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Golden Rectangle In Art And Nature Essay

Fibonacci Numbers and The Golden Section in Art, Architecture and Music

This section introduces you to some of the occurrences of the Fibonacci series and the Golden Ratio inarchitecture, art and music.

Contents of this page

The icon means there is a Things to do investigation at the end of the section.

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The Golden section in architecture

The Parthenon and Greek Architecture

The ancient Greeks knew of a rectangle whose sides are in the goldenproportion (1 : 1.618 which is the same as 0.618 : 1). It occurs naturallyin some of the proportions of the Five Platonic Solids (as we have already seen). A construction for thegolden section point is found in Euclid's Elements.The golden rectangle is supposed to appear in many of the proportions of that famousancient Greek temple, the Parthenon, in the Acropolis in Athens, Greece but there is no original documentary evidence that this was deliberately designed in.(There is a replica of the original building (accurate to one-eighth of an inch!) atNashville which calls itself "The Athens of South USA".)

The Acropolis(see a plan diagram or Roy George's planof the Parthenon with active spots to click on to view photographs), in the centre of Athens, is an outcrop of rock that dominates the ancient city. Its most famous monument is the Parthenon, a temple to the goddessAthena built around430 or 440 BC. It is largely in ruins but is now undergoing some restoration (see the photosat Roy George's site in the link above).
Again there are no original plans of the Parthenon itself. It appears to be built on a design of golden rectangles and root-5 rectangles:
  • the front view (see diagram above): a golden rectangle, Phi times as wide as it is high
  • the plan view: 5 as long as the front is wide so the floor area is a square-root-of-5 rectangle
However, due to the top part being missing and the base being curvedto counteract an optical illusion of level lines appearing bowed, these are only an approximatemeasures but reasonably good ones.
The Panthenon image here shows clear golden sections in the placing of the three horizontal lines but the overall shape and the other prominent features are not golden section ratios.
Pantheon, Libero Patrignani


Modern Architecture

The Eden Project's new Education Building

The Eden Project in St. Austell,between Plymouth and Penzance in SW England and 50 miles from Land's End,has some wonderfully impressive greenhouses based on geodesic domes (called biomes) built in anold quarry. It marks the Millenium in the year 2000 and is now one of the most popular tourist attractions in the SW of England.
A new £15 million Education Centre called The Corehas been designed using Fibonacci Numbers and plant spiralsto reflect the nature of the site - plants fromall over the world. The logo shows the pattern of panels on the roof.
What is 300 million years old, weights 70 tonnes and is the largest of its type in the world? It is the new sculpture called The Seed at the centre of The Core which was unveiled on Midsummer's Day 2007 (June 23). Peter Randall-Paige's design is based on the spirals found in seeds and sunflowers and pinecones.

California Polytechnic Engineering Plaza

The College of Engineering at the CaliforniaPolytechnic State University have plans for anew Engineering Plazabased on the Fibonacci numbers and several geometric diagrams you will alsohave seen on other pages here. There is also a page ofimagesof the new building.
The designer of the Plaza and former student of Cal Poly, Jeffrey Gordon Smith, says
As a guiding element, we selected the Fibonacci series spiral, or golden mean, as the representation of engineering knowledge.
The start of construction is currently planned forlate 2005 or early in 2006.

The United Nations Building in New York

The architect Le Corbusier deliberately incorporated some golden rectangles as the shapes of windows or other aspects of buildings he designed. One of these (not designed by Le Corbusier) is the United Nations building in New York whichis L-shaped. Although you will read in some books that "the upright part of the L has sides in the golden ratio, andthere are distinctive marks on this taller part which divide the heightby the golden ratio", when I looked at photos of the building, I could not find these measurements. Can you?

[With thanks to Bjorn Smestad of Finnmark College, Norway for mentioning these links.]

More Architecture links

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The Golden Section and Art

Luca Pacioli (1445-1517) in his Divina proportione (On Divine Proportion) wrote about the golden section also called the golden mean or the divine proportion:
The line AB is divided at point M so that the ratio of thetwo parts, the smaller MB to the larger AM is thesame as the ratio of the larger part AM to the whole AB.
We have seen on earlier pages at this site that this gives two ratios, AM:AB which is also BM:AM and is 0.618... which we call phi (beginning with a small p). The other ratio is AB:AM = AM:MB = 1/phi= 1.618... or Phi (note the capital P). Both of these are variously called the golden number or golden ratio, golden section, golden mean or the divine proportion. Other pages at this site explain a lot more about it and its amazing mathematical properties and it relation to the Fibonacci Numbers.
Pacioli's work influenced Leonardo da Vinci (1452-1519) and Albrecht Durer (1471-1528) and is seen in some of the work of Georges Seurat, Paul Signac and Mondrian, for instance.

