This post is about two numbers (sequences) and their occurrences in nature. The first one is the **golden ratio**: two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The value of the golden ratio is approximately 1.618.

In other words, an arm is in the golden ratio if the ratio between the length of the full arm and the forearm is equal to the ratio between the forearm and the upper arm. (Assuming that the forearm is longer than the upper arm.) While the golden ratio is employed widely in architecture (Parthenon, Egyptian pyramids, the UN building in New York) and artists (arguably by Leonardo da Vinci as well), the claims of its presence in nature – the “divine proportions” in the branches of trees and plants, veins of leaves, body parts of humans and animals – were found to be inconsistent or with significant variations.

However, the golden ratio is closely related to another mathematical expression: the **Fibonacci sequence **— the ratio of a fibonacci number and its immediate predecessor is *approximately* equal to the golden ratio. And *that* is widely found in nature: flowering of artichoke, branches of trees, leaves on stems, and the family tree of honeybees. (For those who have forgotten, the first two Fibonacci numbers are 0 and 1. And each subsequent number after that is the sum of the last two. Hence, the Fibonacci series is: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…)

Below is an example (from here) how this series can be found in the bee ancestry:

Male honeybees are born from unfertilized eggs. Female honeybees are born from fertilized eggs. Therefore males have only a mother, but females have both a mother and a father.

Take a male honeybee and graph his ancestors. Let B(n) be the number of bees at the nth level of the family tree. At the first level of the tree is our male honeybee all by himself, so B(1) = 1. At the next level of our tree is his mother, all by herself, so B(2) = 1.

Pick one of the bees at level n of the tree. If this bee is male, he has a mother at level n+1, and a grandmother and grandfather at level n+2. If this bee is female, she has a mother and father at level n+1, and one grandfather and two grandmothers at level n+2. In either case, the number of grandparents is one more than the number of parents. Therefore B(n) + B(n+1) = B(n+2).

To summarize, B(1) = B(2) = 1, and B(n) + B(n+1) = B(n+2). These are the initial conditions and recurrence relation that define the Fibonacci numbers. Therefore the number of bees at level n of the tree equals F(n), the nth Fibonacci number.

Another such fascinating series of numbers is: the** ****Prime numbers**. Mathematicians have been pondering over these numbers over thousands of years – most notable among them were Euclid and Ramanujan – and no one has yet discovered a formula that yields all primes . These are the numbers that are fully divisible only by themselves (of course, besides 1). Below is an example (from here) of an occurrence of prime numbers in nature, in the life-cycle of cicadas, to be more specific:

The periodical cicada is one of the world’s longest-living insects, but nobody knows why it times its death with bizarre precision: It either lives for 13 years or 17 years, on the dot. […]

The noisy winged critters spend more than 99 percent of their 13 or 17 years as juveniles, sucking on roots in underground lairs. In the summertime, they crawl out en masse — up to 40,000 can emerge from under a single tree within days. Their subterranean tenures are intriguing not only because 13 and 17 years are long periods over which to remain synchronized, but also because both numbers are prime — divisible only by themselves and the number 1.

And here’s one possible explanation by a mathematician, Glenn Webb:

Our hypothesis is that cicada emergences minimize overlap with the periodic cycles of their predators, like birds and small animals, which are 2 to 5 years. By choosing prime number, through evolution, cicadas avoid meshing with these shorter cycles.

*And, finally a non sequitur:* My car license plate contains a prime number: 1867. And the first three digits of my cell phone number (after the area code) are 618, which are also the first three numbers after the decimal point in the golden ratio (1.618)!

[Picture courtesy: Wired]

Male honeybees are born from unfertilized eggs. Female honeybees are born from fertilized eggs. Therefore males have only a mother, but females have both a mother and a father.

Take a male honeybee and graph his ancestors. Let B(n) be the number of bees at the nth level of the family tree. At the first level of the tree is our male honeybee all by himself, so B(1) = 1. At the next level of our tree is his mother, all by herself, so B(2) = 1.

Pick one of the bees at level n of the tree. If this bee is male, he has a mother at level n+1, and a grandmother and grandfather at level n+2. If this bee is female, she has a mother and father at level n+1, and one grandfather and two grandmothers at level n+2. In either case, the number of grandparents is one more than the number of parents. Therefore B(n) + B(n+1) = B(n+2).

To summarize, B(1) = B(2) = 1, and B(n) + B(n+1) = B(n+2). These are the initial conditions and recurrence relation that define the Fibonacci numbers. Therefore the number of bees at level n of the tree equals F(n), the nth Fibonacci number.