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
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Logarithmic spiral (pitch 10°)
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 
the logarithmic spiral can be written as[2]
or
with 
being the base of natural logarithms, and 
being real constants.
In Cartesian coordinates
The logarithmic spiral with the polar equation
can be represented in Cartesian coordinates 
by
In the complex plane 
:
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 
has the following properties (see Spiral):
- Polar slope:
- with polar slope angle
(see diagram). - (In case of
angle would be 0 and the curve a circle with radius 
.)
- Curvature:
- Arc length:
- Especially:
, if 
. - This property was first realized by Evangelista Torricelli even before calculus had been invented.[5]
- Sector area:
- Inversion: Circle inversion (
) maps the logarithmic spiral 
onto the logarithmic spiral 
Examples for
- Rotating, scaling: Rotating the spiral by angle
yields the spiral 
, which is the original spiral uniformly scaled (at the origin) by 
.
- Scaling by
gives the same curve.
- A scaled logarithmic spiral is congruent (by rotation) to the original curve.
- Example: The diagram shows spirals with slope angle
and . Hence they are all scaled copies of the red one. But they can also be generated by rotating the red one by angles 
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
:
- The polar slope angle 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
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
, resulting in a pitch of
.
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.
External links
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
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Fibonacci |
|---|
 |
| Born | c. 1170
|
|---|
| Died | c. 1250 (aged 79–80)
Pisa, Republic of Pisa |
|---|
| Other names | Leonardo Fibonacci, Leonardo Bonacci, Leonardo Pisano |
|---|
| Occupation | Mathematician |
|---|
| Known for | |
|---|
| Parent(s) | Guglielmo "Bonacci" (father) |
|---|
Fibonacci (;[3] also ,[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
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
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]
- The etymology of Bonacci is "good-natured", so the full name means "son from a good-natured [family]".[7]
References
Further reading
- Devlin, Keith (2012). The Man of Numbers: Fibonacci's Arithmetic Revolution. Walker Books. ISBN 978-0802779083.
- Goetzmann, William N. and Rouwenhorst, K.Geert (2005). The Origins of Value: The Financial Innovations That Created Modern Capital Markets. Oxford University Press Inc., US, ISBN 0-19-517571-9.
- Goetzmann, William N., Fibonacci and the Financial Revolution (October 23, 2003), Yale School of Management International Center for Finance Working Paper No. 03–28
- Grimm, R. E., "The Autobiography of Leonardo Pisano", Fibonacci Quarterly, Vol. 11, No. 1, February 1973, pp. 99–104.
- Horadam, A. F. "Eight hundred years young," The Australian Mathematics Teacher 31 (1975) 123–134.
- Gavin, J., Schärlig, A., extracts of Liber Abaci online and analyzed on BibNum [click 'à télécharger' for English analysis]
External links
- "Fibonacci, Leonardo, or Leonardo of Pisa." Complete Dictionary
of Scientific Biography. 2008. Encyclopedia.com. (April 20, 2015). [1]
- Fibonacci at Convergence
- O'Connor, John J.; Robertson, Edmund F., "Leonardo Pisano Fibonacci", MacTutor History of Mathematics archive, University of St Andrews
- Fibonacci (2 vol., 1857 & 1862) Il liber abaci and Practica Geometriae – digital facsimile from the Linda Hall Library
- Fibonacci, Liber abbaci Bibliotheca Augustana
"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.