Many books on oil painting and water colour in your local library will point out that it is better to position objects not in the centre of the picture but to one side or "about one-third" of the way across, and to use lines which divide the picture into thirds. This seems to make the picture design more pleasing to the eye and relies again on the idea of the golden section being "ideal".

Leonardo's Art

The Uffizi Gallery's Web site in Florence, Italy, has a virtual room of some of Leonardo da Vinci's paintings and drawings. I suggest the following two of Leonardo Da Vinci's paintings to analyse for yourself:
The Annunciation
is a picture that looks like it is in a frame of 1:sqrt(5) shape (a root-5 rectangle). Print it and measure it - is it a root-5 rectangle? Divide it into a square on the left and another on the right. (If it is a root-5 rectangle, these lines mark out two golden-section rectangles as the parts remaining after a square has been removed). Also mark in the lines across the picture which are 0·618 of the way up and 0·618 of the way down it. Also mark in the vertical lines which are 0·618 of the way along from both ends. You will see that these lines mark out significant parts of the picture or go through important objects. You can then try marking lines that divide these parts into their golden sections too.
Leonardo's Madonna with Child and Saints
is in a square frame. Look at the golden section lines (0·618 of the way down and up the frame and 0·618 of the way across from the left and from the right) and see if these lines mark out significant parts of the picture. Do other sub-divisions look like further golden sections?

Modern Art

Graham Sutherland's Tapestry in Coventry Cathedral

Graham Sutherland's (1903-1980) huge tapestry of Christ The King behind the altar in Coventry Cathedral here in a picture takenby Rob Orland.

It seems to have been designed with some clear golden sections as I've shown on Rob's picture:

  • The figure of Christ is framed by an oval with a flattened top. At the golden section point vertically is the navel indicated at the narrowest part of the waist and also the lower edge of the girdle (belt or waist-band), shown by blue arrows.
  • The bottom the the girdle (waist-band) is also at a golden section point for the whole figure from the top of the head to the soles of the feet, shown by purple arrows.
    Since this is also the position of the navel in the human body, this indicates the figure is standing.
  • The top of the girdle and the line of the chest are at golden sections between the base of the girdle and the top of the garment (the shoulders) shown by red arrows.
  • The face also has several golden sections in it, the line of the eyes and the nostrils being at the major golden sections, shown by yellow lines.
  • The two ovals forming the apron and the face are positioned vertically at golden section points apart and at golden sections in size as shown by the green arrows.
  • The other two ovals, the sleeves, have a width that is 0.618 of the distance between the sleeves, shown by grey arrows.
Can you find any more golden sections?

More information on the tapestry.
Take a virtual tour of the Cathedral.
Purchase this print from Rob Orland's Photos website

Links to Art sources

Links specifically related to the Fibonacci numbers or the golden section (Phi):

Links to major sources of Art on the Web:

The work of modern artists using the Golden Section

When I was giving a talk at The Eden Project in Cornwall in July 2007, Patricia Bennetts and Mary Miller of Falmouth introduced me to using Fibonacci Numbers in Quilt design. (Let your mouse rest on their names to see their email addresses.)
Their two designs are based on the pattern in the middle where the strips in the lower half are of widths 1, 2, 3, 5, 8 and 13 in brown which are alternated with lighter strips of the same widths but in decreasing order.
Woolly Thoughts is Steve Plummer and Pat Ashforth's web site with many maths inspired knitting and crochet projects, including designs based on Fibonacci numbers, the golden spiral, pythagorean triangles, flexagons and much much more. They have worked for many years in schools giving a new twist to mathematics with their hands-on approach to design using school maths. An excellent resource for teachers who want to get students involved in maths in a new way and also for mathematicians interested in knitting and crochet.
Billie Ruth Sudduth is a North American artist specialising in basket work that is now internationally known. Her designs are based on the Fibonacci Numbers and the golden section - see her web page JABOBs (Just A Bunch Of Baskets). Mathematics Teaching in the Middle School has a good online introduction to her work (January 1999).
Kees van Prooijen of California has used a similar series to the Fibonacci series - one made from adding the previous three terms, as a basis for his art.

