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Presents, a Life with a Plan. My name is Karen Anastasia Placek, I am the author of this Google Blog. This is the story of my journey, a quest to understanding more than myself. The title of my first blog delivered more than a million views!! The title is its work as "The Secret of the Universe is Choice!; know decision" will be the next global slogan. Placed on T-shirts, Jackets, Sweatshirts, it really doesn't matter, 'cause a picture with my slogan is worth more than a thousand words, it's worth??.......Know Conversation!!!

Wednesday, November 10, 2021

The Title Is Known

 

 

Cantore mathematics is a complicated arithmetic leaving the metric to the venue for the opening to introduce the understanding of how Albert Einstein found MC2 at E.  To engage that only by name "Albert Einstein" I broaden the Fibonacci to encompass the seashell found as a sand dollar at Ocean beach when I was a child, this is important as the 'Conch shell' is mentioned in the conical design of his sequence called the Fibonacci Number (sequence) of the Fibonacci.  Fibonacci himself is also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages":  See below.

 


 

These simple numbers in designed are in representation and demonstrate nothing more to cantore mathematics than the sand dollar as I was born and raised in San Francisco, California and the downtown of my city represents by number and address the sand dollar in multiply the sight to thought and never represent that as a discount to the work.  As that is done the Band of Holes in Peru of more than to the last post is of demonstration to a possible volcanic chain due to the design and local story to the mystery of "ley lines" (Nazca Lines) that can only be seen from the air.  These basic foundation would by design introduce the same as the planet to planet representing the interest of just the magma to planet introducing the meteor as the projection to the landing of the Fibonacci by both gravity and temperature.

reference:  Scuba diving at  Lake Tahoe

Logarithmic spiral

From Wikipedia, the free encyclopedia
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"Spira mirabilis" redirects here. For the orchestra, see Spira Mirabilis (orchestra). For for the Italian film, see Spira Mirabilis (film).
Logarithmic spiral (pitch 10°)
A section of the Mandelbrot set following a logarithmic spiral

A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige lini").[1] More than a century later, the curve was discussed by Descartes (1638), and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous spiral".

The logarithmic spiral can be distinguished from the Archimedean spiral by the fact that the distances between the turnings of a logarithmic spiral increase in geometric progression, while in an Archimedean spiral these distances are constant.

Definition

In polar coordinates {\displaystyle (r,\varphi )} the logarithmic spiral can be written as[2]

{\displaystyle r=ae^{k\varphi },\quad \varphi \in \mathbb {R} ,}

or

{\displaystyle \varphi ={\frac {1}{k}}\ln {\frac {r}{a}},}

with e being the base of natural logarithms, and {\displaystyle a>0,k\neq 0} being real constants.

In Cartesian coordinates

The logarithmic spiral with the polar equation

{\displaystyle \;r=ae^{k\varphi }}

can be represented in Cartesian coordinates {\displaystyle (x=r\cos \varphi ,\,y=r\sin \varphi )} by

  • {\displaystyle x=ae^{k\varphi }\cos \varphi ,\qquad y=ae^{k\varphi }\sin \varphi .}

In the complex plane {\displaystyle (z=x+iy,\,e^{i\varphi }=\cos \varphi +i\sin \varphi )}:

  • {\displaystyle z=ae^{(k+i)\varphi }.}

Spira mirabilis and Jacob Bernoulli

Spira mirabilis, Latin for "miraculous spiral", is another name for the logarithmic spiral. Although this curve had already been named by other mathematicians, the specific name ("miraculous" or "marvelous" spiral) was given to this curve by Jacob Bernoulli, because he was fascinated by one of its unique mathematical properties: the size of the spiral increases but its shape is unaltered with each successive curve, a property known as self-similarity. Possibly as a result of this unique property, the spira mirabilis has evolved in nature, appearing in certain growing forms such as nautilus shells and sunflower heads. Jacob Bernoulli wanted such a spiral engraved on his headstone along with the phrase "Eadem mutata resurgo" ("Although changed, I shall arise the same."), but, by error, an Archimedean spiral was placed there instead.[3][4]

Properties

Definition of slope angle and sector

The logarithmic spiral {\displaystyle \;r=ae^{k\varphi }\;,\;k\neq 0,\;} has the following properties (see Spiral):

