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 in architecture, art and music.

Contents of this page

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The Golden section in architecture

The Parthenon and Greek Architecture

Even from the time of the Greeks, a rectangle whose sides are in the "golden proportion" (1 : 1.618 which is the same as 0.618 : 1) has been known since it occurs naturally in some of the proportions of the Five Platonic Solids (as we have already seen). This rectangle is supposed to appear in many of the proportions of that famous ancient Greek temple, the Parthenon, in the Acropolis in Athens, Greece. (There is a replica of the original building (accurate to one-eighth of an inch!) at Nashville which calls itself "The Athens of South USA".)
The Acropolis, in the centre of Athens, is an outcrop of rock that dominates this ancient city. Its most famous monument, now largely ruined, is the Parthenon, a temple to the goddess "Athena" built around 430 or 440 BC.
Though no original plans of the temple exist, it appears that the temple was built on a square-root-of-5 rectangle, that is, it is sqrt5 times as long as it is wide. These are also the dimensions of the longest side view of the temple. Also, the front elevation is built on a Golden Rectangle, that is, it is Phi times as wide as it is tall.

Links

WWW: There is a wonderful collection of pictures of the Parthenon and the Acropolis at Indiana University's web site

Modern Architecture

The architect LeCorbusier deliberately incorporated some golden rectangles as the shapes of windows or other aspects of buildings he designed. One of these (not designed by LeCorbusier) is the United Nations building in New York which is L-shaped. Although you will read in some books that "the upright part of the L has sides in the golden ratio, and there are distinctive marks on this taller part which divide the height by the golden ratio", when I looked at photos of the building, I could not find these measurements. The United Nations Headquarters On-line Tour has an aerial view of the building (with thanks to Ralph Bechtolt for alerting me to this link). [With thanks to Bjorn Smestad of Finnmark College, Norway for mentioning these links.]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! The artist says "it is a metaphor of the human quest for order and harmony among chaos."

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

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:
     A     M        B
     | 1-x |    x   |
The line AB is divided at point M so that the ratio of the two parts, the smaller to the larger (AM and MB), is the same as the ratio of the larger part (MB) to the whole AB.
If AB is of length 1 unit, and we let MB have length x, then the definition (in bold) above becomes
the ratio of 1-x to x is the same as the ratio of x to 1 or, in symbols:
     1 - x   =  x  which simplifies to 1-x = x2
       x        1
This gives two values for x, (-1-sqrt5)/2 and (sqrt5-1)/2.
The first is negative, so does not apply here. The second is just phi (which has the same value as 1/Phi and as Phi-1).
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. Here are two for you to analyse for yourself. [The pictures are links to the Uffizi Gallery site and the pictures are copyrighted by the Gallery.]
( image: 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.
This image: Madonna with Child and Saints
is in a square frame. Print it out and mark on it 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?


Links to Art sources

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Fibonacci and Poetry

Martin Gardner, in the chapter "Fibonacci and Lucas Numbers" in "Mathematical Circus" (Penguin books, 1979) mentions Prof 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 is called 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 - more details in Garland's book.

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 Kay. He reports on the analysis of many of Mozart's sonatas and finds they divide into two parts exactly at the golden section point in almost all cases. Was this a conscious choice (his sister said he was always playing with numbers and was fascinated by mathematics) or did he do this intuitively?

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

Beethoven's Fifth

Article: In an interesting little article in Mathematics Teaching volume 84 in 1978, Derek Haylock writes about The Golden Section in Beethoven's Fifth on pages 56-57.
He finds that the famous opening "motto" appears not only in the first and last bars (bar 601 before the Coda) but also exactly at the golden mean point 0·618 of the way through the symphony (bar 372) and also at the start of the recapitulation which is phi or 0·382 of the way through the piece! He poses the question:
Was this by design or accident?

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:
Article: Duality and Synthesis in the Music of Bela Bartók E Lendvai
pages 174-193 of Module, Proportion, Symmetry, Rhythm G Kepes (editor), George Brazille, 1966;
Article: Some striking Proportions in the Music of Bela Bartók
in Fibonacci Quarterly Vol 9, part 5, 1971, pages 527-528 and 536-537.
Book: 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.
Book: Debussy in Proportion - a musical analysis by Roy Howat,
Cambridge Univ. Press,1983, ISBN = 0 521 23282 1. After its first publication in 1986, this book is now (February 2000) back in print.
WWW: See also Roy Howat's Web site for more information.
Article: 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
Book: Schubert Studies, (editor Brian Newbould) London: Ashgate Press, 1998
has a chapter by Roy Howat Architecture as drama in late Schubert, pages 168 - 192, about Schubert's golden sections in his late A major sonata (D.959).
Article: 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.
Article: 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 organisation.


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

There are many books and articles that say that the golden rectangle is the most pleasing shape and point out how it was used in the shapes of famous buildings, in the structure of some music and in the design of some famous works of art. Indeed, people such as Corbusier and Bartók have deliberately and consciously used the golden section in their designs.
However, the "most pleasing shape" idea is open to criticism. The golden section as 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 does not seem to have any written texts (ancient Greek, Egyptian or Babylonian) as supporting hard evidence.
At best, the golden section used in design is just one 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 ancient Egyptians really use it as the main "number" for the shapes of the Pyramids? We do not know. Usually the shapes of such buildings are not truly square and perhaps, as with the pyramids and the Parthenon, parts of the buildings have 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 lines on 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!

Important article that point out the weaknesses in parts of "the golden-section is the most pleasing shape" theory:
Article: George Markowsky's Misconceptions about the Golden ratio in The College Mathematics Journal Vol 23, January 1992, pages 2-19.
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 this are explained on earlier pages at this site.
Article: How to Find the "Golden Number" without really trying Roger Fischler, Fibonacci Quarterly, 1981, Vol 19, pp 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 foundaton whatever".
Article: 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.
Since almost all of the material at this site is about Mathematics, then 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

Key:
book: a book
article: an article in a magazine or
a paper in an academic journal
WWW: a website

Music

Book: 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.
Article: 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
WWW:Gamelan
is the percussion oriented music of Indonesia.
WWW: 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.
WWW: Some CDs
on Gamelan music of Central Java (the country not the software!).
Other music
WWW: The Fibonacci Sequence
is the name of a classical music ensemble of internationally famous soloists, who are the musicians in residence at Kingston University (Kingston-upon-Thames, Surrey, UK). Based in the London (UK) area, their current programme of events is on the Web site link above.
Art
Book: A Mathematical History of the Golden Section ISBN 0486400077.
Book: Education through Art (3rd edition) H Read,
Pantheon books,1956, pages 14-22;
Book: The New Landscape in Art and Science G Kepes
P Theobald and Co, 1956, pages 329 and 294;
Book: H E Huntley's, The Divine Proportion: A study in mathematical beauty,
ISBN 0-486-22254-3 is a 1970 Dover reprint of an old classic.
Book: C. F. Linn, The Golden Mean: Mathematics and the Fine Arts,
Doubleday 1974.
Book: Gyorgy Doczi, The Power of Limits: Proportional Harmonies in Nature, Art, and Architecture
Shambala Press, (new edition 1994).
Book: M. Boles, The Golden Relationship: Art, Math, Nature, 2nd ed.,
Pythagorean Press 1987. 
The "Golden Cut" or beauty and design using the golden section, through the eyes of a florist.

 
 
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© 1996-2001 Dr Ron Knott      R.Knott@surrey.ac.uk      updated: 23 April 2001