Editing Fibonacci sequence (section)

== Applications ==

=== Mathematics ===

[[File:Pascal triangle fibonacci.svg|thumb|upright=1.2|The Fibonacci numbers are the sums of the diagonals (shown in red) of a left-justified [[Pascal's triangle]].]]
The Fibonacci numbers occur as the sums of [[binomial coefficient]]s in the "shallow" diagonals of [[Pascal's triangle]]:{{Sfn | Lucas | 1891 | p = 7}}
<math display=block>F_n = \sum_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor} \binom{n-k-1}{k}.</math>
This can be proved by expanding the generating function
<math display=block>\frac{x}{1-x-x^2} = x + x^2(1+x) + x^3(1+x)^2 + \dots + x^{k+1}(1+x)^k + \dots = \sum\limits_{n=0}^\infty F_n x^n</math>
and collecting like terms of <math>x^n</math>.

To see how the formula is used, we can arrange the sums by the number of terms present:
:{|
| {{math|5}}
| {{math|1== 1+1+1+1+1}}
|-
|
| {{math|1== 2+1+1+1}}
| {{math|1== 1+2+1+1}}
| {{math|1== 1+1+2+1}}
| {{math|1== 1+1+1+2}}
|-
|
| {{math|1== 2+2+1}}
| {{math|1== 2+1+2}}
| {{math|1== 1+2+2}}
|}

which is <math>\textstyle \binom{5}{0}+\binom{4}{1}+\binom{3}{2}</math>, where we are choosing the positions of {{mvar|k}} twos from {{math|''n''−''k''−1}} terms.

[[File:Fibonacci climbing stairs.svg|thumb|right|Use of the Fibonacci sequence to count {{nowrap|{1,&thinsp;2}-restricted}} compositions]]
These numbers also give the solution to certain enumerative problems,<ref>{{citation|last=Stanley|first=Richard|title=Enumerative Combinatorics I (2nd ed.)|year=2011|publisher=Cambridge Univ. Press|isbn=978-1-107-60262-5|page=121, Ex 1.35}}</ref> the most common of which is that of counting the number of ways of writing a given number {{mvar|n}} as an ordered sum of 1s and 2s (called [[composition (combinatorics)#Number of compositions|compositions]]); there are {{math|''F''<sub>''n''+1</sub>}} ways to do this (equivalently, it's also the number of [[domino tiling]]s of the <math>2\times n</math> rectangle). For example, there are {{math|1=''F''<sub>5+1</sub> = ''F''<sub>6</sub> = 8}} ways one can climb a staircase of 5 steps, taking one or two steps at a time:
:{|
| {{math|5}}
| {{math|1== 1+1+1+1+1}}
| {{math|1== 2+1+1+1}}
| {{math|1== 1+2+1+1}}
| {{math|1== 1+1+2+1}}
| {{math|1== 2+2+1}}
|-
|
| {{math|1== 1+1+1+2}}
| {{math|1== 2+1+2}}
| {{math|1== 1+2+2}}
|}
The figure shows that 8 can be decomposed into 5 (the number of ways to climb 4 steps, followed by a single-step) plus 3 (the number of ways to climb 3 steps, followed by a double-step). The same reasoning is applied [[recursion|recursively]] until a single step, of which there is only one way to climb.