Fibonacci and Phi for fashioning Furniture

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Fibonacci in Films

The Russian Sergie Eisenstein directed the classic silent film of 1925 The Battleship Potemkin(a DVDor videoversion of this 75 minute film is now available, both in PAL format).He divided the film up using golden section points to start important scenes in the film, measuringthese by length on the celluloid film.
Jonathan Berger of Stanford University's Center for Computer Research in Music and Acoustics used this as an illustration of Fibonacci numbers in a lecture course.
Dénes Nagy, in a fascinating article entitled Golden Section(ism): From mathematics to the theory of art and musicology, Part 1 in Symmetry, Culture and Science, volume 7, number 4, 1996, pages 337-448 talks about whether we can percieve a golden section point in time without being initially aware of the whole time interval. He gives a reference to his own work on golden section perception in video art too (page 418 of the above article).

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Fibonacci Numbers and Poetry

The first section here is inspired by Dr Rachel Hall's Multicultural mathematics course syllabus at St Joseph's University in Philadelphia, USA. (Read more about it with some nice maths puzzles in this pdf document.)

Stress, Meter and Sanskrit Poetry

In English, we tend to think of poetry as lines of text that rhyme,that is, lines that end with similar sounds as in this children's song:
Twinkle twinkle little star
How I wonder what you are.
Up above the world, so high
Like a diamond in the sky
Also we have the rhythm of the separate sounds (called syllables). Words liketwinkle have two syllables: twin- and -kle whereas words such as star have just one. Some syllables are emphasized or stressedmore than others so that they soundlouder (such as TWIN- in twinkle), whereas others are unstressed and quieter(such as -kle in twinkle). Dictionaries will often show how to pronounce a word by separating it intosyllables, the stressed parts shown in capital as we have done here, e.g.DIC-tion-ar-y to show it has 4 syllables with the first one only being stressed.
If we let S stand for a stressed syllable and s anunstressed one, then the stress-pattern of each line of the song or poemis its meter (rhythm). In the song above each line has the meterSsSsSsS.

In Sanskrit poetry syllables are are either long or short.
In English we notice this in some words but not generally - all the syllables in the songabove take about the same length of time to say whether they are stressed or not, so all thelines take the same amount of time to say.
Howevercloudy sky has two words and three syllables CLOW-dee SKY, but the first and third syllables are stressed and take a longer to say then the other syllable.
Let's assume that long syllables take just twice as long to say as short ones.
So we can ask the question:

in Sanskrit poetry, if all lines take the same amount of time to say, what combinations ofshort (S) and long (L) syllables can we have?

This is just another puzzle of the same kind as on the Simple Fibonacci Puzzlespage at this site.

For one time unit, we have only one short syllable to say: S = 1 way
For two time units, we can have two short or one long syllable: SS and L = 2 ways
For three units, we can have: SSS, SL or LS = 3 ways
Any guesses for lines of 4 time units? Four would seem reasonable - but wrong! It's five!
the general answer is that lines that take n time units to say can be formed in Fib(n)ways.

This was noticed by Acarya Hemacandra about 1150 AD or 70 yearsbefore Fibonacci published his first edition of Liber Abaci in 1202!

Acarya Hemacandra and the (so-called) Fibonacci NumbersInt. J. of Mathematical Education vol 20 (1986) pages 28-30.

Virgil's Aeneid

Martin Gardner, in the chapter "Fibonacci and Lucas Numbers" in Mathematical Circus(Penguin books, 1979 or Mathematical Assoc. of America 1996) mentionsProf George Eckel Duckworth's book Structural patterns and proportions in Virgil's Aeneid : a study in mathematical composition (University of Michigan Press, 1962). Duckworth argues that Virgil consciously used Fibonacci numbers to structure his poetry and so did other Roman poets of the time.

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Fibonacci and Music

Trudi H Garland's [see below] points out that on the 5-tone scale (the black notes on the piano), the 8-tone scale (the white notes on the piano) and the 13-notes scale (a complete octave in semitones, with the two notes an octave apart included). However, this is bending the truth a little, since to get both 8 and 13, we have to count the same note twice (C...C in both cases). Yes, it iscalled an octave, because we usually sing or play the 8th note which completes the cycle by repeating the starting note "an octave higher" and perhaps sounds more pleasing to the ear.But there are really only 12 different notes in our octave, not 13!

