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The Best

Fibonacci Numbers and the Golden Section

Far and away the best single source for Fibonacci and Golden Ratio browsing.  Truly remarkable breadth and detail.  Take the time to peruse the myriad of links; you will be rewarded.  Kudos are definitely in order for Ron Knott!



Getting Started

The Life and Numbers of Fibonacci

Brief history and a quick walk through the concepts, this web site addresses the basic and more advanced issues elegantly and concisely.  Written by Dr. Ron Knott and D. A. Quinney.

Who was Fibonacci?

Dr. Ron Knott's excellent resources at our disposal again, describing the man and his contributions to mathemtatics.  Also be sure to visit his other pages, specifically his Fibonacci Numbers and the Golden Section page.

Relation between the Fibonacci Sequence and the Golden Ratio

Dr. Math's discussion of the Golden Ratio, Rectangle and Fibonacci sequence.  Simple layout and concise graphics aid the initial learning experience.

Ask Dr. Math

Another Dr. Math web site, this one containing all the questions gathered pertaining to Fibonacci and Golden Ratio.

Rabbit Numerical Series

Ed Stephen's page has some cute rabbits and quickly describes the derivation of the Fibonacci Series and Golden Ratio.

Recommended Reading...

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The Man and His Life

Leonardo Pisano Fibonacci

A brief history of the man, Leonardo of Pisa.  Contains the best written and most extensive account of the great mathematician's life.

Fibonacci (c.1175 - c.1240) Mathematician

An history of Fibonacci, with a beautiful color picture of his statue at Fortezza Camp Santo.  Includes Dr. A. F. Horadam's article "Eight Hundred Years Young".

Recommended Reading...

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The Fibonacci Numbers

What the Hell is the Fibonacci Series?

A short, elegant Macromedia Flash animation of the Fibonacci numbers.  Not as irreverent as the name implies...  ;o)

Fibonacci and the Golden Ratio

Excellent, excellent web site from Cynthia Conner for her math term project at Queen's University.  This is a fantastic one-stop web site, giving great information on each of the elements including art, music, nature, the golden ratio, and Fibonacci.

The First 1,001 Fibonacci Numbers - Project Gutenberg

Dr. Simon Plouffe publishes the first 1,001 Fibonacci Numbers.

Fibonacci Primes

Brief discussion of the Fibonacci numbers that are prime.  To date the 9,311st is the largest known prime Fibonacci Number, while the 81,839th is the largest proposed Fibonacci Prime.

Fibonacci Number Java Applet

Deutsch site with a Java applet calculating the requested Fibonacci number.  Slow once you get past 40 or so, I haven't discovered it's maximum but it appears to be around there.

Fibonacci

Good web site portraying numerous aspects of the Fibonacci numbers, including checkers, the stock market, fractions, numerical representation, squares, etc.

Fibonacci Numbers

Stephanie J. Morris' great web page goes through the exercise adeptly.

Platonic Realms Fibonacci Sequence

Platonic Realms web site displaying the Fibonacci Series.  Touches on the relation of the Fibonacci Series to Pascal's Triangle, something extremely profound yet we suspect Pascal wasn't aware of it.

Fibonacci Numbers Formulae

Rajesh Ram's excellent source of summation formulae for the Fibonacci Series.

The Fibonacci Association

Official website of The Fibonacci Association.  Wealth of information and links to many great topics related to all things Fibonacci.

The Fibonacci Quarterly

Official publication of The Fibonacci Association.  Great links and subscription information.

Fibonacci Numbers Spelled Out

Ivan Galkin presents materials for 91.503 Algorithms course work at U.MASS Lowell.  This page indicates that the "Spelled Out" series contains all derivations usually omitted by gurus and left for the reader to agonize over.

Fibonacci Sequence in Mandelbrot

Robert L. Devaney of Boston University shows the Fibonacci Numbers in the periodicity of the bulbs in the Mandelbrot set.

Fibonacci - Man and Sequence

Text not in English, a good web site sowing the Fibonacci Numbers and "gulden snede".

Ivars Peterson - Divakar Viswanath's New Constant

Ivars Peterson's page (now dark) from Science News Online describing Divakar Viswanath's new constant 1.13198824... based on the Fibonacci series combined with an element of randomness.  His new constant is the exponential rate at which the average absolute value of a random Fibonacci sequence increases.

Radoslav Jovanovic's Fibonacci Numbers Page

Radoslav Jovanovic is professor of Computing Science in the middle school, in Lajkovac, Yugoslavia.  His field of research is the relations between Fibonacci Numbers and the Pascal Triangle.