  • Polar slope: {\displaystyle \ \tan \alpha =k\quad ({\color {red}{\text{constant !}}})}
with polar slope angle \alpha (see diagram).
(In case of k=0 angle \alpha would be 0 and the curve a circle with radius a.)
  • Curvature: {\displaystyle \;\kappa ={\frac {1}{r{\sqrt {1+k^{2}}}}}={\frac {\cos \alpha }{r}}}
  • Arc length:{\displaystyle \ L(\varphi _{1},\varphi _{2})={\frac {\sqrt {k^{2}+1}}{k}}{\big (}r(\varphi _{2})-r(\varphi _{1}){\big )}={\frac {r(\varphi _{2})-r(\varphi _{1})}{\sin \alpha }}}
Especially: {\displaystyle \ L(-\infty ,\varphi _{2})={\frac {r(\varphi _{2})}{\sin \alpha }}\quad ({\color {red}{\text{finite !}}})\;}, if k>0 .
This property was first realized by Evangelista Torricelli even before calculus had been invented.[5]
  • Sector area: {\displaystyle \ A={\frac {r(\varphi _{2})^{2}-r(\varphi _{1})^{2}}{4k}}}
  • Inversion: Circle inversion ({\displaystyle r\to 1/r}) maps the logarithmic spiral {\displaystyle \;r=ae^{k\varphi }\;} onto the logarithmic spiral {\displaystyle \;r={\tfrac {1}{a}}e^{-k\varphi }\ .}
Examples for {\displaystyle a=1,2,3,4,5}
  • Rotating, scaling: Rotating the spiral by angle \varphi _{0} yields the spiral {\displaystyle r=ae^{-k\varphi _{0}}e^{k\varphi }}, which is the original spiral uniformly scaled (at the origin) by {\displaystyle e^{-k\varphi _{0}}}.
Scaling by {\displaystyle \;e^{kn2\pi }\;,n=\pm 1,\pm 2,...,\;} gives the same curve.
A scaled logarithmic spiral is congruent (by rotation) to the original curve.
Example: The diagram shows spirals with slope angle {\displaystyle \alpha =20^{\circ }} and {\displaystyle a=1,2,3,4,5}. Hence they are all scaled copies of the red one. But they can also be generated by rotating the red one by angles {\displaystyle -109^{\circ },-173^{\circ },-218^{\circ },-253^{\circ }} resp.. All spirals have no points in common (see property on complex exponential function).
  • Relation to other curves: Logarithmic spirals are congruent to their own involutes, evolutes, and the pedal curves based on their centers.
  • Complex exponential function: The exponential function exactly maps all lines not parallel with the real or imaginary axis in the complex plane, to all logarithmic spirals in the complex plane with centre at {\displaystyle 0}:
{\displaystyle z(t)=\underbrace {(kt+b)\;+it} _{\text{line}}\quad \to \quad e^{z(t)}=e^{kt+b}\cdot e^{it}=\underbrace {e^{b}e^{kt}(\cos t+i\sin t)} _{\text{log. spiral}}\ }
The polar slope angle \alpha of the logarithmic spiral is the angle between the line and the imaginary axis.

Special cases and approximations

The golden spiral is a logarithmic spiral that grows outward by a factor of the golden ratio for every 90 degrees of rotation (polar slope angle about 17.03239 degrees). It can be approximated by a "Fibonacci spiral", made of a sequence of quarter circles with radii proportional to Fibonacci numbers.

In nature

Further information: Patterns in nature § Spirals
An extratropical cyclone over Iceland shows an approximately logarithmic spiral pattern
The arms of spiral galaxies often have the shape of a logarithmic spiral, here the Whirlpool Galaxy
Cutaway of a nautilus shell showing the chambers arranged in an approximately logarithmic spiral. The plotted spiral (dashed blue curve) is based on growth rate parameter {\displaystyle b=0.1759}, resulting in a pitch of {\displaystyle \arctan b\approx 10^{\circ }}.