The Fibonacci numbers can be found in different ways among the set of [[binary numeral system|binary]] [[String (computer science)|strings]], or equivalently, among the [[subset]]s of a given set.
* The number of binary strings of length {{mvar|n}} without consecutive {{math|1}}s is the Fibonacci number {{math|''F''<sub>''n''+2</sub>}}. For example, out of the 16 binary strings of length 4, there are {{math|1=''F''<sub>6</sub> = 8}} without consecutive {{math|1}}s—they are 0000, 0001, 0010, 0100, 0101, 1000, 1001, and 1010.  Such strings are the binary representations of [[Fibbinary number]]s.  Equivalently, {{math|''F''<sub>''n''+2</sub>}} is the number of subsets {{mvar|S}} of {{math|{{mset|1, ..., ''n''}}}} without consecutive integers, that is, those {{mvar|S}} for which {{math|{{mset|''i'', ''i'' + 1}} ⊈ ''S''}} for every {{mvar|i}}. A [[bijection]] with the sums to {{math|''n''+1}} is to replace 1 with 0 and 2 with 10, and drop the last zero.
* The number of binary strings of length {{mvar|n}} without an odd number of consecutive {{math|1}}s is the Fibonacci number {{math|''F''<sub>''n''+1</sub>}}. For example, out of the 16 binary strings of length 4, there are {{math|1=''F''<sub>5</sub> = 5}} without an odd number of consecutive {{math|1}}s—they are 0000, 0011, 0110, 1100, 1111. Equivalently, the number of subsets {{mvar|S}} of {{math|{{mset|1, ..., ''n''}}}} without an odd number of consecutive integers is {{math|''F''<sub>''n''+1</sub>}}. A bijection with the sums to {{mvar|n}} is to replace 1 with 0 and 2 with 11.
* The number of binary strings of length {{mvar|n}} without an even number of consecutive {{math|0}}s or {{math|1}}s is {{math|2''F''<sub>''n''</sub>}}. For example, out of the 16 binary strings of length 4, there are {{math|1=2''F''<sub>4</sub> = 6}} without an even number of consecutive {{math|0}}s or {{math|1}}s—they are 0001, 0111, 0101, 1000, 1010, 1110. There is an equivalent statement about subsets.
* [[Yuri Matiyasevich]] was able to show that the Fibonacci numbers can be defined by a [[Diophantine equation]], which led to [[Matiyasevich's theorem|his solving]] [[Hilbert's tenth problem]].<ref>{{citation|title=Review of Yuri V. Matiyasevich, ''Hibert's Tenth Problem''|journal=Modern Logic|first=Valentina|last=Harizanov|author-link=Valentina Harizanov|volume=5|issue=3|year=1995|pages=345–55|url=https://projecteuclid.org/euclid.rml/1204900767}}</ref>
* The Fibonacci numbers are also an example of a [[complete sequence]]. This means that every positive integer can be written as a sum of Fibonacci numbers, where any one number is used once at most.
* Moreover, every positive integer can be written in a unique way as the sum of ''one or more'' distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as [[Zeckendorf's theorem]], and a sum of Fibonacci numbers that satisfies these conditions is called a Zeckendorf representation. The Zeckendorf representation of a number can be used to derive its [[Fibonacci coding]].
* Starting with 5, every second Fibonacci number is the length of the [[hypotenuse]] of a [[right triangle]] with integer sides, or in other words, the largest number in a [[Pythagorean triple]], obtained from the formula <math display=block>(F_n F_{n+3})^2 + (2 F_{n+1}F_{n+2})^2 = {F_{2 n+3}}^2.</math> The sequence of Pythagorean triangles obtained from this formula has sides of lengths (3,4,5), (5,12,13), (16,30,34), (39,80,89), ...&thinsp;. The middle side of each of these triangles is the sum of the three sides of the preceding triangle.<ref>{{citation
 | last = Pagni | first = David
 | date = September 2001
 | issue = 4
 | journal = Mathematics in School
 | jstor = 30215477
 | pages = 39–40
 | title = Fibonacci Meets Pythagoras
 | volume = 30}}</ref>
* The [[Fibonacci cube]] is an [[undirected graph]] with a Fibonacci number of nodes that has been proposed as a [[network topology]] for [[parallel computing]].
* Fibonacci numbers appear in the [[ring lemma]], used to prove connections between the [[circle packing theorem]] and [[conformal map]]s.<ref>{{citation|last=Stephenson|first=Kenneth|isbn=978-0-521-82356-2|mr=2131318|publisher=Cambridge University Press|title=Introduction to Circle Packing: The Theory of Discrete Analytic Functions|title-link=Introduction to Circle Packing|year=2005}}; see especially Lemma 8.2 (Ring Lemma), [https://books.google.com/books?id=38PxEmKKhysC&pg=PA73 pp. 73–74], and Appendix B, The Ring Lemma, pp. 318–321.</ref>

=== Computer science ===
[[File:Fibonacci Tree 6.svg|thumb|upright=1.2|Fibonacci tree of height 6. [[AVL tree#Balance factor|Balance factor]]s green; heights red.<br />The keys in the left spine are Fibonacci numbers.]]

