Fibonacci Numbers

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811 ...

In mathematics, the Fibonacci numbers form a sequence defined by the following recurrence relation:

The Fibonacci numbers are named after Leonardo of Pisa, known as Fibonacci, although they had been described earlier in India. The Fibonacci numbers first appeared, under the name *matrameru* (mountain of cadence), in the work of the Sanskrit grammarian Pingala (Chandah-shastra, the Art of Prosody, 450 or 200 BC).

Prosody was important in ancient Indian ritual because of an emphasis on the purity of utterance. The Indian mathematician Virahanka (6th century AD) showed how the Fibonacci sequence arose in the analysis of metres with long and short syllables. Subsequently, the Jain philosopher Hemachandra (c.1150) composed a well known text on these. A commentary on Virahanka by Gopala in the 12th c. also revisits the problem in some detail.

Fibonacci Numbers --- Golden Ratio

The brain is an electrochemical machine that processes through binary code - zeroes and ones that create patterns of experiences and realities. Reality is a consciousness hologram (algorithm) that uses the Fibonacci Sequence to create experiences in which we vicariously experience.

Ancient Pueblo Used Golden Ratio to Build the Sun Temple Live Science - February 13, 2017

The Great Pyramids in Giza, the Parthenon in Athens and Chichen Itza in Mexico have something in common. Besides attracting hordes of tourists, all of these architectural wonders appear to use the golden ratio. This mathematical number is often written as 1.618, the first few digits of its infinite decimal form. Expressed another way, two quantities - let's call the larger one "a" and the smaller "b" - are in the golden ratio if "a is to b" as "a + b is to a." The result is a composition with aesthetically pleasing proportions. Now, shapes with the golden ratio, as well as other geometric shapes, have been found in another, unexpected site: the Sun Temple at Mesa Verde National Park in Colorado, built by the ancient Pueblo people who lived in what is now the modern-day Southwest; they had no known written language or written number system.

Logarithmic Spirals NASA May 17, 2008

A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jakob Bernoulli, who called it *Spira mirabilis,* "the marvelous spiral".

Scientists find clues to the formation of Fibonacci spirals in nature PhysOrg - May 1, 2007

While the aesthetics and symmetry of Fibonacci spiral patterns has often attracted scientists, a mathematical or physical explanation for their common occurrence in nature is yet to be discovered. Recently, scientists have successfully produced Fibonacci spiral patterns in the lab, and found that an elastically mismatched bi-layer structure may cause stress patterns that give rise to Fibonacci spirals. The discovery may explain the widespread existence of the pattern in plants. Researchers produced their Fibonacci spiral pattern by manipulating the stress on inorganic microstructures made of a silver core and a silicon dioxide shell. The spontaneous assembly of Fibonacci patterns has rarely been realized in the laboratory, and the scientistsŐ results suggest that plant patterns might be modeled by mutually repulsive entities for both spherical and conical surfaces.