Recommended Reading...

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The Golden Ratio

The Golden Ratio and The Fibonacci Numbers

From the proceedings of the Friesian School, Fourth Series, comes a well-rounded, aesthetically pleasing and robust presentation of the relationship between the Golden Ratio and the Fibonacci Numbers.  Thanks to Todd Saylor for bringing this page to my attention.

The Golden Mean

Rashomon's Golden Mean pages.  Excellent, straight forward analysis of the mathematics and geometry deriving phi.

The Greeks and the Golden Mean

A paper relating the early discovery of the Golden Mean by the ancient Greeks and their methods of constructing it.

Mathematical Elements In Plato's Thought

Analysis of Plato, Socrates and the Greek philosophers and how their writings relate to mathematical concepts.

The Golden Mean

More detail about the Golden Ratio, related to the above link.  Thank you to Steve Finch for providing the link.

The Golden Mean

Very nice detailed web site taking us through many of the mathematical jewels adorning the crown that is the Golden Ratio.

The Golden Section

Short page describing the Golden Section.  Good description of various constructions and history.

Phi: The Golden Number

Please visit Mr. Meisner's extensive and detailed pages describing all aspects of the Golden Number, Phi.  Excellent use of graphics and organization make this a great one-stop Internet resource.

The Golden Mean in Number Theory

Thank you again to Dr. Volkmar Weiss for bringing this web page to my attention.  It covers books, articles and lectures relating to the Golden Ratio in Number Theory.

The Golden Ratio - Infinite Series

Brian Roselle investigates expressing Phi in an infinite series.  Thank you to Mr. Roselle for bringing his research to my attention.

Extensions of The Golden Section and Fibonacci Series

Satoshi Omori from the Omori Patent Office in Japan emailed this link interpreting Phi and Fibonacci Series in a novel way, developed for college students.  Thank you to Mr. Omori for bringing this to my attention.

Le Nombre d'Or

French pages detailing the Golden ratio. Good graphics and encompassing a variety of aspects of Phi.  Even if you don't speak French, please take a look at all these great sites as the Golden Ratio transcends all dialectic boundaries.

The Golden Mean and ASIS

Access Solutions Information Services logo is constructed using the Golden Mean, which is a symbol of the company's philosophy.

Irrational Constants

A quick study on Euler's Number, square root of 2, Fibonacci Numbers, and of course our favorite friend Phi.

Recommended Reading...

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The 2nd Dimension

The Golden Section

Great web site demonstrating the classic constructions relating to the Golden Ratio.  Must have a Java enabled browser to enjoy watching the steps unfold in each of three constructions.  Thanks to Jill Britton for bringing this web site to my attention.

Phi: That Golden Number, by Mark Freitag

Nice web page deriving phi, clear and concise, great graphics, with a proof showing that the ratio of consecutive large Fibonacci Numbers approaches phi.

Ratio of a Line Segment

Susie Lanier's work at the University of Georgia is quite elegant.  View some of her classwork here.

The Golden Pentagon

MathPages depiction of tiling a plane using a progression of equilateral triangles dividing a pentagon.  Most interesting is that this pentagon is uniquely and recursively divided by triangles, just as is seen in the more common recursive division of the Golden Rectangle.

Division in Mean and Extreme Ratio

Depiction of the construction of the Sri Yantra, a diagram of the continuous process of Creative Generation in Tantric art.

Sacred Geometry Home Page

The Sphere, Circle, Point, Square Root of 2 and 3, Spirals, Toroids, Fractals, Solids, Stellations, and of course, the Golden Ratio are described on these excellent and concise pages.

The Irrationality of the Pentagon and the Pentagram

Author and Physicist Michael Lahanas describes the construction of the Pentagram and shows that it is the most irrational of all irrational numbers.  Dr. Lahanas includes links to other excellent resources, some provided on this link page.  One link discusses occurrences of Pentagram on Greek coins.

Golden Bowls and the Logarithmic Spiral

Analysis of the symmetric properties of the log spiral having a base relating to the Fibonacci series.  Here the author relates Phi to a great number of mathematical constants of physics.  Here the author presents the Platonic Line.

Recommended Reading...

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The 3rd Dimension

Jill Britton's Polyhedra Pastimes

Jill Britton created and maintains this tremendous web site exploring the many intriguing facets of polyhedra, including stellations, kites, history, and of course the Golden Ratio.  Thanks go to Jill for bringing her wonderful sites to my attention!