* The Fibonacci numbers are important in [[Analysis of algorithms|computational run-time analysis]] of [[Euclidean algorithm|Euclid's algorithm]] to determine the [[greatest common divisor]] of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers.<ref>{{Citation| first= Donald E |last= Knuth| author-link= Donald Knuth | year =1997|title=The Art of Computer Programming | volume = 1: Fundamental Algorithms|edition= 3rd | publisher = Addison–Wesley |isbn=978-0-201-89683-1 | page = 343}}</ref>
* Fibonacci numbers are used in a polyphase version of the [[merge sort]] algorithm in which an unsorted list is divided into two lists whose lengths correspond to sequential Fibonacci numbers—by dividing the list so that the two parts have lengths in the approximate proportion {{mvar|φ}}. A tape-drive implementation of the [[polyphase merge sort]] was described in ''[[The Art of Computer Programming]]''.
* {{anchor|Fibonacci Tree}}A Fibonacci tree is a [[binary tree]] whose child trees (recursively) differ in [[Tree height|height]] by exactly 1. So it is an [[AVL tree]], and one with the fewest nodes for a given height—the "thinnest" AVL tree. These trees have a number of vertices that is a Fibonacci number minus one, an important fact in the analysis of AVL trees.<ref>{{citation|last1=Adelson-Velsky|first1=Georgy|last2=Landis|first2=Evgenii|year=1962|title=An algorithm for the organization of information|journal=[[Proceedings of the USSR Academy of Sciences]]|volume=146|pages=263–266|language=ru}} [https://zhjwpku.com/assets/pdf/AED2-10-avl-paper.pdf English translation] by Myron J. Ricci in ''Soviet Mathematics - Doklady'', 3:1259–1263, 1962.</ref>
* Fibonacci numbers are used by some [[pseudorandom number generator]]s.<!-- Knuth vol. 2 -->
* Fibonacci numbers arise in the analysis of the [[Fibonacci heap]] data structure.
* A one-dimensional optimization method, called the [[Fibonacci search technique]], uses Fibonacci numbers.<ref>{{Citation| first1 = M | last1 = Avriel | first2 = DJ | last2 = Wilde | title= Optimality of the Symmetric Fibonacci Search Technique |journal=Fibonacci Quarterly|year=1966 |issue=3 |pages= 265–69| doi = 10.1080/00150517.1966.12431364 }}</ref>
* The Fibonacci number series is used for optional [[lossy compression]] in the [[Interchange File Format|IFF]] [[8SVX]] audio file format used on [[Amiga]] computers. The number series [[companding|compands]] the original audio wave similar to logarithmic methods such as [[μ-law]].<ref>{{Citation | title = Amiga ROM Kernel Reference Manual | publisher = Addison–Wesley | year = 1991}}</ref><ref>{{Citation | url = https://wiki.multimedia.cx/index.php?title=IFF#Fibonacci_Delta_Compression | contribution = IFF | title = Multimedia Wiki}}</ref>
* Some Agile teams use a modified series called the "Modified Fibonacci Series" in [[planning poker]], as an estimation tool. Planning Poker is a formal part of the [[Scaled agile framework|Scaled Agile Framework]].<ref>{{citation|author=Dean Leffingwell |url=https://www.scaledagileframework.com/story/ |title=Story |publisher=Scaled Agile Framework |date=2021-07-01 |accessdate=2022-08-15}}</ref>
* [[Fibonacci coding]]
* [[Negafibonacci coding]]

=== Nature ===
{{Further|Patterns in nature}}
{{see also|Golden ratio#Nature}}
[[File:FibonacciChamomile.PNG|thumb|[[Yellow chamomile]] head showing the arrangement in 21 (blue) and 13 (cyan) spirals. Such arrangements involving consecutive Fibonacci numbers appear in a wide variety of plants.]]

