Why is it everywhere?
In: Mathematics
A sequence in general is an ordered list of numbers. Extending that, sequences of interest have some kind of generating formula. For example, let *n* refer to the term number of a sequence. Then 2*n* + 1 generates the sequence {3, 5, 7, 9, … }.
Some sequences have a recursive formula, which means that the formula is based on other terms in the sequence. The classic Fibonacci sequence is one such sequence: it says that the *n*th term is the sum of the (*n*-1)th term and the (*n*-2)th term, starting with the first and second terms as 1. This generates the sequence {1, 1, 2, 3, 5, 8, 13, 21, … }.
It should be noted that “Fibonacci sequence” can refer to two things: the original one in the prior paragraph, or as a general term for similarly structured sequences. For example, the Lucas sequence is the same, but the first two terms are 2 and 1. This generates {2, 1, 3, 4, 7, 11, … }. It is not *the* Fibonacci sequence when speaking of the famous one, but it is *a* Fibonacci sequence under the general meaning.
It has various uses in number theory, geometry, and analysis. It is also one of multiple ways to find the “golden ratio,” a constant which pops up with several uses much like pi or *e*. This ratio leads to a lot of geometric designs which can be used in powerful ways. For example, placing objects at an angle of the golden ratio apart is among the more efficient ways to distribute those objects for structural soundness or sun exposure or packing, which has caused a lot of numerology about it being some mystical number that “appears everywhere” (and by extension the sequence) because plant leaves or shell spirals or seed packings often follow such a pattern. It should be noted that many are just approximations or entirely different sequences, so I’d personally advise against getting deep into the superstition a lot of people hold over it, but it is a pretty neat number nonetheless; likewise, the sequence(s) is(are) pretty neat, too. It’s just not as “magical” as a lot of people like to believe.
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