Fibonacci numbers

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

You can indeed see in the margin how we operated, namely that we added the first number to the second, namely the 1 to the 2, and the second to the third, and the third to the fourth, and the fourth to the fifth, and thus one after another until we added the tenth to the eleventh, namely the 144 to the 233, and we had the abovewritten sum of rabbits, namely 377, and thus you can in order find it for an unending number of months. ~ Leonardo Fibonacci

$1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\;\ldots \;$

or (often, in modern usage):

$0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\;\ldots \;$

The Fibonacci sequence is named after Leonardo Fibonacci.  His 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics.  Fibonacci numbers are intimately connected with the golden ratio.

Quotes

The Fibonacci numbers go 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144... and when plotted as squares, a "golden spiral" can be drawn.  …  The number of syllables in each line in "Lateralus" are all Fibonacci numbers.  "Black (1) Then (1) White are (2) All I see (3) In my infancy (5) Red and yellow then came to be (8) Reaching out to me (5) Let's me see (3)".  "Ride the spiral" most likely refers to the "golden spiral". ~ Sam Wolfe

• 1, 2, 3, 5, 8, who do we decapitate?
  13, 21, rinse, repeat, regret more fun
  34, 55, shiny girls, collapse the hive
  89, 144, Vermin, honey, we want more
  • Angelspit, "Vermin," Hello My Name Is (2011).

• Now everybody hop on the one, the sounds of the two
  It's the third eye vision, five-side dimension
  The eighth light, is gonna shine bright tonight
  • Black Star, "Astronomy (8th Light)", Mos Def & Talib Kweli Are Black Star (1998).

• 1, 1, 2, 3, 5, 8, 13, 21
  Mathematics is the language of nature
  • BT, "Fibonacci Sequence," Movement in Still Life (1999).

• It ["Lateralus"] was originally titled 9-8-7.  For the time signatures.  Then it turned out that 987 was the 17th step of the Fibonacci sequence (in which each integer is equal to the sum of the preceding two).  So that was cool.
  • Danny Carey, recorded by Chris Norris in "Hammer Of The Gods," Spin (2001).

• Qvidam posuit unum par cuniculorum in quodam loco, qui erat undique pariete circundatus, ut sciret, quot ex eo paria germinarentur in uno anno: cum natura eorum sit per singulum mensem aliud par germinare; et in secundo mense ab eorum natiuitate germinant. Quia suprascriptum par in primo mense germinat, duplicabis ipsum, erunt paria duo in uno mense. Ex quibus unum, scilicet primum, in secundo mense geminat; et sic sunt in secundo mense paria 3 ; ex quibus in uno mense duo pregnantur; et geminantur in tercio mense paria 2 coniculorum ; et sic sunt paria 5 in ipso mense; ex quibus in ipso pregnantur paria 3; et sunt in quarto mense paria 8; ex quibus paria 5 geminant alia paria 5: quibus additis cum parijs 8, faciunt paria 13 in quinto mense; ex quibus paria 5, que geminata fuerunt in ipso mense, non concipiunt in ipso mense, sed alia 8 paria pregnantur; et sic sunt in sexto mense paria 21; cum quibus additis parijs 13, que geminantur in septimo , erunt in ipso paria 34 ; cum quibus additis parijs 21, que geminantur in octauo mense, erunt in ipso paria 55; cum quibus additis parjis ^[sic] 34, que geminantur in nono mense, erunt in ipso paria 89; cum quibus additis rursum parijs 55, que geminantur in decimo mense 144; cum quibus additis rursum parijs 89, que geminantur in undecimo mense, erunt in ipso paria 233. Cum quibus etiam additis parijs 144 , que geminantur in ultimo mense, erunt paria 377; et tot paria peperit suprascriptum par in prefato loco in capite unius anni. Potes enim uidere in hac margine, qualiter hoc operati fuimus, scilicet quod iunximus primum numerum cum secundo, uidelicet 1 cum 2; et secundum cum tercio; et tercium cum quarto; et quartum cum quinto, et sic deinceps, donec iunximus decimum cum undecimo, uidelicet 144 cum 233; et habuimus suprascriptorum cuniculorum summam, uidelicet 377 ; et sic posses facere per ordinem de infinitis numeris mensibus.
  • A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also.  Because the abovewritten pair in the first month bore, you will double it; there will be two pairs in one month.  One of these, namely the first, bears in the second month, and thus there are in the second month 3 pairs; of these in one month two are pregnant, and in the third month 2 pairs of rabbits are born, and thus there are 5 pairs in the month; in this month 3 pairs are pregnant, and in the fourth month there are 8 pairs, of which 5 pairs bear another 5 pairs; these are added to the 8 pairs making 13 pairs in the fifth month; these 5 pairs that are born in this month do not mate in this month, but another 8 pairs are pregnant, and thus there are in the sixth month 21 pairs; to these are added the 13 pairs that are born in the seventh month; there will be 34 pairs in this month; to this are added the 21 pairs that are born in the eighth month; there will be 55 pairs in this month; to these are added the 34 pairs that are born in the ninth month; there will be 89 pairs in this month; to these are added again the 55 pairs that are born in the tenth month; there will be 144 pairs in this month; to these are added again the 89 pairs that are born in the eleventh month; there will be 233 pairs in this month.  To these are still added the 144 pairs that are born in the last month; there will be 377 pairs, and this many pairs are produced from the abovewritten pair in the mentioned place at the end of the one year.  You can indeed see in the margin how we operated, namely that we added the first number to the second, namely the 1 to the 2, and the second to the third, and the third to the fourth, and the fourth to the fifth, and thus one after another until we added the tenth to the eleventh, namely the 144 to the 233, and we had the abovewritten sum of rabbits, namely 377, and thus you can in order find it for an unending number of months.
  • Leonardo Fibonacci (1202) Liber abbaci, translated by Laurence E. Sigler