Various composers have used the Fibonacci numbers when composing music, and some authors find the golden section as far back as the Middle Ages (10th century) ( see, for instance, The Golden Section In The Earliest Notated Western Music P Larson Fibonacci Quarterly 16 (1978) pages 513-515 ).

Golden sections in Violin construction

The section on "The Violin" in The New Oxford Companion to Music, Volume 2, shows how Stradivari was aware of the golden section and used it to place the f-holes in his famous violins.

Baginsky's method of constructing violins is also based on golden sections.

Did Mozart use the Golden mean?

This is the title of an article in the American Scientist of March/April 1996 by Mike May. He reports on John Putz's analysis of many of Mozart's sonatas. John Putz found that there was considerable deviation from goldensection division and that any proximity to golden sections can beexplained by constraints of the sonata form itself, rather than purposefuladherence to golden section division.

The Mathematics Magazine Vol 68 No. 4, pages 275-282, October 1995 hasan article by Putz on Mozart and the Golden section in his music.

Phi in Beethoven's Fifth Symphony?

In Mathematics Teaching volume 84 in 1978, Derek Haylockwrites about The Golden Section in Beethoven's Fifth on pages 56-57.

He claims that the famous opening "motto" (click on the music to hear it)occurs exactly at the golden mean point 0·618 in bar 372 of 601and again at bar 228 which is the other golden section point (0.618034 from the end of the piece)but he has to use 601 bars to get these figures. This he does by ignoring the final 20 bars that occur after the final appearance of the motto and also ignoring bar387.
Have a look at the full score for yourself at The Hector Berlioz websiteon the Berlioz: Predecessors and Contemporariespage, if you follow the Scores Available link. A browser plug-in enables you to hear it also. Note that the repeated 124 bars at the beginning are not included in the bar counts on the musical score.

Tim Benjamin for points out thatBut there are 626 bars and not 601!
Therefore the golden section points actually occur at bars 239 (shown as bar 115 as the counts do not include the repeat)and 387 (similarly marked as bar 263).

As UK composer Tim Benjamin points out:

The 626 bars are comprised of a repeated section of 124 bars - so that's the first 248 bars in the repeated section, the "exposition" - followed by 354 of "development" section, then a 24 bar "recapitulation" (standard "first movement form"). Therefore there can't really be anything significant at 239, because that moment happens twice. However at 387, there is something pretty odd - this inversion of the main motto. You have some big orchestral activity, then silence, then this quiet inversion of the motto, then silence, then big activity again.

Also you have to bear in mind that bar numbers start at 1, and not 0, so you would need to look for something happening at 387.9 (rounding to 1dp) and not 386.9. This is in fact what happens - the strange inversion runs from 387.25 to 388.5.

But bar 387 is precisely one that Haylock singles out to ignore!
So is it Beethoven's "phi-fth" or just plain old "Fifth"?

Bartók, Debussy, Schubert, Bach and Satie

There are some fascinating articles and books which explain how these composers may have deliberately used the golden section in their music:
Duality and Synthesis in the Music of Bela Bartók
by E Lendvai on pages 174-193 of Module, Proportion, Symmetry, Rhythm G Kepes (editor), George Brazille, 1966;
Some striking Proportions in the Music of Bela Bartók
in Fibonacci Quarterly Vol 9, part 5, 1971, pages 527-528 and 536-537.
Bela Bartók: an analysis of his music
by Erno Lendvai, published by Kahn & Averill, 1971; has a more detailed look at Bartók's use of the golden mean.
Debussy in Proportion - a musical analysis
by Roy Howat, Cambridge Univ. Press,1983, ISBN = 0 521 23282 1.
Concert pianist Roy Howat's Web site has more information on his Debussy in Proportion book and others works and links.
Adams, Coutney S. Erik Satie and Golden Section Analysis.
in Music and Letters, Oxford University Press,ISSN 0227-4224, Volume 77, Number 2 (May 1996), pages 242-252
Schubert Studies, (editor Brian Newbould) London: Ashgate Press, 1998
has a chapter Architecture as drama in late Schubert by Roy Howat, pages 168 - 192, about Schubert's golden sections in his late A major sonata (D.959).
The Proportional Design of J.S. Bach's Two Italian Cantatas,
Tushaar Power, Musical Praxis, Vol.1, No.2. Autumn 1994, pp.35-46.
This is part of the author's Ph D Thesis J.S. Bach and the Divine Proportion presented at Duke University's Music Department in March 2000.
Proportions in Music by Hugo Norden in Fibonacci Quarterly vol 2 (1964) pages 219-222
talks about the first fugue in J S Bach's The Art of Fugue and shows how both the Fibonacci and Lucas numbers appear in its organization.
Per Nørgård's 'Canon' by Hugo Norden in Fibonacci Quarterly vol 14 (1976), pages 126-128 says the title piece is an "example of music based entirely and to the minutest detail on the Fibonacci Numbers".
The Fibonacci Series in Twentieth Century Music J Kramer, Journal of Music Theory 17 (1973), pages 110-148