Fibonacci sequences appear in biological settings,<ref>{{Citation |first1=S |last1=Douady |first2=Y |last2=Couder |title=Phyllotaxis as a Dynamical Self Organizing Process |journal=Journal of Theoretical Biology |year=1996 |issue=3 |pages=255–74 |url=http://www.math.ntnu.no/~jarlet/Douady96.pdf |doi=10.1006/jtbi.1996.0026 |volume=178 |url-status=dead |archive-url=https://web.archive.org/web/20060526054108/http://www.math.ntnu.no/~jarlet/Douady96.pdf |archive-date=2006-05-26 }}</ref> such as branching in trees, [[Phyllotaxis|arrangement of leaves on a stem]], the fruitlets of a [[pineapple]],<ref>{{Citation | first1=Judy |last1=Jones | first2=William | last2=Wilson |title=An Incomplete Education |publisher=Ballantine Books |year=2006 |isbn=978-0-7394-7582-9 |page=544 |chapter=Science}}</ref> the flowering of [[artichoke]], the arrangement of a [[pine cone]],<ref>{{Citation| first=A | last=Brousseau |title=Fibonacci Statistics in Conifers | journal=[[Fibonacci Quarterly]] |year=1969 |issue=7 |pages=525–32}}</ref> and the family tree of [[honeybee]]s.<ref>{{citation|url = https://www.cs4fn.org/maths/bee-davinci.php |work = Maths | publisher = Computer Science For Fun: CS4FN |title = Marks for the da Vinci Code: B–}}</ref><ref>{{Citation|first1=T.C.|last1=Scott|first2=P.|last2=Marketos| url = http://www-history.mcs.st-andrews.ac.uk/Publications/fibonacci.pdf | title = On the Origin of the Fibonacci Sequence | publisher = [[MacTutor History of Mathematics archive]], University of St Andrews| date = March 2014}}</ref> [[Kepler]] pointed out the presence of the Fibonacci sequence in nature, using it to explain the ([[golden ratio]]-related) [[pentagon]]al form of some flowers.{{sfn|Livio|2003|p=110}} Field [[Leucanthemum vulgare|daisies]] most often have petals in counts of Fibonacci numbers.{{sfn|Livio|2003|pp=112–13}} In 1830, [[Karl Friedrich Schimper]] and [[Alexander Braun]] discovered that the [[Parastichy|parastichies]] (spiral [[phyllotaxis]]) of plants were frequently expressed as fractions involving Fibonacci numbers.<ref>{{Citation |first =Franck |last = Varenne |title = Formaliser le vivant - Lois, Théories, Modèles | accessdate = 2022-10-30| url = https://www.numilog.com/LIVRES/ISBN/9782705670894.Livre | page = 28 | date = 2010| isbn = 9782705678128|publisher = Hermann|quote = En 1830, K. F. Schimper et A. Braun [...]. Ils montraient que si l'on représente cet angle de divergence par une fraction reflétant le nombre de tours par feuille ([...]), on tombe régulièrement sur un des nombres de la suite de Fibonacci pour le numérateur [...].|lang = fr}}</ref>

[[Przemysław Prusinkiewicz]] advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on [[free group]]s, specifically as certain [[L-system|Lindenmayer grammars]].<ref>{{Citation|first1 = Przemyslaw |last1 = Prusinkiewicz | first2 = James | last2 = Hanan| title = Lindenmayer Systems, Fractals, and Plants (Lecture Notes in Biomathematics) |publisher= [[Springer Science+Business Media|Springer-Verlag]] |year=1989 |isbn=978-0-387-97092-9}}</ref>

[[File:SunflowerModel.svg|thumb|Illustration of Vogel's model for {{math|''n'' {{=}} 1 ... 500}}]]
A model for the pattern of [[floret]]s in the head of a [[sunflower]] was proposed by {{ill|Helmut Vogel|de|Helmut Vogel (Physiker)}} in 1979.<ref>{{Citation | last =Vogel | first =Helmut | title =A better way to construct the sunflower head | journal = Mathematical Biosciences | issue =3–4 | pages = 179–89 | year = 1979 | doi = 10.1016/0025-5564(79)90080-4 | volume = 44}}</ref>  This has the form

