More Fibonacci

Fibonacci sequence is one of life’s most interesting mathematical mysteries. The basic mathematical definition of the Fibonacci sequence is basically is that it starts with 0 and 1 and that every number after that equals the sum of the previous two numbers {0, 1, 1, 2, 3, 5, 8, 13, 21…}. The Fibonacci sequence is one of several recursive sequences, which means simply that it follows the rule that to get the next number one simply finds the sum of the previous two. The name Fibonacci comes from Leonardo Pisano Bigollo the son of Bonacci, a merchant trader with Africa. His most important contribution to mathematics was not the Fibonacci sequence that he is famous for; his greatest achievement was bringing Arabic Numerals to Europe. His father was a merchant and in Bigollo’s travels with him, he learned that Arabic Numerals functioned much better than Roman ones. Overtime, people remembered him only as the son of Bonacci or Fibonacci and not by his proper name which did not even have Bonacci in it. Fibonacci, although it was not the proper name was still the first mathematician to bring to light how the sequence reoccurred in nature.
The Fibonacci first noticed in the mating system of rabbits. Fibonacci noticed how rabbits multiplied and that the first month there would be 1 pair of rabbits, then the second month the female would be pregnant but there’d still only be 1 pair of rabbits. By the second month there would be 2 pairs of rabbits, the third month only one set of rabbits would be able to reproduce so there would only be 3 pairs of rabbits, so on and so on always following the sequence where a number would always add up to the previous two numbers. Since this discovery and actually even before it, mathematicians have found the Fibonacci sequence recurring in different places throughout nature.
It had been for a long time observed even in the doctrines of the Ancient Greeks, a certain aesthetically pleasing ratio that can be found in nature using the Fibonacci sequence. The ratio is formed by taken the current number over the previous number in the sequence {1/1, 2/1, 3/2, 5/3, 8/5…}. This sequence converges meaning that there is a single real number which the terms of this sequence approach more and more closely, eventually arbitrarily close. If we impose limits and assume that this sequence converges on a real number (this would be a fact that would require a proof). If we follow these rules we find that there is a simple quadratic equation to solve for x. or the golden ratio. Many things such as plant rings or the growth patterns of a snail shell fit in almost perfectly to this ratio. Even the sum of Pascal’s oblique triangles fits right into this pattern.
I think it was seeing that it fit into Pascal’s triangle that made me wonder, is this really a magical divine number for beauty or is it a sort of coincidence that these numbers follow this pattern. Another words, are these numbers simply golden because they are proportionate or is that not sufficient enough an explanation? There are some things that don’t fit the golden ratio, some flower’s petals do not fit the ratio perfectly. However, I guess I’ve never truly thought about whether everything has a mathematical explanation before or not. To me, if the very makings of the universe can somehow all be worked out into perhaps several mathematical sequences, well I’m not sure if I’d think that was divine or not. It remains amazing to me that everything from the Pantheon, to the works of Leonardo Di Vinci, to the United Nations building in New York used the golden ratio and the Fibonacci numbers in its creation.