In several natural phenomena one may find curves that are close to being logarithmic spirals. Here follow some examples and reasons:

  • The approach of a hawk to its prey in classical pursuit, assuming the prey travels in a straight line. Their sharpest view is at an angle to their direction of flight; this angle is the same as the spiral's pitch.[6]
  • The approach of an insect to a light source. They are used to having the light source at a constant angle to their flight path. Usually the sun (or moon for nocturnal species) is the only light source and flying that way will result in a practically straight line.[7]
  • The arms of spiral galaxies.[8] Our own galaxy, the Milky Way, has several spiral arms, each of which is roughly a logarithmic spiral with pitch of about 12 degrees.[9]
  • The nerves of the cornea (this is, corneal nerves of the subepithelial layer terminate near superficial epithelial layer of the cornea in a logarithmic spiral pattern).[10]
  • The bands of tropical cyclones, such as hurricanes.[11]
  • Many biological structures including the shells of mollusks.[12] In these cases, the reason may be construction from expanding similar shapes, as is the case for polygonal figures.
  • Logarithmic spiral beaches can form as the result of wave refraction and diffraction by the coast. Half Moon Bay (California) is an example of such a type of beach.[13]

In engineering applications

A kerf-canceling mechanism leverage the self similarity of the logarithmic spiral to lock in place under rotation, independent of the kerf of the cut.[14]
A logarithmic spiral antenna
  • Logarithmic spiral antennas are frequency-independent antennas, that is, antennas whose radiation pattern, impedance and polarization remain largely unmodified over a wide bandwidth.[15]
  • When manufacturing mechanisms by subtractive fabrication machines (such as laser cutters), there can be a loss of precision when the mechanism is fabricated on a different machine due to the difference of material removed (that is, the kerf) by each machine in the cutting process. To adjust for this variation of kerf, the self-similar property of the logarithmic spiral has been used to design a kerf cancelling mechanism for laser cutters.[16]
  • Logarithmic spiral bevel gears are a type of spiral bevel gear whose gear tooth centerline is a logarithmic spiral. A logarithmic spiral has the advantage of providing equal angles between the tooth centerline and the radial lines, which gives the meshing transmission more stability.[17]

See also

References


  • Albrecht Dürer (1525). Underweysung der Messung, mit dem Zirckel und Richtscheyt, in Linien, Ebenen unnd gantzen corporen.
    1. Jiang, Jianfeng; Luo, Qingsheng; Wang, Liting; Qiao, Lijun; Li, Minghao (2020). "Review on logarithmic spiral bevel gear". Journal of the Brazilian Society of Mechanical Sciences and Engineering. 42 (8): 400. doi:10.1007/s40430-020-02488-y. ISSN 1678-5878.

    External links

    Languages


  • Priya Hemenway (2005). Divine Proportion: Φ Phi in Art, Nature, and Science. Sterling Publishing Co. ISBN 978-1-4027-3522-6.

  • Livio, Mario (2002). The Golden Ratio: The Story of Phi, The World's Most Astonishing Number. New York: Broadway Books. ISBN 978-0-7679-0815-3.

  • Yates, R. C.: A Handbook on Curves and Their Properties, J. W. Edwards (1952), "Evolutes". p. 206.

  • Carl Benjamin Boyer (1949). The history of the calculus and its conceptual development. Courier Dover Publications. p. 133. ISBN 978-0-486-60509-8.

  • Chin, Gilbert J. (8 December 2000), "Organismal Biology: Flying Along a Logarithmic Spiral", Science, 290 (5498): 1857, doi:10.1126/science.290.5498.1857c

  • John Himmelman (2002). Discovering Moths: Nighttime Jewels in Your Own Backyard. Down East Enterprise Inc. p. 63. ISBN 978-0-89272-528-1.

  • G. Bertin and C. C. Lin (1996). Spiral structure in galaxies: a density wave theory. MIT Press. p. 78. ISBN 978-0-262-02396-2.

  • David J. Darling (2004). The universal book of mathematics: from Abracadabra to Zeno's paradoxes. John Wiley and Sons. p. 188. ISBN 978-0-471-27047-8.

  • C. Q. Yu CQ and M. I. Rosenblatt, "Transgenic corneal neurofluorescence in mice: a new model for in vivo investigation of nerve structure and regeneration," Invest Ophthalmol Vis Sci. 2007 Apr;48(4):1535-42.

  • Andrew Gray (1901). Treatise on physics, Volume 1. Churchill. pp. 356–357.

  • Michael Cortie (1992). "The form, function, and synthesis of the molluscan shell". In István Hargittai and Clifford A. Pickover (ed.). Spiral symmetry. World Scientific. p. 370. ISBN 978-981-02-0615-4.

  • Allan Thomas Williams and Anton Micallef (2009). Beach management: principles and practice. Earthscan. p. 14. ISBN 978-1-84407-435-8.

  • "kerf-canceling mechanisms". hpi.de. Retrieved 2020-12-26.