There is a very useful set of mathematical links to Art and Music web resources from Mathematics Archives that is worth looking at.

The Golden String as Music

The Golden String is a fractal string of 0s and 1s that grows ina Fibonacci-like way as follows:
After the first two lines, all the others are made from the two latest lines in a similar wayto each Fibonacci numbers being a sum of the two before it.Each string (list of 0s and 1s) here is a copy of the one above it followed by the one above that.The resulting infintely long string is the Golden String or Fibonacci Word or Rabbit Sequence.It is interesting to hear it in musical form and I give two ways in the sectionHear the Golden sequence on that page. In that same section I mention the London based group Perfect Fifth whohave used it in a piece called Fibonacci that you can hear online too .

Other Fibonacci and Phi related music

John Biles, a computer scientist at Rochester university in New York State used the series which isthe number of sets of Fibonacci numbers whose sum is n to make a piece of music.He wrote about it and has a link to hear the piece online. The series looks like this:
It has some fractal properties in that the graph can be seen in sections, eachbeginning and ending when the graph dips down to lowest points on the y=1 line. Each section begins and ends with a copy of the section two before it (and moved up a bit), and in between them isa copy of the previous section again moved up.
I've written more about this series in a section calledSumthing about Fibonacci Numbers on theFibonacci Bases and other ways of representing integers.

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Miscellaneous, Amusing and Odd places to find Phi and the Fibonacci Numbers

TV Stations in Halifax, Canada

In Halifax, Nova Scotia, there are 4 non-cable TV channels and they are numbered 3, 5, 8 and 13! Prof. Karl Dilcher reported this coincidence at the Eighth International Conference on Fibonacci Numbers and their Applications in summer 1998.

Turku Power Station, Finland

Joerg Wiegels of Duesseldorf told me that he was astonished to see the Fibonacci numbers glowing brightly in the night sky on a visit to Turku in Finland. The chimney of the Turku power station has the Fibonacci numbers on it in 2 metre high neon lights! It was the first commission of the Turku City Environmental Art Project in 1994. The artist, Mario Merz (Italy) calls it Fibonacci Sequence 1-55 and says "it is a metaphor of the human quest for order and harmony among chaos."

The picture here was taken by Dr. Ching-Kuang Shene of Michigan Technological University and is reproduced here with his kind permission from his page of photos of his Finland trip.

Designed in?

  • If you measure a credit card, you'll find it is a perfect golden rectangle.
  • The golden rectangle icon of National Geographic also seems to be a golden-section rectangle too.
  • Brian Agron of Fairfax, California, found the golden section in the design of his mountain bike, a Trek Fuel 90shown above with golden sections marked.
  • Brian also says the shape of the large doors in hospitals seem to be a golden rectangle.
  • John Harrison MA has found a golden rectangle in the shape of a Kit-Kat chocolate wafer - the larger 4 finger bar in its older wrapping as shown above.

Two myths about clocks and golden ratio time

Sometimes you will read that clocks and watches set at ten to two have their hands positioned so as to form a golden rectangle and that this is "aesthetically pleasing".
But it is easy to calculate that the angle between the hands at this time is 0.3238 of a turn (or, the larger angle is 0.6762 of a turn) both of which are nowhere near the golden ratio angles of 0.618 and 0.382 (= 1–0.618) of a turn.

There are eleven distinct times in any 12 hour period when the hands of a clock mark out a golden ratio on the circumference.
What times are they?

Which is the most symmetrical arrangement?
Which is the easiest to remember?
Which is closest to a multiple of 5 minutes?