<math display=block>\theta = \frac{2\pi}{\varphi^2} n,\  r = c \sqrt{n}</math>

where {{mvar|n}} is the index number of the floret and {{mvar|c}} is a constant scaling factor; the florets thus lie on [[Fermat's spiral]]. The divergence [[angle]], approximately 137.51°, is the [[golden angle]], dividing the circle in the golden ratio.  Because this ratio is irrational, no floret has a neighbor at exactly the same angle from the center, so the florets pack efficiently.  Because the rational approximations to the golden ratio are of the form {{math|''F''(&thinsp;''j''):''F''(&thinsp;''j'' + 1)}}, the nearest neighbors of floret number {{mvar|n}} are those at {{math|''n'' ± ''F''(&thinsp;''j'')}} for some index {{mvar|j}}, which depends on {{mvar|r}}, the distance from the center. Sunflowers and similar flowers most commonly have spirals of florets in clockwise and counter-clockwise directions in the amount of adjacent Fibonacci numbers,{{sfn|Livio|2003|p=112}} typically counted by the outermost range of radii.<ref>{{Citation | last1 = Prusinkiewicz | first1 = Przemyslaw | author1-link = Przemyslaw Prusinkiewicz | author2-link = Aristid Lindenmayer | last2 = Lindenmayer | first2 = Aristid | title = The Algorithmic Beauty of Plants | publisher = Springer-Verlag | year = 1990 | pages = [https://archive.org/details/algorithmicbeaut0000prus/page/101 101–107] | chapter = 4 | chapter-url = https://algorithmicbotany.org/papers/#webdocs | isbn = 978-0-387-97297-8 | url = https://archive.org/details/algorithmicbeaut0000prus/page/101 }}</ref>

Fibonacci numbers also appear in the ancestral pedigrees of [[bee]]s (which are [[haplodiploid]]s), according to the following rules:
* If an egg is laid but not fertilized, it produces a male (or [[Drone (bee)|drone bee]] in honeybees).
* If, however, an egg is fertilized, it produces a female.

Thus, a male bee always has one parent, and a female bee has two. If one traces the pedigree of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, and so on. This sequence of numbers of parents is the Fibonacci sequence. The number of ancestors at each level, {{math|''F''<sub>''n''</sub>}}, is the number of female ancestors, which is {{math|''F''<sub>''n''−1</sub>}}, plus the number of male ancestors, which is {{math|''F''<sub>''n''−2</sub>}}.<ref>{{Citation | url = https://www.fq.math.ca/Scanned/1-1/basin.pdf | title = The Fibonacci sequence as it appears in nature | journal = The Fibonacci Quarterly | volume = 1 | number = 1 | pages = 53–56 | year = 1963| doi = 10.1080/00150517.1963.12431602 | last1 = Basin | first1 = S. L. }}</ref><ref>Yanega, D. 1996. Sex ratio and sex allocation in sweat bees (Hymenoptera: Halictidae). J. Kans. Ent. Soc. 69 Suppl.: 98-115.</ref> This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.

[[File:X chromosome ancestral line Fibonacci sequence.svg|thumb|upright=1.2|The number of possible ancestors on the X chromosome inheritance line at a given ancestral generation follows the Fibonacci sequence. (After Hutchison, L. "Growing the Family Tree: The Power of DNA in Reconstructing Family Relationships".<ref name="xcs"/>)]]

It has similarly been noticed that the number of possible ancestors on the human [[X chromosome]] inheritance line at a given ancestral generation also follows the Fibonacci sequence.<ref name="xcs">{{citation|last=Hutchison|first=Luke|date=September 2004|title=Growing the Family Tree: The Power of DNA in Reconstructing Family Relationships|url=http://fhtw.byu.edu/static/conf/2005/hutchison-growing-fhtw2005.pdf|journal=Proceedings of the First Symposium on Bioinformatics and Biotechnology (BIOT-04)|access-date=2016-09-03|archive-date=2020-09-25|archive-url=https://web.archive.org/web/20200925132536/https://fhtw.byu.edu/static/conf/2005/hutchison-growing-fhtw2005.pdf|url-status=dead}}</ref> A male individual has an X chromosome, which he received from his mother, and a [[Y chromosome]], which he received from his father. The male counts as the "origin" of his own X chromosome (<math>F_1=1</math>), and at his parents' generation, his X chromosome came from a single parent {{nowrap|(<math>F_2=1</math>)}}. The male's mother received one X chromosome from her mother (the son's maternal grandmother), and one from her father (the son's maternal grandfather), so two grandparents contributed to the male descendant's X chromosome {{nowrap|(<math>F_3=2</math>)}}. The maternal grandfather received his X chromosome from his mother, and the maternal grandmother received X chromosomes from both of her parents, so three great-grandparents contributed to the male descendant's X chromosome {{nowrap|(<math>F_4=3</math>)}}. Five great-great-grandparents contributed to the male descendant's X chromosome {{nowrap|(<math>F_5=5</math>)}}, etc. (This assumes that all ancestors of a given descendant are independent, but if any genealogy is traced far enough back in time, ancestors begin to appear on multiple lines of the genealogy, until eventually a [[Founder effect|population founder]] appears on all lines of the genealogy.)
[[File:BerlinVictoryColumnStairs.jpg|thumb|The Fibonacci sequence can also be found in man-made construction, as seen when looking at the staircase inside the Berlin Victory Column.]]