  • Mayes, P.E. (1992). "Frequency-independent antennas and broad-band derivatives thereof". Proceedings of the IEEE. 80 (1): 103–112. doi:10.1109/5.119570.

  • Roumen, Thijs; Apel, Ingo; Shigeyama, Jotaro; Muhammad, Abdullah; Baudisch, Patrick (2020-10-20). "Kerf-Canceling Mechanisms: Making Laser-Cut Mechanisms Operate across Different Laser Cutters". Proceedings of the 33rd Annual ACM Symposium on User Interface Software and Technology. Virtual Event USA: ACM: 293–303. doi:10.1145/3379337.3415895. ISBN 978-1-4503-7514-6.

    • Fibonacci

    From Wikipedia, the free encyclopedia
    Jump to navigation Jump to search
    For the number sequence, see Fibonacci number. For the Prison Break character, see Otto Fibonacci.
    Fibonacci
    Leonardo da Pisa.jpg
    Statue of Fibonacci (1863) by Giovanni Paganucci in the Camposanto di Pisa[a]
    Bornc. 1170
    Pisa,[2] Republic of Pisa
    Diedc. 1250 (aged 79–80)
    Pisa, Republic of Pisa
    Other namesLeonardo Fibonacci, Leonardo Bonacci, Leonardo Pisano
    OccupationMathematician
    Known for
    Parent(s)Guglielmo "Bonacci" (father)

    Fibonacci (/ˌfɪbəˈnɑːtʃi/;[3] also US: /ˌfiːb-/,[4][5] Italian: [fiboˈnattʃi]; c. 1170 – c. 1240–50),[6] also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'[7]), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages".[8]

    The name he is commonly called, Fibonacci, was made up in 1838 by the Franco-Italian historian Guillaume Libri[9][10] and is short for filius Bonacci ('son of Bonacci').[11][b] However, even earlier in 1506 a notary of the Holy Roman Empire, Perizolo mentions Leonardo as "Lionardo Fibonacci".[12]

    Fibonacci popularized the Hindu–Arabic numeral system in the Western world primarily through his composition in 1202 of Liber Abaci (Book of Calculation).[13][14] He also introduced Europe to the sequence of Fibonacci numbers, which he used as an example in Liber Abaci.[15]

    Biography

    Fibonacci was born around 1170 to Guglielmo, an Italian merchant and customs official.[7] Guglielmo directed a trading post in Bugia, the capital of the Hammadid empire.[16] Fibonacci travelled with him as a young boy, and it was in Bugia (Algeria) where he was educated that he learned about the Hindu–Arabic numeral system.[17][6]

    Fibonacci travelled around the Mediterranean coast, meeting with many merchants and learning about their systems of doing arithmetic.[18] He soon realised the many advantages of the Hindu-Arabic system, which, unlike the Roman numerals used at the time, allowed easy calculation using a place-value system. In 1202, he completed the Liber Abaci (Book of Abacus or The Book of Calculation),[19] which popularized Hindu–Arabic numerals in Europe.[6]

    Fibonacci was a guest of Emperor Frederick II, who enjoyed mathematics and science. In 1240, the Republic of Pisa honored Fibonacci (referred to as Leonardo Bigollo)[20] by granting him a salary in a decree that recognized him for the services that he had given to the city as an advisor on matters of accounting and instruction to citizens.[21] [22]

    Fibonacci is thought to have died between 1240[23] and 1250,[24] in Pisa.

    Liber Abaci

    Main article: Liber Abaci
    A page of Fibonacci's Liber Abaci from the Biblioteca Nazionale di Firenze showing (in box on right) the Fibonacci sequence with the position in the sequence labeled with Latin numbers and Roman numerals and the value in Hindu-Arabic numerals.