It is much easier to compute the times if :
  • we measure hours as a decimal so that 2:30 is 2.5 hours and 12:00 and 0:00 are 0.0 hours and
  • if we measure angles from 12 o'clock in fractions of a turn and not in radians or degrees so that, for example, the hour hand is at 0.25 of a turn at 3 o'clock
At the following times the hands form a golden angle of exactly 0.6180339... turns or 222.492...° (or 137.508° if you prefer):

Other authors say the hands at 1:50 or 10:08 form a golden rectangle using the points onthe rim.
This also is not true even if one could imagine them projected on to the rim and thenmaking a rectangle - not an easy visual exercise!
Here are the clocks with hands extended to the rim and a golden rectanglesuperimposed on the clocks. When the hour hand points at the right place, it is about 10:04 and whenthe minute hand gets to the correct position, it is about 10h 9m 35s but then the hour hand does notpoint to the right place.
The time when the hands are exactly symmetrical is10 hours 9 minutes and 13.8462... seconds and also1 hours, 50 minutes and 46.1538 seconds.So 10:09 and 1:51 are both reasonably close, but even with the visual gymnastics, it seems unlikely that the eye recognizes sucha golden rectangle construction at those times, in my mathematical opinion!

Things to do

  1. What other logos can you find that are golden rectangles?
  2. Where else have you found the golden rectangle?
Email me with any answers to these questions and I'll try to include them on this page.

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A Controversial Issue

There are many books and articles that say that the golden rectangle is the most pleasing shapeand point out how it was used in the shapes of famous buildings, in the structure of some music and inthe design of some famous works of art. Indeed, people such as Corbusier and Bartók have deliberatelyand consciously used the golden section in their designs.
However, the "most pleasing shape" idea is open to criticism. The golden sectionas a concept was studied by the Greek geometers several hundred years before Christ, as mentioned on earlier pages at this site,But the concept of it as a pleasing or beautiful shape only originated in the late 1800's and doesnot seem to have any written texts (ancient Greek, Egyptian or Babylonian) as supporting hardevidence.
At best, the golden section used in design is justone of several possible "theory of design" methods which help people structure what they are creating. At worst,some people have tried to elevate the golden section beyond what we can verify scientifically. Did the ancientEgyptians really use it as the main "number" for the shapes of the Pyramids? We do not know. Usually the shapesof such buildings are not truly square and perhaps, as with the pyramids and the Parthenon, parts of the buildingshave been eroded or fallen into ruin and so we do not know what the original lengths were. Indeed, if you look at where I have drawn the lineson the Parthenon picture above, you can see that they can hardly be called precise so any measurements quoted by authors are fairly rough!

So this page has lots of speculative material on it and would make a good Project for a Science Fair perhaps, investigating if the golden section does account for some major design features in important works of art, whether architecture, paintings, sculpture, music or poetry. It's over to you on this one!

George Markowsky's Misconceptions about the Golden ratio
in The College Mathematics Journal Vol 23, January 1992, pages 2-19 is an important article that points out the weaknesses in parts of "the golden-section is the most pleasing shape" theory.
This is readable and well presented. Perhaps too many people just take the (unsupportable?) remarks of others and incorporate them in their works? You may or may not agree with all that Markowsky says, but this is a good article which tries to debunk a simplistic and unscientific "cult" status being attached to Phi, seeing it where it really is not! This is not to deny that Phi certainly is genuinely present in much of botany and the mathematical reasons for thisare explained on earlier pages at this site.
How to Find the "Golden Number" without really trying
Roger Fischler, Fibonacci Quarterly, 1981, Vol 19, pages 406 - 410.
Another important paper that points out how taking measurements and averaging them will almost always produce an average near Phi. Case studies are data about the Great Pyramid of Cheops and the various theories propounded to explain its dimensions, the golden section in architecture, its use by Le Corbusier and Seurat and in the visual arts. He concludes that several of the works that purport to show Phi was used are, in fact, fallacious and "without any foundation whatever".
The Fibonacci Drawing Board Design of the Great Pyramid of Gizeh Col. R S Beard in Fibonacci Quarterly vol 6, 1968, pages 85 - 87;
has three separate theories (only one of which involves the golden section) which agree quite well with the dimensions as measured in 1880.
Golden Section(ism): From mathematics to the theory of art and musicology, Part 1, Dénes Nagy in Symmetry, Culture and Science, volume 7, number 4, 1996, pages 337-448
Section 2.1 says there are at least nine different theories about the shape of the Great Pyramid of Pharoah Khufu (the Great Pyramid of Cheops), two of which refer to the golden section:
The angle of the slope of the faces is
  • the angle whose cosine is 0·618... which is about 51·82°
  • the angle whose tangent is twice 0·618... which is about 51·027°
although a better fit is provided by a mathematical problem in the Rhind Papyrus which, in our notation is
  • the angle whose tangent is 28/22 which is about 51·84°
All of the material at this site is about Mathematics so this page is definitely the odd one out! All the other material is scientifically (mathematically) verifiable and this page (and the final part of the Links page) is the only speculative material on these Fibonacci and Phi pages.

