The FibPhi Link Page
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 onestop 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
9,311^{st} is the largest known prime Fibonacci Number, while the
81,839^{th} 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 wellrounded, 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 onestop 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.
Phil Fraley's Golden Ratio art/geometry page
Physical and computer art from Phil Fraley. Thank you to the artist for emailing me this link.
Mondimat
A fun web site to create Mondrianesque 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 EyePlacement 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 equalsided 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 "TriTen" 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 highend 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 IQTrap: 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 selforganization. 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 GodThumper 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 ManyWorld's
Quantum Theory, D4D5E6 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 Phibased 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 DoublePrecision 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 OnLine Encyclopedia of Integer Sequences
Online 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 pi3 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, June 8, 2008, 14:23:31, PST
