Fibonacci and His Rabbits

Gabrielle Birchak/ January 28, 2025/ Early Modern History, Middle Ages, Post Classical/ 0 comments

What do rab­bits, nature’s cutest fluff­balls, have to do with one of the most famous pat­terns in math­e­mat­ics? Well, imag­ine this: a sin­gle pair of rab­bits start multiplying—just two at first, but soon, the field is hop­ping with Rab­bit DeNiros, Luke Sky­hop­pers, Mar­i­lyn Bun-roes, and Jes­si­cas. Before you know it, you’re ask­ing your­self: How many rab­bits are there?’ And boom—you’ve just stum­bled into the genius of Fibonacci!

Today, we’re div­ing into the hypo­thet­i­cal cot­ton-tail farm that intro­duced the world to one of math’s most fas­ci­nat­ing sequences. Spoil­er alert: it’s more than just cute bun­nies. This sto­ry con­nects every­thing from sun­flow­ers to seashells, even galax­ies, to a sim­ple prob­lem that was pro­posed in medieval Italy. So, keep read­ing because Fibonacci’s rab­bits are about to mul­ti­ply their way into your mind. 

This sto­ry isn’t just about cute bun­nies; it’s about uncov­er­ing the math­e­mat­i­cal DNA of life itself. From sun­flower spi­rals to stock mar­ket trends, Fibonacci’s thought exper­i­ment about a rab­bit prob­lem paved the way for con­nec­tions that res­onate cen­turies lat­er. It turns out that his thought exper­i­ment turned out to be a for­mu­la for under­stand­ing the cosmos.

Fibonac­ci was a thir­teenth-cen­tu­ry math­e­mati­cian who lived in Pisa. Actu­al­ly, his name was not Fibonac­ci. Instead, it is record­ed that his name was Leonar­do. In thir­teenth-cen­tu­ry Italy, peo­ple didn’t have last names. They were asso­ci­at­ed with the areas that they were born in. So, he was referred to as Leonar­do of Pisa or Leonar­do Pisano. The actu­al ori­gin of his name is still debat­able. Sources list him as Leonar­do Figlio di Bonac­ci, which means Leonar­do son of Bonac­ci or Leonar­do Pisano Bigol­lo. One of my favorite his­to­ri­ans, David Eugene Smith, notes that the name Bigol­lo was used in Tus­cany and meant trav­el­er.[1] This makes sense because Fibonacci’s father, Gugliel­mo, was a cus­toms offi­cial and an Ital­ian mer­chant. As a young boy, Fibonac­ci trav­eled with Gugliel­mo through­out the Mediter­ranean coast, where he learned about the mer­chants and their math­e­mat­ics. Then, in 1838, a Fran­co-Ital­ian his­to­ri­an named Guil­laume Lib­ri ref­er­enced him as Fibonac­ci, and the name stuck.[2]

Who Was Fibonacci?

Dur­ing his trav­els to North Africa in the ear­ly 1200s, Fibonac­ci was exposed to the Ara­bic math­e­mat­i­cal sys­tem, which was far more advanced than what was com­mon­ly used in Europe. The Arabs had made sig­nif­i­cant strides in alge­bra, dec­i­mal place val­ue, the use of zero, and geom­e­try, all of which had been heav­i­ly influ­enced by Indi­an math­e­mat­ics. These ideas were unfa­mil­iar to Euro­peans, who still relied on Roman numer­als and old­er forms of calculations.

Through his inter­ac­tions with Arab schol­ars and traders, Fibonac­ci absorbed these new con­cepts and rec­og­nized their prac­ti­cal appli­ca­tions, espe­cial­ly for com­merce and trade. This knowl­edge pro­found­ly impact­ed him as a kid, and he con­tin­ued to refine these appli­ca­tions of math­e­mat­ics over the years as Fibonac­ci incor­po­rat­ed these ideas into his work. Then, in 1202, at about 32, he pub­lished his ground­break­ing book Liber Abaci, which means The Book of Cal­cu­la­tion.

