. When the egg is fertilized, a female worker bee and a male drone if the egg is not fertilized. The drone subsequently has one parent (the queen), but the female bee has two parents (the queen and the drone). If you trace a male drone family back, you have a good Fibonacci sequence. He has 1 parent (a female). He has 2 grandparents (a female and a male, since his grandmother was a female who had two parents). He has 3 great-grandparents, 5 great-great-grandparents, etc. Each generation has the number equal to the product of the two before it, and simply a biological definition of the Fibonacci sequence. Why does it happen so frequently?
Not through a half-mythical "force of life" but through the plain arithmetic of evolutionary advantage and gearwork construction. Plants that evolved a Fibonacci-phased phyllotaxis grew more rapidly, had less chance of becoming diseased in bud clusters, and had more seeds. And they learned their blueprints for this type of growth from their parents. Natural selection preferred this numerical approach over hundreds of millions of years merely because it operated so effectively. It's an amazing energy-saving gadget in the genetic coding of millions of living organisms. Remember that nature is not perfect, and the Fibonacci series is a trend, not an absolute.
Four-leaf clover flowers and six-petaled flowers do happen. The sequence is the ideal theoretical shape of some theoretical growth process. The omnipresence of the Fibonacci sequence throughout nature is an emerald witness to the creation of the universe according to mathematical principles. The intertwined arrangement of a galaxy, the form of an erupting wave, and the distribution of seeds on a sunflower are all the result of the same rules for growth and filling of space. The Fibonacci sequence is a lesson in how the pattern we observe in a shell, in a flower, isn't random; it's our perception of an ordered arrangement of numbers that underlies them, a secret code for survival and efficiency growing, branching, curling its way around the globe for half a billion years.