    In the Liber Abaci (1202), Fibonacci introduced the so-called modus Indorum (method of the Indians), today known as the Hindu–Arabic numeral system.[25][26] The manuscript advocated numeration with ten digits including a zero and positional notation. The book showed the practical use and value of this by applying the numerals to commercial bookkeeping, converting weights and measures, calculation of interest, money-changing, and other applications. The book was well-received throughout educated Europe and had a profound impact on European thought. Replacing Roman numerals, its ancient Egyptian multiplication method, and using an abacus for calculations, was an advance in making business calculations easier and faster, which assisted the growth of banking and accounting in Europe.[27][28]

    The original 1202 manuscript is not known to exist.[29] In a 1228 copy of the manuscript, the first section introduces the numeral system and compares it with others, such as Roman numerals, and methods to convert numbers to it. The second section explains uses in business, for example converting different currencies, and calculating profit and interest, which were important to the growing banking industry. The book also discusses irrational numbers and prime numbers.[29][27][28]

    Fibonacci sequence

    Main article: Fibonacci number

    Liber Abaci posed and solved a problem involving the growth of a population of rabbits based on idealized assumptions. The solution, generation by generation, was a sequence of numbers later known as Fibonacci numbers. Although Fibonacci's Liber Abaci contains the earliest known description of the sequence outside of India, the sequence had been described by Indian mathematicians as early as the sixth century.[30][31][32][33]

    In the Fibonacci sequence, each number is the sum of the previous two numbers. Fibonacci omitted the "0" and first "1" included today and began the sequence with 1, 2, 3, ... . He carried the calculation up to the thirteenth place, the value 233, though another manuscript carries it to the next place, the value 377.[34][35] Fibonacci did not speak about the golden ratio as the limit of the ratio of consecutive numbers in this sequence.

    Legacy

    In the 19th century, a statue of Fibonacci was set in Pisa. Today it is located in the western gallery of the Camposanto, historical cemetery on the Piazza dei Miracoli.[1][36]

    There are many mathematical concepts named after Fibonacci because of a connection to the Fibonacci numbers. Examples include the Brahmagupta–Fibonacci identity, the Fibonacci search technique, and the Pisano period. Beyond mathematics, namesakes of Fibonacci include the asteroid 6765 Fibonacci and the art rock band The Fibonaccis.

    Works

    See also

    Notes


  • Fibonacci's actual appearance is not known.[1]
    1. The etymology of Bonacci is "good-natured", so the full name means "son from a good-natured [family]".[7]

    References


    1. Devlin, Keith (2010). "The Man of Numbers: In Search of Leonardo Fibonacci" (PDF). Mathematical Association of America. pp. 21–28.

    Further reading

    External links

    Languages


  • "Fibonacci's Statue in Pisa". Epsilones.com. Retrieved 2010-08-02.

  • Smith, David Eugene; Karpinski, Louis Charles (1911), The Hindu–Arabic Numerals, Boston and London: Ginn and Company, p. 128.

  • "Fibonacci, Leonardo". Lexico UK Dictionary. Oxford University Press. Retrieved 23 June 2019.

  • "Fibonacci series" and "Fibonacci sequence". Collins English Dictionary. HarperCollins. Retrieved 23 June 2019.

  • "Fibonacci number". Merriam-Webster Dictionary. Retrieved 23 June 2019.

  • MacTutor, R. "Leonardo Pisano Fibonacci". www-history.mcs.st-and.ac.uk. Retrieved 2018-12-22.

  • Livio, Mario (2003) [2002]. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (First trade paperback ed.). New York City: Broadway Books. pp. 92–93. ISBN 0-7679-0816-3.

  • Eves, Howard. An Introduction to the History of Mathematics. Brooks Cole, 1990: ISBN 0-03-029558-0 (6th ed.), p. 261.

  • Devlin, Keith (2017). Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World. Princeton University Press. p. 24.

  • Colin Pask (7 July 2015). Great Calculations: A Surprising Look Behind 50 Scientific Inquiries. Prometheus Books. p. 35. ISBN 978-1-63388-029-0.

  • Keith Devlin, The Man of Numbers: Fibonacci's Arithmetic Revolution,A&C Black, 2012 p. 13.

  • Drozdyuk, Andriy; Drozdyuk, Denys (2010). Fibonacci, his numbers and his rabbits. Toronto: Choven Pub. p. 18. ISBN 978-0-9866300-1-9. OCLC 813281753.

  • "Fibonacci Numbers". www.halexandria.org.

  • Leonardo Pisano: "Contributions to number theory". Encyclopædia Britannica Online, 2006. p. 3. Retrieved 18 September 2006.

  • Singh, Parmanand. "Acharya Hemachandra and the (so called) Fibonacci Numbers". Math. Ed. Siwan, 20(1):28–30, 1986. ISSN 0047-6269]

  • G. Germano, New editorial perspectives in Fibonacci's Liber abaci, «Reti medievali rivista» 14, 2, pp. 157–173.