Geometry Junkyard

Excellent resource on a myriad of subjects.  Before leaving be sure to check out the general index.  Following are links of special interest:

George Hart's Sculpture, Books and Art

George Hart's incredible web site depicting his wonderful sculpting abilities in constructing Polyhedra, including stellations and truncations, and his books on Multidimensional Analysis, Zome Geometry, Polyhedra and Art through History, and other great explorations.  I highly recommend this site!

The Zometool Ball

An interesting "toy" called the Zome object, based intrinsically around phi and pentagonal 3D geometry, with the capability to make thousands of geometric shapes.

Recommended Reading...

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Art

Jill Britton's Golden Section in Art and Architecture

Excellent portrayal of the use of the Golden Section in art and architecture.  Sites depicting this most interesting and fulfilling relationship between mathematics and human aesthetics are surprisingly rare.

The Golden Section In Art, Architecture and Music

Ron Knott's dicussion of the Golden Ratio in art, architecture, and music.

Mondimat

A fun web site to create Mondrian-esque pictures.  Give it a try - it's fun!

An Angular Perspective of Leonardo da Vinci's Vitruvian Man

Author Joseph Turbeville's analysis of da Vinci's Vitruvian Man as it relates to the Golden Ratio.  Thank you to Mr. Turbeville for bringing his treatise to my attention.

An Eye-Placement Principle In 500 Years Of Portraits

Christopher W. Tyler's study of the placement of eyes in paintings over 500 years of portraits.

Matjuška Teja Krašek

Art, lectures, and exhibits from Teja Krasek of Ljubljana, Slovenija.  Thank you to the author for emailing me this link.

Recommended Reading...

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Architecture

Jill Britton's Golden Section in Art and Architecture

Excellent portrayal of the use of the Golden Section in art and architecture.  Sites depicting this most interesting and fulfilling relationship between mathematics and human aesthetics are surprisingly rare.

The Golden Section In Art, Architecture and Music

Ron Knott's dicussion of the Golden Ratio in art, architecture, and music.

Deutsch Fibonacci Number and Golden Ratio web site

This site in Deutsch examines all aspects of the Golden Ratio and Fibonacci Numbers, but I am placing it here in architecture as this site offers the good, difficult to come by, presentation.   The link appears to be broken last time I checked.

Deep Secrets - The Great Pyramid, The Golden Ratio and The Royal Cubit

"This site provides a new, and perhaps for some a controversial, explanation for the rationale behind the exterior design parameters of the Great Pyramid of Giza.  Learn here: the historical significance of both the "golden ratio" and the equal-sided pentagon (and pentagram); a new theory for the derivation of the ancient Egyptian Royal Cubit; a diagrammatic method by which the square root of any number can be derived; how to diagrammatically derive a trigonometric table; and an easy to follow presentation of Euclid's derivation of the 36 angle."

Glimmers of Light From the Eye of a Giant

Joseph Turbeville discusses the similarity in measurements of the Great pyramid at Khufu with those we find in nature.  Note the concept of "Tabularization of the Fibonacci Series".  Thank you to Mr. Turbeville for emailing me his link.

The LIX Unit

Joseph Edward Batter proposes the "LIX" unit, Phi^16 and 17 divided by 1000, noting its conversion to other known measurements such as the Planetary mean distances, Light Speed and the Great Pyramid at Giza.  The results suggest the existence of a Planetary System of Measure.  LIX he calls the "Tri-Ten" system.  L is for Light Speed, I is the Planetary mean distance Set and X is for Time.

The Great Pyramid

The American Institute of Pyramidology explores the the origin and purpose of the Great Pyramid.

Recommended Reading...

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Music

Did Mozart Use the Golden Section?

March - April 1996 issue of Science Observer contained this article by Mike May (now archived).

The Golden Section In Art, Architecture and Music

Ron Knott's dicussion of the Golden Ratio in art, architecture, and music.

Casey Mongoven's Music

Casey Mongoven emailed me a link to this web site - his official hompage. Mongoven works with Fibonacci numbers, related sequences and the Golden Ratio.  His site contains scores, writings, WAVs and MP3s available for download.

VoiceSync Tool

This page provides a description and a link to download a program called the VoiceSync tool, which analyses and generates balancing tones to improve and harmonize the quality of your voice.  They use "PHI (golden mean)" analysis to measure the aesthetic sounds of voice.

George Cardas - Audio

George Cardas uses the Golden Ratio in his high-end audio equipment cables.