References and Links on the golden section in Music and Art

a book
an article in a magazine or
a paper in an academic journal
a website


Fascinating Fibonaccis by Trudi Hammel Garland,
Dale Seymours publications, 1987 is an excellent introduction to the Fibonacci series with lots of useful ideas for the classroom. Includes a section on Music.
An example of Fibonacci Numbers used to Generate Rhythmic Values in Modern Music
in Fibonacci Quarterly Vol 9, part 4, 1971, pages 423-426;

Links to other Music Web sites

Gamelan music
is the percussion oriented music of Indonesia. The American Gamelan Institute has lots of information including a Gongcast recorded online music so you can hear Gamelan music for yourself.
New music
from David Canright of the Maths Dept at the Naval Postgraduate School in Monterey, USA; combining the Fibonacci series with Indonesian Gamelan musical forms.
Some CDs
on Gamelan music of Central Java (the Indonesian island not the software!).
Other music Art
A Mathematical History of the Golden Number by Roger Herz-Fischler, Dover 1998, ISBN 0486400077. A scholarly study of all major references in an attempt to trace the earliest references to the "golden section", its names, etc.
Education through Art (3rd edition) H Read,
Pantheon books,1956, pages 14-22;
The New Landscape in Art and Science G Kepes
P Theobald and Co, 1956, pages 329 and 294;
H E Huntley's, The Divine Proportion: A study in mathematical beauty,
is a 1970 Dover reprint of an old classic.
C. F. Linn, The Golden Mean: Mathematics and the Fine Arts,
Doubleday 1974.
Gyorgy Doczi, The Power of Limits: Proportional Harmonies in Nature, Art, and Architecture
Shambala Press, (new edition 1994).
M. Boles, The Golden Relationship: Art, Math, Nature, 2nd ed.,
Pythagorean Press 1987.
© 1996-2010 Dr Ron Knott
updated 22 September 2010

Use of the Golden Ratio in Our World Essay

595 Words3 Pages

Leonardo of Pisa, better known as Fibonacci, was born in Pisa, Italy, about 1175 AD. He was known as the greatest mathematician of the middle ages. Completed in 1202, Fibonacci wrote a book titled Liber abaci on how to do arithmetic in the decimal system. Although it was Fibonacci himself that discovered the sequence of numbers, it was French mathematician, Edouard Lucas who gave the actual name of "Fibonacci numbers" to the series of numbers that was first mentioned by Fibonacci in his book. Since this discovery, it has been shown that Fibonacci numbers can be seen in a variety of things today.

He began the sequence with 0,1,… and then calculated each successive number from the sum of the previous two. This sequence of numbers is…show more content…

Throughout history the length to width ratio for rectangles was one to 1.61803 39887 49894 84820. This ratio has always been considered most pleasing to the eye. This ratio was named the golden ratio by the Greeks. In the world of mathematics, the numeric value is called "phi", named for the Greek sculptor Phidias. The space between the columns form golden rectangles. There are golden rectangles throughout this structure which is found in Athens, Greece. He sculpted many things including the bands of sculpture that run above the columns of the Parthenon. Phidias widely used the golden ratio in his works of sculpture. The exterior dimensions of the Parthenon in Athens, built in about 440BC, form a perfect golden rectangle.

Many artists who lived after Phidias have used this proportion. Piet Mondrian and Leonardo da Vinci both thought that art should manifest itself in continuous movement and beauty. Therefore, they both expressed movement by incorporating the golden rectangle into their paintings. The golden ratio expresses movement because it keeps on spiraling to infinity. They showed beauty in their paintings by using the golden ratio because it is pleasing to the eye. To express the Fibonacci Sequence in art one must pay close attention to beauty,

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