Fibonacci Numbers In Nature Essayist

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In mathematics, the Fibonacci numbers are the numbers in the following integer sequence, called the Fibonacci sequence, and characterized by the fact that every number after the first two is the sum of the two preceding ones:

Often, especially in modern usage, the sequence is extended by one more initial term:

.[3]

By definition, the first two numbers in the Fibonacci sequence are either 1 and 1, or 0 and 1, depending on the chosen starting point of the sequence, and each subsequent number is the sum of the previous two.

The sequence Fn of Fibonacci numbers is defined by the recurrence relation:

with seed values

or

The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics.[7][8][9] The sequence described in Liber Abaci began with F1 = 1.

Fibonacci numbers are closely related to Lucas numbers in that they form a complementary pair of Lucas sequences and . They are intimately connected with the golden ratio; for example, the closest rational approximations to the ratio are 2/1, 3/2, 5/3, 8/5, ... .

Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. They also appear in biological settings,[10] such as branching in trees, phyllotaxis (the arrangement of leaves on a stem), the fruit sprouts of a pineapple,[11] the flowering of an artichoke, an uncurling fern and the arrangement of a pine cone's bracts.[12]

Origins[edit]

The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody.[8][13] In the Sanskrit tradition of prosody, there was interest in enumerating all patterns of long (L) syllables that are 2 units of duration, and short (S) syllables that are 1 unit of duration. Counting the different patterns of L and S of a given duration results in the Fibonacci numbers: the number of patterns that are m short syllables long is the Fibonacci number Fm + 1.[9]

Susantha Goonatilake writes that the development of the Fibonacci sequence "is attributed in part to Pingala (200 BC), later being associated with Virahanka (c. 700 AD), Gopāla (c. 1135), and Hemachandra (c. 1150)".[7] Parmanand Singh cites Pingala's cryptic formula misrau cha ("the two are mixed") and cites scholars who interpret it in context as saying that the cases for m beats (Fm+1) is obtained by adding a [S] to Fm cases and [L] to the Fm−1 cases. He dates Pingala before 450 BC.[14]

However, the clearest exposition of the sequence arises in the work of Virahanka (c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135):

Variations of two earlier meters [is the variation]... For example, for [a meter of length] four, variations of meters of two [and] three being mixed, five happens. [works out examples 8, 13, 21]... In this way, the process should be followed in all mātrā-vṛttas [prosodic combinations].[15]

The sequence is also discussed by Gopala (before 1135 AD) and by the Jain scholar Hemachandra (c. 1150).

Outside India, the Fibonacci sequence first appears in the book Liber Abaci (1202) by Fibonacci. Fibonacci considers the growth of an idealized (biologically unrealistic) rabbit population, assuming that: a newly born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one year?

  • At the end of the first month, they mate, but there is still only 1 pair.
  • At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
  • At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
  • At the end of the fourth month, the original female has produced yet another new pair, and the female born two months ago also produces her first pair, making 5 pairs.

At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs (that is, the number of pairs in month n − 2) plus the number of pairs alive last month (that is, n − 1). This is the nth Fibonacci number.[16]

The name "Fibonacci sequence" was first used by the 19th-century number theorist Édouard Lucas.[17]

List of Fibonacci numbers[edit]

The first 21 Fibonacci numbers Fn for n = 0, 1, 2, …, 20 are:[18]

F0F1F2F3F4F5F6F7F8F9F10F11F12F13F14F15F16F17F18F19F20
011235813213455891442333776109871597258441816765

The sequence can also be extended to negative index n using the re-arranged recurrence relation

which yields the sequence of "negafibonacci" numbers[19] satisfying

Thus the bidirectional sequence is

F−8F−7F−6F−5F−4F−3F−2F−1F0F1F2F3F4F5F6F7F8
−2113−85−32−1101123581321

Use in mathematics[edit]

The Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's triangle (see binomial coefficient):

These numbers also give the solution to certain enumerative problems.[21] The most common such problem is that of counting the number of compositions of 1s and 2s that sum to a given total n: there are Fn+1 ways to do this.

For example, if n = 5, then Fn+1 = F6 = 8 counts the eight compositions:

1+1+1+1+1 = 1+1+1+2 = 1+1+2+1 = 1+2+1+1 = 2+1+1+1 = 2+2+1 = 2+1+2 = 1+2+2,

all of which sum to 5.