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Nature

Jill Britton's Fibonacci Numbers in Nature

Excellent portrayal of the appearance of Fibonacci Numbers in nature.  Contains great step by step examples depicting flower petals, stems, seeds and pineapple scales.

The IQ-Trap: Intelligence, Social Structure, and Politics

Thank you to Dr. Volkmar Weiss for bringing his web page to my attention.  It covers books and articles relating to psychology, intelligence, sociology, and genetics.  Of specific interest to this link page: "short term memory capacity, attention, EEG, quantum mechanics of intelligence - The golden mean as clock cycle of brain waves" .

Web site text also available in Dutch.

The Golden Section: a Morphological Law of Living Matter

Thank you to Dr. Boris Rozin for drawing my attention to his web page article on Fibonacci numbers, the Golden Section and self-organization.  From his epilogue: "The Golden section is one of many manifestations of the asymmetry of animate nature.  Together with Fibonacci numbers, it is only one of the steps towards a pyramid of morphology.  On the top of this pyramid, there is a mysterious number.  That, however, is the topic of my future book."
Sounds very intriguing!

Web site text also available in Russian.

Marquardt Beauty Analysis, Inc.

Dr. Stephen Marquardt of Marquardt Beauty Analysis, Inc., brought this web site to my attention.  It hosts a number of very interesting Golden Ratio animations.  His site is dedicated to the study of Beauty in the human facial form and how the Golden Ratio relates to aesthetics through what Dr. Marquardt refers to as the "decagon complex" and "golden decagon".

Phyllotaxis

An interactive site for the mathematical study of plant pattern formation.

Recommended Reading...

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Space and Time

Greek Symbols: Ancient Artifacts

Kent Steadman's phenomenal website with sounds like God-Thumper and various subjects, including phi crop spirals, UFO sound analysis, and the innate power of fluorite.

CyberSpace ORBIT

The Orbit Cover page.  Here you can see the mythstream of events, discoveries and news under current investigation.  Includes SolWatch, Nemo Theory, magnetic storms, EM/RF/MW rings, chemtrails, Atlantis, pyramids, scalar rings, earthquakes, and much more!

Tony Smith's WWW Home Page

Tony's very informative web page contains various diversions in Andrew Gray's Many-World's Quantum Theory, D4-D5-E6 Physics Model, and of course the Golden Ratio.

Solar System Geometry

A case for Divine Design in the Solar System.  Alan Bennett tracks the evidence for Divine creation in our Solar System.

Spira Solaris

Look for the evidence of exponential order in the planetary framework, modeled best by a Phi-based complex equiangular spiral.

The Evolution of Truth

Easy to read discussion of the occurance of the Golden Ratio in our universe, leading to a proposed order resulting in the Divine conclusion.

Generalized Golden Section and the Time Theory

Albert R. Timashev's report at the June 19, 1996 International Scientific Conference in St. Petersburg, Russia, regarding stable quantities and velocities of time.

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Digits, ah yes, wonderful digits

The LARGEST computation of the Square Root of 5: 10,000,000 places!

Here it is!  My own efforts paying off as I completed calculating the largest number of digits of the square root of 5 as of December 20, 1999.  I wrote a program in C without the use of any special 3rd party math programs like Maple or Mathematica.

The LARGEST computation of Phi: 1.5 Billion places!

Here is the largest known calculation of Phi.  Xavier Gourdon and Pascal Sebah performed the calculation in 3 hours in May, 2000, on a 700MHz Pentium.  If anyone knows of any longer, please email me.  The above Web page is Gourdon and Sebah's.  It includes the current records for various popular constants, including pi, Phi, e, and the square root of 2.

Golden Ratio to 1,000,000 decimal digits

The author's page contains a "mere" one million digits of Phi, due to space limitations.  If anyone would like more, please email me and I'll be happy to provide up to 15,000,000 that I have computed.  I have since run out of space on my server, so please email me to receive the digits.

ICON: phi, the golden ratio, to 50,000 places

This page is within a site dedicated to the University of Arizona's, et. al., programming language, ICON.  It also contains 50,000 primes, as well as pi and e to 50,000 digits.  I have verified the correctness of this phi to 50,000 places.

RJN's More Digits of Irrational Numbers Page

Robert Nemiroff's excellent site showcasing millions and millions of digits of many popular irrational constants, including sqrt(5) and a whopping 10 million digits of sqrt(2).  To date I have not been able to verify the correctness of sqrt(5).

Phi to 2000 decimal places

Ron Knott's contribution to the digit quest. Be sure to check out his entire, incredible, site.