Though the orig­i­nal work does not exist, a copy from 1228 has been ana­lyzed and trans­lat­ed. In his pio­neer­ing work, Liber Abaci, Fibonac­ci intro­duced the Hin­du-Ara­bic numer­al sys­tem to the West. He explained algo­rithms, equa­tions, and prac­ti­cal prob­lem-solv­ing using these meth­ods. His expo­sure to Ara­bic math­e­mat­ics trans­formed how cal­cu­la­tions were per­formed and helped lay the foun­da­tion for the mod­ern dec­i­mal sys­tem and the Fibonac­ci sequence. Through this book, Fibonac­ci bridged the gap between East­ern and West­ern math­e­mat­i­cal tra­di­tions, fun­da­men­tal­ly shap­ing Euro­pean math­e­mat­ics for centuries.

In his book Liber Abaci, Fibonacci’s Hin­du-Ara­bic numer­al sys­tem includ­ed ten dig­its, zero through nine. It used posi­tion­al nota­tion, which made cal­cu­la­tions more effi­cient and man­age­able. In the sec­ond sec­tion of his book, Fibonac­ci demon­strat­ed the prac­ti­cal val­ue of these numer­als by apply­ing them to tasks like com­mer­cial book­keep­ing, con­vert­ing weights and mea­sures, con­vert­ing dif­fer­ent cur­ren­cies, inter­est cal­cu­la­tions, and mon­ey exchange. Liber Abaci also includes infor­ma­tion about irra­tional num­bers and prime numbers.

His work was wide­ly embraced across Europe and sig­nif­i­cant­ly impact­ed Euro­pean think­ing. It replaced the old sys­tem of Roman numer­als, ancient Egypt­ian mul­ti­pli­ca­tion meth­ods, and the use of an aba­cus for cal­cu­la­tions, mak­ing busi­ness cal­cu­la­tions more man­age­able and quick­er. This advance­ment played a nec­es­sary role in the growth of bank­ing and account­ing in Europe, help­ing to stream­line finan­cial process­es and dri­ve eco­nom­ic devel­op­ment. Addi­tion­al­ly, his book did more than assist in finan­cial pro­cess­ing. It had a con­sid­er­able influ­ence on Euro­pean sci­ence and mathematics.

The Fibonacci Sequence

Liber Abaci is an incred­i­ble body of work that includes one of my favorite thought exper­i­ments. Fibonac­ci writes, “A man has one pair of rab­bits at a cer­tain place entire­ly sur­round­ed by a wall. We wish to know how many pairs will be bred from it in one year, if the nature of these rab­bits is such that they breed every month one oth­er pair and begin to breed in the sec­ond month after their birth.”[3] This hypo­thet­i­cal rab­bit pop­u­la­tion prob­lem intro­duces the Fibonac­ci sequence.

Why did he choose rab­bits? Because rab­bits are a friend­ly bunch. Like super friend­ly. Like turn on the slow jazz and pour the wine, kind of friend­ly. They are so friend­ly that they have the entire Bar­ry White playlist blast­ing from the bush­es while they make babies, who then go on to make more babies.

So, Fibonacci’s solu­tion shows that, gen­er­a­tion by gen­er­a­tion, there becomes a sequence of num­bers. The sequence is known as the Fibonac­ci Sequence, and it is a series of num­bers where­in each num­ber is the sum of the two pre­ced­ing num­bers. The sim­plest exam­ple would be 1,1,2,3,5,8,13.

So, the sequence starts with the num­ber 1. The sec­ond num­ber is also 1 because the pre­ced­ing num­bers that we add are 0 and 1. The third num­ber is 2 because the pre­ced­ing num­bers that we add are 1 and 1. The fourth num­ber is 3 because the pre­ced­ing num­bers that we add are 1 and 2. The fifth num­ber is 5 because the pre­ced­ing num­bers that we add are 2 and 3. The sixth num­ber is 8 because the two pre­ced­ing num­bers that we add are 3 and 5. And the sev­enth num­ber is 13 because the two pre­ced­ing num­bers that we add are 5 and 8.