The Fibonacci numbers can be found in different ways among the set of binarystrings, or equivalently, among the subsets of a given set.

  • The number of binary strings of length n without consecutive 1s is the Fibonacci number Fn+2. For example, out of the 16 binary strings of length 4, there are F6 = 8 without consecutive 1s – they are 0000, 0001, 0010, 0100, 0101, 1000, 1001 and 1010. By symmetry, the number of strings of length n without consecutive 0s is also Fn+2. Equivalently, Fn+2 is the number of subsets S ⊂ {1,...,n} without consecutive integers: {i, i+1} ⊄ S for every i. The symmetric statement is: Fn+2 is the number of subsets S ⊂ {1,...,n} without two consecutive skipped integers: that is, S = {a1 < … < ak} with ai+1 ≤ ai + 2.
  • The number of binary strings of length n without an odd number of consecutive 1s is the Fibonacci number Fn+1. For example, out of the 16 binary strings of length 4, there are F5 = 5 without an odd number of consecutive 1s – they are 0000, 0011, 0110, 1100, 1111. Equivalently, the number of subsets S ⊂ {1,...,n} without an odd number of consecutive integers is Fn+1.
  • The number of binary strings of length n without an even number of consecutive 0s or 1s is 2Fn. For example, out of the 16 binary strings of length 4, there are 2F4 = 6 without an even number of consecutive 0s or 1s – they are 0001, 0111, 0101, 1000, 1010, 1110. There is an equivalent statement about subsets.

Relation to the golden ratio[edit]

Closed-form expression[edit]

Like every sequence defined by a linear recurrence with constant coefficients, the Fibonacci numbers have a closed-form solution. It has become known as "Binet's formula", even though it was already known by Abraham de Moivre:[22]

where

is the golden ratio (A001622), and

Since , this formula can also be written as

To see this, note that φ and ψ are both solutions of the equations

so the powers of φ and ψ satisfy the Fibonacci recursion. In other words,

and

It follows that for any values a and b, the sequence defined by

satisfies the same recurrence

If a and b are chosen so that U0 = 0 and U1 = 1 then the resulting sequence Un must be the Fibonacci sequence. This is the same as requiring a and b satisfy the system of equations:

which has solution

producing the required formula.

Taking U0 and U1 to be variables, a more general solution can be found for any starting values:

where

.

Computation by rounding[edit]

Since

for all n ≥ 0, the number Fn is the closest integer to . Therefore, it can be found by rounding, that is by the use of the nearest integer function:

or in terms of the floor function:

Similarly, if we already know that the number F > 1 is a Fibonacci number, we can determine its index within the sequence by

where can be computed using logarithms to other usual bases. For example, .

Limit of consecutive quotients[edit]

Johannes Kepler observed that the ratio of consecutive Fibonacci numbers converges. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that these ratios approach the golden ratio .[25][26]

This convergence holds regardless of the starting values, excluding 0 and 0, or any pair in the conjugate golden ratio . This can be derived from Binet's formula. For example, the initial values 3 and 2 generate the sequence 3, 2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, …, etc. The ratio of consecutive terms in this sequence shows the same convergence towards the golden ratio.

Another consequence is that the limit of the ratio of two Fibonacci numbers offset by a particular finite deviation in index corresponds to the golden ratio raised by that deviation. Or, in other words:

Decomposition of powers of the golden ratio[edit]

Since the golden ratio satisfies the equation

this expression can be used to decompose higher powers as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of and 1. The resulting recurrence relationships yield Fibonacci numbers as the linear coefficients:

This equation can be proved by induction on n.

This expression is also true for n < 1 if the Fibonacci sequence Fn is extended to negative integers using the Fibonacci rule

Matrix form[edit]

A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is

alternatively denoted

A tiling with squares whose side lengths are successive Fibonacci numbers
The Fibonacci spiral: an approximation of the golden spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling;[4] this one uses squares of sizes 1, 1, 2, 3, 5, 8, 13 and 21.
The number of rabbit pairs form the Fibonacci sequence
The Fibonacci numbers are the sums of the "shallow" diagonals (shown in red) of Pascal's triangle.
Animated GIF file showing successive tilings of the plane, and a graph of approximations to the Golden Ratio calculated by dividing successive pairs of Fibonacci numbers, one by the other. Uses the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144

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