20,000 Digits

The Project Gutenberg Etext of The Golden Mean [Golden Ratio].

10,000 Digits

Ed Stephen's claims the first web page with phi. Unfortunately it only goes to 10,000 places, although he got this number from the Digit Warehouse which carries sqrt(5) out to 1,000,000.

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Computing Algorithms

Xavier Gourdon and Pascal Sebah Algorithm pages

Various algorithm pages from the masters!  Note the programs linked in the section beneath as well.

A Library Routine for Computing Square Roots of Double-Precision Vectors on CRAY MPP

Assembly code to compute square roots.  The link appears to have gone dark, but I leave it here for completeness.

ICS 161: Design and Analysis of Algorithms - Introduction to analysis of algorithms

Lecture notes from January 9, 1996, using the Fibonacci sequence as a problem for teaching algorithmic solutions.

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References/Resources

CRC Reference

The excellent reference manual published by CRC Press contains a wealth of information on all things mathematical.  It is back online now - after a year's struggle.  Thanks go to Eric Weisstein of Wolfram Research for his trials and tribulations in dealing with the publishing issues.
Areas of interest include:

Fibonacci Resource - Systematic Fibonacci Identity Matrix

Bob Johnson's excellent Fibonacci Resource site detailing Systematic discovery of Fibonacci identities by matrix methods.  And a warm "Thank You!" to Dr. Johnson for including a link back to my link page.

ScienceDirect - Digital library of the future

A resource for research journals, abstract databases and reference works.

The On-Line Encyclopedia of Integer Sequences

On-line Encyclopedia of Integer Sequences.  Enter an integer sequence, separated by commas, and it will search for the sequence in its extensive database.

Britannica on Golden Ratio

Encyclopedia Britannica search tool.

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Miscellany

School of Pythagoras

This site is dedicated to the mysteries of number and proportion.  The work of Pythagoras provides a starting point for ongoing explorations of these topics.  Contemporary developments in sacred geometry, cosmology, music, sound and acoustics along with new interpretations of ancient architecture are some of the other areas of focus.

Pi: The Movie

The excellent and fascinating film of the movie pi, released in 1998, contained a theme of psychosis mixed with the digits of an irrational but valued constant.  The author of the page that you are currently reading had begun work on such a project built around a fascination with his favorite number, phi, a few years before pi.  His film remains unfinished.

Museum of Harmony and Golden Section

Now dark, but kept for completeness.  Professor Alexey Stakhov and his daughter Anna Sluchenkova created the idea and now this web site contains extensive pages covering all aspects of the Golden Section in culture, nature, art, math, Fibonacci, classrooms, and clubs.  Text available in English and Russian.

Sacred Geometry

Domain for sale - bidding starts at $800.  I am not affiliated with the sellers in any way.

Pi

Shane's Pi web site, proposing the Silver Ratio, Qui, which I'd not seen before.  Shane's also a member of my Yahoo Golden Ratio Group.

The Silver Ratio

A site describing the Silver Ratio.  A different interpretation from Shane's above.  Thanks go to Edel for bringing this one to my attention.

Rescaled range analysis and the fractal dimension of pi

Some interesting analysis on pi-3 and a mention of an algorithm to calculate pi.

The Uselessness of Pi and its irrational friends

Really fun site with links poking fun at all this number cranking.  Link appears to be broken...

Animation of Binet's formula for Reals

Interesting Applet's showing Binet's formula for Fibonacci numbers extended for Reals.

Stock Market

Great web page describing Fibonacci series and Golden Ratio as they relate to the stock market.

Ermanometry Stock Market Research

Web site devoted to the investigation of dynamic growth patterns in freely traded markets, including both price and time.  They claim that Ermanometry principles are equally accurate and effective in all cash and futures markets.

Elliot Wave

Bob Prechter's Elliot Wave stock market and economy analysis.

Elliot Wave Chart Page

Elliot Wave charts and technical analysis on stock market forecasting.

Fibonacci Solution

Software designed to help people become successful Fibonacci stock traders.  Thank you to Yuriy Zubarev for emailing the link to me.  mGlider seems to have suplanted the Fib solution, but its link is dark.

Golden Ratio Applet

Sebastian Egner's Java applet.  Move the leg, and the three end points maintain the Golden Ratio.

The Numbers of Life

Nik Walter tells interesting text story regarding his personal discovery of the beauty hidden within Fibonacci Numbers and the Golden Ratio.

Recommended Reading...

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Changes last made on: Sunday, May 25, 2008, 21:17:31, PST