These hypo­thet­i­cal breed­ing rab­bits show that the num­ber of rab­bits increased through a pat­tern after each gen­er­a­tion, which, in the­o­ry, could extend to infin­i­ty. Infi­nite rabbits!

Fibonac­ci also showed that as the sequence pro­gress­es, the ratio of each num­ber to the num­ber before it approach­es the Gold­en Ratio, which is approx­i­mate­ly 1.618. It is denot­ed by the Greek let­ter phi. So, for exam­ple, when we get to the 16th and 17th dig­its, which are 610 and 987, and we divide 987 by 610, we get 1.618033. The equa­tion that defines the Gold­en Ratio is

\varphi=\frac{1+\sqrt{5}}{2} \approx 1.618

The Gold­en Ratio can be applied in sev­er­al ways to enhance bal­ance, har­mo­ny, and effi­cien­cy. In archi­tec­ture, the Gold­en Ratio is often used to cre­ate aes­thet­i­cal­ly pleas­ing pro­por­tions, as seen in the design of build­ings like the Great Pyra­mid of Giza (con­struct­ed in 2570 BC) and the Stu­pa of Borobudur in Java, Indone­sia, which has the dimen­sion of the square base relat­ed to the diam­e­ter of the largest cir­cu­lar ter­race as 1.618:1.

In art, artists like Leonar­do da Vin­ci have incor­po­rat­ed the Gold­en Ratio into their com­po­si­tions, such as in the posi­tion­ing of fig­ures in “The Vit­ru­vian Man,” ensur­ing a visu­al­ly pleas­ing arrange­ment. Graph­ic design­ers, pho­tog­ra­phers, and hob­by­ists use the Gold­en Ratio to cre­ate lay­outs that nat­u­ral­ly guide the viewer’s eye. This results in more effec­tive and engag­ing visu­al com­mu­ni­ca­tion, such as in brochures or web page designs.

For my math nerds, in fur­ther math­e­mat­i­cal advances, we also learn that the Fibonac­ci sequence is a sec­ond-order lin­ear homo­ge­neous recur­rence rela­tion with con­stant coef­fi­cients, which sat­is­fies the recur­rence rela­tion. In short, a sec­ond-order lin­ear homo­ge­neous recur­rence rela­tion is a rule that tells you how to cal­cu­late the fol­low­ing num­ber in a sequence based on the two pre­vi­ous num­bers. This spe­cif­ic rule involves fixed num­bers, also known as con­stants, that don’t change as the sequence continues.

Here’s an anal­o­gy: imag­ine you are build­ing a tow­er of blocks. The height of each new lev­el depends on the height of the two lev­els below it. And you will use a rule that tells you how to stack the blocks. The rule would say, “To make the next lev­el, dou­ble the height of the one just below it, then sub­tract the height of the lev­el that is two steps down.” Thus, if the first lev­el is one block high and the sec­ond lev­el is two blocks high, you can fig­ure out the third lev­el. You do this by dou­bling the sec­ond lev­el, which is 2 x 2 = 4. Then you sub­tract the first lev­el, which is 4 — 1 = 3. So the third lev­el is three blocks high. And you keep fol­low­ing this rule to find the fol­low­ing heights. Thus, a recur­rence rela­tion is like build­ing a tow­er where you fol­low the rule to find the next number. 

What fol­lows is a rule with two dis­tinct roots that show how the Fibonac­ci sequence sat­is­fies the recur­rence relation. 

The rela­tion says you can use this rule repeat­ed­ly to build the sequence, which fol­lows the same pattern.