===Other===
* In [[optics]], when a beam of light shines at an angle through two stacked transparent plates of different materials of different [[refractive index]]es, it may reflect off three surfaces: the top, middle, and bottom surfaces of the two plates. The number of different beam paths that have {{mvar|k}} reflections, for {{math|''k'' > 1}}, is the {{mvar|k}}-th Fibonacci number. (However, when {{math|1=''k'' = 1}}, there are three reflection paths, not two, one for each of the three surfaces.){{sfn|Livio|2003|pp=98–99}}
* [[Fibonacci retracement]] levels are widely used in [[technical analysis]] for financial market trading.
* Since the [[conversion of units|conversion]] factor 1.609344 for miles to kilometers is close to the golden ratio, the decomposition of distance in miles into a sum of Fibonacci numbers becomes nearly the kilometer sum when the Fibonacci numbers are replaced by their successors. This method amounts to a [[radix]] 2 number [[processor register|register]] in [[golden ratio base]] {{mvar|φ}} being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead.<ref>{{Citation | url = https://www.encyclopediaofmath.org/index.php/Zeckendorf_representation | contribution = Zeckendorf representation | title = Encyclopedia of Math}}</ref>
* The measured values of voltages and currents in the infinite resistor chain circuit (also called the [[resistor ladder]] or infinite series-parallel circuit) follow the Fibonacci sequence. The intermediate results of adding the alternating series and parallel resistances yields fractions composed of consecutive Fibonacci numbers. The equivalent resistance of the entire circuit equals the golden ratio.<ref>{{citation
 | last1 = Patranabis | first1 = D.
 | last2 = Dana | first2 = S. K.
 | date = December 1985
 | doi = 10.1109/tim.1985.4315428
 | issue = 4
 | journal = [[IEEE Transactions on Instrumentation and Measurement]]
 | pages = 650–653
 | title = Single-shunt fault diagnosis through terminal attenuation measurement and using Fibonacci numbers
 | volume = IM-34| bibcode = 1985ITIM...34..650P
 | s2cid = 35413237
 }}</ref>
* Brasch et al. 2012 show how a generalized Fibonacci sequence also can be connected to the field of [[economics]].<ref name="Brasch et al. 2012">{{Citation| first1 =T. von | last1 = Brasch | first2 = J. | last2 = Byström | first3 = L.P. | last3 = Lystad| title= Optimal Control and the Fibonacci Sequence |journal = Journal of Optimization Theory and Applications |year=2012 |issue=3 |pages= 857–78 |doi = 10.1007/s10957-012-0061-2
|volume=154 | hdl = 11250/180781 | s2cid = 8550726 | url = https://urn.kb.se/resolve?urn=urn:nbn:se:ltu:diva-24073 | hdl-access = free }}</ref>  In particular, it is shown how a generalized Fibonacci sequence enters the control function of finite-horizon dynamic optimisation problems with one state and one control variable. The procedure is illustrated in an example often referred to as the Brock–Mirman economic growth model.
* [[Mario Merz]] included the Fibonacci sequence in some of his artworks beginning in 1970.{{sfn|Livio|2003|p=176}}
* [[Joseph Schillinger]] (1895–1943) developed [[Schillinger System|a system of composition]] which uses Fibonacci intervals in some of its melodies; he viewed these as the musical counterpart to the elaborate harmony evident within nature.{{sfn|Livio|2003|p=193}} See also {{slink|Golden ratio|Music}}.