  • Thomas F. Glick; Steven Livesey; Faith Wallis (2014). Medieval Science, Technology, and Medicine: An Encyclopedia. Routledge. p. 172. ISBN 978-1-135-45932-1.

  • In the Prologus of the Liber abaci he said: "Having been introduced there to this art with an amazing method of teaching by means of the nine figures of the Indians, I loved the knowledge of such an art to such an extent above all other arts and so much did I devote myself to it with my intellect, that I learned with very earnest application and through the technique of contradiction anything to be studied concerning it and its various methods used in Egypt, in Syria, in Greece, in Sicily, and in Provence, places I have later visited for the purpose of commerce" (translated by G. Germano, New editorial perspectives in Fibonacci's Liber abaci, «Reti medievali rivista» 14, 2, pp. 157–173.

  • The English edition of the Liber abaci was published by L.E. Sigler, Leonardo Pisano’s book of calculation, New York, Springer-Verlag, 2003

  • See the incipit of Flos: "Incipit flos Leonardi bigolli pisani..." (quoted in the MS Word document Sources in Recreational Mathematics: An Annotated Bibliography by David Singmaster, 18 March 2004 – emphasis added), in English: "Here starts 'the flower' by Leonardo the wanderer of Pisa..."
    The basic meanings of "bigollo" appear to be "bilingual" or "traveller". A. F. Horadam contends a connotation of "bigollo" is "absent-minded" (see first footnote of "Eight hundred years young"), which is also one of the connotations of the English word "wandering". The translation "the wanderer" in the quote above tries to combine the various connotations of the word "bigollo" in a single English word.

  • Keith Devlin (7 November 2002). "A man to count on". The Guardian. Retrieved 7 June 2016.

  • «Considerantes nostre civitatis et civium honorem atque profectum, qui eis, tam per doctrinam quam per sedula obsequia discreti et sapientis viri magistri Leonardi Bigolli, in abbacandis estimationibus et rationibus civitatis eiusque officialium et aliis quoties expedit, conferuntur; ut eidem Leonardo, merito dilectionis et gratie, atque scientie sue prerogativa, in recompensationem laboris sui quem substinet in audiendis et consolidandis estimationibus et rationibus supradictis, a Comuni et camerariis publicis, de Comuni et pro Comuni, mercede sive salario suo, annis singulis, libre xx denariorum et amisceria consueta dari debeant (ipseque pisano Comuni et eius officialibus in abbacatione de cetero more solito serviat), presenti constitutione firmamus». F. Bonaini, Memoria unica sincrona di Leonardo Fibonacci, novamente scoperta, «Giornale storico degli archivi toscani» 1, 4, 1857, pp. 239–246.

  • Koshy, Thomas (2011), Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, p. 3, ISBN 9781118031315.

  • Tanton, James Stuart (2005), Encyclopédia of Mathematics, Infobase Publishing, p. 192, ISBN 9780816051243.

  • Fibonacci's Liber Abaci, translated by Sigler, Laurence E., Springer-Verlag, 2002, ISBN 0-387-95419-8

  • Grimm 1973

  • "Fibonacci: The Man Behind The Math". NPR.org. Retrieved 2015-08-29.

  • Devlin, Keith. "The Man of Numbers: Fibonacci's Arithmetic Revolution [Excerpt]". Retrieved 2015-08-29.

  • Gordon, John Steele. "The Man Behind Modern Math". Retrieved 2015-08-28.

  • Singh, Pamanand (1985). "The so-called fibonacci numbers in ancient and medieval India". Historia Mathematica. 12 (3): 229–244. doi:10.1016/0315-0860(85)90021-7.

  • Goonatilake, Susantha (1998). Toward a Global Science. Indiana University Press. p. 126. ISBN 978-0-253-33388-9. Virahanka Fibonacci.

  • Knuth, Donald (2006). The Art of Computer Programming: Generating All Trees – History of Combinatorial Generation; Volume 4. Addison-Wesley. p. 50. ISBN 978-0-321-33570-8.

  • Hall, Rachel W. Math for poets and drummers Archived 2012-02-12 at the Wayback Machine. Math Horizons 15 (2008) 10–11.

  • Sloane, N. J. A. (ed.). "Sequence A000045 (Fibonacci Numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

  • Pisanus, Leonardus; Boncompagni, Baldassarre (1 January 1857). Scritti: Il Liber Abbaci. Tip. delle Scienze Fisiche e Matematiche. p. 231 – via Google Books.

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