For instance, if we were to look at the equation

x^{2}-x-1=0

we can deter­mine that the equa­tion has two dis­tinct roots. For instance, by the qua­drat­ic for­mu­la, we find that its roots are:

x=\frac{1 \pm \sqrt{1-4(-1)}}{2}=\frac{1+\sqrt{5}}{2} \text { and } \frac{1-\sqrt{5}}{2}

As a result, by the dis­tinct roots the­o­rem, the Fibonac­ci sequence sat­is­fies the following:

F_{n}=C \cdot\left(\frac{1+\sqrt{5}}{2}\right)^{n}+D \cdot\left(\frac{1-\sqrt{5}}{2}\right)^{n} \text { for all intergers } n \geq 0

Here, the val­ues of C and D are found through Fibonacci’s con­di­tions that 

F_{0}=F_{1}=1

Thus, we find C and D as fol­lows:

F_{0}=C \cdot\left(\frac{1+\sqrt{5}}{2}\right)^{0}+D \cdot\left(\frac{1-\sqrt{5}}{2}\right)^{0}=C \cdot 1+D \cdot 1=C+D . \tag{1}

And

F_{1}=C \cdot\left(\frac{1+\sqrt{5}}{2}\right)^{1}+D \cdot\left(\frac{1-\sqrt{5}}{2}\right)^{1}=C \cdot\left(\frac{1+\sqrt{5}}{2}\right)+D \cdot\left(\frac{1-\sqrt{5}}{2}\right)

We want to find num­bers where C+D=1 and

C \cdot\left(\frac{1+\sqrt{5}}{2}\right)+D \cdot\left(\frac{1-\sqrt{5}}{2}\right)=1 \text { and } C(1+\sqrt{5})+D(1-\sqrt{5})=2  \tag{2}

Tak­ing equa­tions (1) and (2), we find the following:

D(1-\sqrt{5})-D(1+\sqrt{5})=2-1-\sqrt{5} \Rightarrow D=\frac{\sqrt{5}-1}{2 \sqrt{5}}

Sub­sti­tut­ing D back into the equa­tion, we find the following:

C=\frac{2 \sqrt{5}-\sqrt{5}+1}{2 \sqrt{5}}=\frac{\sqrt{5}+1}{2 \sqrt{5}}

Thus, the solu­tion of the recur­rence rela­tion is

F_{n}=\left(\frac{1+\sqrt{5}}{2 \sqrt{5}}\right) \cdot\left(\frac{1+\sqrt{5}}{2}\right)^{n}+\left(\frac{-(1-\sqrt{5})}{2 \sqrt{5}}\right) \cdot\left(\frac{1-\sqrt{5}}{2}\right)^{n}

And when sim­pli­fied, we get

F_{n}=\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^{n+1}-\left(\frac{1-\sqrt{5}}{2}\right)^{n+1}\right)

For all integers 

n\ge0

 Thus, it is shown that the Fibonac­ci sequence sat­is­fies the recur­rence relation.

FIBONACCI SPIRAL

The Fibonac­ci sequence is bril­liant because it even lends itself to the Fibonac­ci Spi­ral. You can visu­al­ize the sequence through the Fibonac­ci Spi­ral. The Fibonac­ci Spi­ral is a geo­met­ric pat­tern approx­i­mat­ing the Gold­en Spi­ral, a curve that grows pro­por­tion­al­ly as it winds out­ward. So, let’s take the Fibonac­ci Sequence, which starts with the series of num­bers 1, 1, 2, 3, 5, 8, 13. Now, draw a one-by-one square and then anoth­er one-by-one square next to it. Then, add a two-by-two square adja­cent to the first two one-by-one squares. Then you add a three-by-three square fol­lowed by a five-by-five square, and so on and so on. Then, in each square, you can draw a quar­ter-cir­cle arc that con­nects oppo­site cor­ners. So, as you move from one square to the next, the arcs cre­ate a con­tin­u­ous spiral. 

The spi­ral grows larg­er as it moves out­ward, and the shape becomes smoother and could the­o­ret­i­cal­ly expand to infin­i­ty. It has no end. It is so amaz­ing how nature is con­nect­ed to the Fibonac­ci number. 

You can also see the spi­ral in an arrange­ment of sun­flower seeds, pinecones, and Nau­tilus shells. The Fibonac­ci sequence and the Fibonac­ci spi­ral are a won­der­ful reminder that we are all very con­nect­ed to math­e­mat­ics; it is in nature, around us, and with­in us.