• First person:  He started with one, and adding one to itself, got two.
  Second person:  Yeah, that sounds pretty simple.
  First person:  Ah, but then he had one next to one next to two, so he said to himself, "I'll take the last two numbers in the list, add one next to the two"—
  Second person:  The one next to the two.
  First person:  —and that'll give him three.
  Second person:  Oh.
  First person:  And then the three next to the two would give him five—
  Second person:  Five, yeah.
  First person:  —the five next to the three would give him eight—
  Second person:  Eight.
  First person:  —and he kept playing with the numbers.  Thirteen, twenty-one, thirty-four, fifty-five, eighty-nine—
  Second person:  [laughs]  Sounds like a waste of time.
  First person:  Well, that's what I thought, y'know, that he didn't have anything better to do.  But it turns out that the curiosity that had him playing with this kind of "adding up of numbers"—what's now remembered as the Fibonacci numbers—turns out to have a curious and very surprising relationship to botany and classical art.
  Second person:  How so?
  First person:  I don't lie, but there seems to be examples of Fibonacci numbers all over the place in nature.
  • Ken Nordine, "Fibonacci Numbers," A Transparent Mask (2001).

• Make me, one, copy and paste
  Make me, one, copy and paste
  Make me, two, copy and paste
  Make me, Fibonacci
  Make me, three, copy and paste
  Make me, five, copy and paste
  Make me, eight, copy and paste
  Make me, Fibonacci
  • Doctor Steel, "Fibonacci Sequence," People of Earth (2002)

• Black
  Then
  White are
  All I see
  In my infancy
  Red and yellow then came to be
  Reaching out to me
  Lets me see

  There is
  So
  Much
  More and
  Beckons me
  To look through to these
  Infinite possibilities
  As below, so above and beyond, I imagine
  Drawn outside the lines of reason
  Push the envelope
  Watch it bend

  • Tool, "Lateralus", Lateralus (2001).

• The title track "Lateralus" makes use of the "Fibonacci sequence".  The Fibonacci numbers go 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144... and when plotted as squares, a "golden spiral" can be drawn.  Fibonacci numbers are prevalent in nature: in pine cones, arrangement of leaves, the centre of a sunflower, and so on.  The number of syllables in each line in "Lateralus" are all Fibonacci numbers.  "Black (1) Then (1) White are (2) All I see (3) In my infancy (5) Red and yellow then came to be (8) Reaching out to me (5) Let's me see (3)".  "Ride the spiral" most likely refers to the "golden spiral".
  • Sam Wolfe, "Interesting Facts About Tool's Music," SamWolfe.com (Saturday, 30 March 2013).

See also

• Golden ratio