The Fibonac­ci sequence and the Fibonac­ci Spi­ral are evi­dent in art and archi­tec­ture. As well as the rhythm that we lis­ten to in music. The scales on the piano reflect Fibonac­ci pat­terns as well. Algo­rithms cre­at­ed for search func­tions and data struc­tures are also based on the Fibonac­ci sequence.

In com­put­er sci­ence, Fibonac­ci-inspired algo­rithms play a role in effi­cient data orga­ni­za­tion, such as Fibonac­ci heaps used in graph opti­miza­tion prob­lems. A Fibonac­ci heap is a type of data struc­ture used in com­put­er sci­ence to man­age a col­lec­tion of items, espe­cial­ly when you need to find and update the small­est val­ue. For exam­ple, imag­ine you have a lot of files in a fil­ing cab­i­net. These files have num­bers or pri­or­i­ties, like a task with a dead­line. And you want to put the files in the fil­ing cab­i­net. Instead of putting all the files in a spe­cif­ic order in the fil­ing cab­i­net, you group them into piles, each fol­low­ing a loose struc­ture. A Fibonac­ci heap allows you to find the file with the small­est num­ber or the low­est pri­or­i­ty. And it’s always a fast process because the file with the small­est num­ber or low­est pri­or­i­ty is always on top of one of the piles inside the cab­i­net. And when you add, remove, or change the files around, the cab­i­net will reor­ga­nize itself so it can still find the small­est file quick­ly. That’s the con­cept behind a Fibonac­ci heap.

Also, in engi­neer­ing, many designs ben­e­fit from Fibonac­ci pat­terns to opti­mize effi­cien­cy and aes­thet­ics. In biol­o­gy and med­i­cine, the sequence helps researchers under­stand growth pat­terns and struc­tures in organ­isms. We see the spi­ral in the arrange­ment of seeds in sun­flow­ers, which offers insights into devel­op­ments in nature. We also see the spi­ral pat­terns in our DNA, which lead us to under­stand poten­tial med­ical appli­ca­tions. When you look at the Dou­ble Helix, the ratio of the width of the helix to its length for one com­plete turn is about 1 to 1.618, which is the Gold­en Ratio. So, as you can see, math is in our DNA. That is such a cool con­cept to realize.

But, while the Fibonac­ci Sequence and the Gold­en Ratio are often cel­e­brat­ed as uni­ver­sal rules of beau­ty and nature, they are not as all-encom­pass­ing as com­mon­ly believed. Many nat­ur­al pat­terns, like the arrange­ment of leaves or spi­rals in shells, are only approx­i­ma­tions of the Fibonac­ci sequence, and some devi­ate entire­ly. Sim­i­lar­ly, the Gold­en Ratio is often exag­ger­at­ed in its role in art and archi­tec­ture. Claims that it was inten­tion­al­ly used in struc­tures like the Parthenon or famous art­works like the Mona Lisa are spec­u­la­tive or coin­ci­den­tal. More­over, the Fibonac­ci Sequence is just one exam­ple of a broad­er math­e­mat­i­cal study of pat­terns in nature, which includes oth­er forms like log­a­rith­mic spi­rals, frac­tals, and tes­sel­la­tions. While fas­ci­nat­ing and impor­tant, nei­ther the Fibonac­ci Sequence nor the Gold­en Ratio holds a monop­oly on explain­ing the com­plex­i­ty and beau­ty of the world around us.

Still, Fibonac­ci was a bril­liant man whose con­tri­bu­tions to our under­stand­ing of math in nature con­tin­ue to astound us! Fibonac­ci pub­lished three oth­er works, includ­ing Prac­ti­ca Geome­tri­ae in 1220, the Liber Quadra­to­rum in 1225, and the Flos in 1225. The Flos result­ed from Fibonacci’s meet­ing with Emper­or Fred­er­ick II and his court. Through this meet­ing, he was intro­duced to Johannes of Paler­mo, also a math­e­mati­cian. The Emper­or pro­posed that Fibonac­ci exhib­it his math­e­mat­i­cal insights by solv­ing math­e­mat­i­cal ques­tions writ­ten and pre­sent­ed by Johannes of Paler­mo. Hence, Fibonac­ci set forth to prove to Johannes his capa­bil­i­ties through the con­struc­tion of the Flos (Horadam). He pub­lished two oth­er works that were lost in the annals of his­to­ry; these include Di Mino Guisa (On Com­mer­cial Arith­metic) and Com­men­tary on Book X of Euclid’s Elements.

Sym­me­try teach­es us a lot. It teach­es us how bal­ance in life, rela­tion­ships, and work leads to greater effi­cien­cy and har­mo­ny. Whether in nature, art, our per­son­al and work­ing rela­tion­ships, or our soci­o­log­i­cal struc­tures, sym­me­try reminds us that find­ing bal­ance is key to thriv­ing and growth.

Rab­bits have this fig­ured out. They know the impor­tance of growth, which explains why they breed so well. So, the next time you see Fluffer­nut­ter McFluffy­face, Rab­bit Downey Junior, or the entire Bun­ny band, Thumper­nick­el­back, scout­ing the yard for their Lovin’ Lago­morphs, know that those rab­bits are liv­ing their best lives in math­e­mat­i­cal (ahem) explo­ration. I won­der if that’s what Fibonac­ci was think­ing when he devel­oped this hypo­thet­i­cal bun­ny farm and inad­ver­tent­ly invent­ed a rab­bit herd of alge­bra­ic romance.

So, next time you’re mar­veling at a sun­flower, plot­ting Fibonac­ci retrace­ments in the stock mar­ket, or just pon­der­ing the sym­me­try of life, give a lit­tle nod to those con­cu­pis­cent cot­ton bot­toms. Thanks for reading! 

Until next time, carpe diem, my friends.

Ref­er­ences

Devlin, K. J. (2011). The man of num­bers: Fibonacci’s arith­metic rev­o­lu­tion. New York, NY: Walk­er & Co.

Epp, S. S. (1995). Dis­crete math­e­mat­ics, with appli­ca­tions, 2nd ed (2nd ed.). Boston, MA: PWS Publishing.

Gard­ner, H., De C., Tansey, R. G., & Kirk­patrick, D. (1991). Gardner’s Art Through the Ages (1st ed.). San Diego, FL: Har­court Brace Jovanovich.

Guisepi, R. (n.d.). The His­to­ry of Edu­ca­tion. Retrieved from http://history-world.org/history_of_education.htm

Horadam, A. F. (1975). Eight hun­dred years young. The Aus­tralian Math­e­mat­ics Teacher, 31, 123–134. Retrieved from http://faculty.evansville.edu/ck6/bstud/fibo.html

Smith, D. E. (1958). The Occi­dent from 1000 to 1500. In the his­to­ry of math­e­mat­ics: Gen­er­al sur­vey of the his­to­ry of ele­men­tary math­e­mat­ics (pp. 194–265). New York, NY: Dover Publications.

Suter, H. (1887). Die Math­e­mat­ic auf den Uni­ver­si­tat­en des Mit­te­lal­ters. Zurich, Switzer­land: Fest-schrift der Kan­ton­schule in Zurich.

[1] Devlin, Kei­th. Find­ing Fibonac­ci: The Quest to Redis­cov­er the For­got­ten Math­e­mat­i­cal Genius Who Changed the World. Prince­ton Uni­ver­si­ty Press, 2017. https://doi.org/10.2307/j.ctvc77n03.

[2] Drozdyuk, Andriy, and Denys Drozdyuk. Fibonac­ci, His Num­bers and His Rab­bits. Toron­to: Choven Pub, 2010.

[3] Fibonac­ci, Leonar­do. Liber Abaci — O Livro Do Cál­cu­lo — Leonar­do Fibonac­ci. Trans­lat­ed by Lau­rence Sigler. New York: Springer, 2003. http://archive.org/details/liber-abaci-o-livro-do-calculo-leonardo-fibonacci.