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

AMB

Links:

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

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

• A ray traced image based on Fibonacci spirals and rectangles
• the Web Museum pages on Durer, Famous Painting Virtual Exhibition. their long list of famous artists and their works.
• There is a very useful set of mathematical links to Art and Music web resources from Mathematics Archives that is worth looking at.

Links to major sources of Art on the Web:

• Top9.com's List of the top art sources on the web is an excellent place for links to good art sources on the web. Highly recommended!
• The Metropolitan Museum of Art in New York houses more than 2 million works of art.
• The Fine Arts Museums of San Francisco site has an Image base of 65,000 works of art. It includes art from Ancient to Modern, from paintings to ceramics and textiles, from all over the world as well as America.
• A Guide to Art Collections in the UK
• Michelangelo is famous for his paintings (such as the ceiling in the Sistine Chapel) and his sculptures (for instance David). This site has links to several sources and images of his works and some links to sites on the golden section.
  Using the picture of his David sculpture, measure it and see if he has used Phi - eg is the navel ("belly button") 0·618 of the David's height?
• Why not visit the Leonardo Museum in the town of Vinci (Italy) itself from which town Leonardo is named, of course.
  There are many sketches and paintings of Leonardo's at The WebMuseum, Paris too.

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

• Pietro Malusardi and Karen Wallace have a web page showing some elegant applications of the golden section in furniture design.
• Custom Furniture Solutions have a Media cabinet designed using golden section proportions.
• A recent edition (Jan/Feb 2003) of the Ancient Egypt Magazine contained an article on Woodworking in Ancient Egypt where the author, Geoffrey Killen, explains how a box (chest) exhibits the golden section in its design but is not sure if this is coincidence or design.
• Fletcher Cox is a craftsman in wood who has used the golden section in his birds-eye maple wooden plate.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..

Fibonacci in Films

The Russian Sergie Eisenstein directed the classic silent film of 1925 The Battleship Potemkin (a DVD or video version 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, measuring these 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 like twinkle have two syllables: twin- and -kle whereas words such as star have just one. Some syllables are emphasized or stressed more than others so that they sound louder (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 into syllables, 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 an unstressed one, then the stress-pattern of each line of the song or poem is its meter (rhythm). In the song above each line has the meter SsSsSsS.

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 song above take about the same length of time to say whether they are stressed or not, so all the lines take the same amount of time to say.
However cloudy 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:

This is just another puzzle of the same kind as on the Simple Fibonacci Puzzles page 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!
SSSS, SSL, SLS, LSS and LL;
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 years before Fibonacci published his first edition of Liber Abaci in 1202!

Acarya Hemacandra and the (so-called) Fibonacci Numbers Int. 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) 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, 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 golden section division and that any proximity to golden sections can be explained by constraints of the sonata form itself, rather than purposeful adherence to golden section division.

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.

Phi in Beethoven's Fifth Symphony?

In Mathematics Teaching volume 84 in 1978, Derek Haylock writes 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 601 and 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 bar 387.
Have a look at the full score for yourself at The Hector Berlioz website on the Berlioz: Predecessors and Contemporaries page, 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 that But 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),
• 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 in a Fibonacci-like way as follows:

1
10
101
10110
10110101
1011010110110
101101011011010110101
...

After the first two lines, all the others are made from the two latest lines in a similar way to 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 section Hear the Golden sequence on that page. In that same section I mention the London based group Perfect Fifth who have 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 is the 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, each beginning 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 is a copy of the previous section again moved up.
I've written more about this series in a section called Sumthing about Fibonacci Numbers on the Fibonacci 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?

Click on the images to find out more in a new window

• 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 90 shown 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.

Other authors say the hands at 1:50 or 10:08 form a golden rectangle using the points on the rim.
This also is not true even if one could imagine them projected on to the rim and then making a rectangle - not an easy visual exercise!
Here are the clocks with hands extended to the rim and a golden rectangle superimposed on the clocks. When the hour hand points at the right place, it is about 10:04 and when the minute hand gets to the correct position, it is about 10h 9m 35s but then the hour hand does not point to the right place.
The time when the hands are exactly symmetrical is 10 hours 9 minutes and 13.8462... seconds and also 1 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 such a golden rectangle construction at those times, in my mathematical opinion!

• Chris Carlson's Mathematica Blog and Mathematica code was used to draw the clocks in this section

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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!

• 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 this are 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

Music

• 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

• Gamelan
  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

• Martin Morgenstern has a large and interesting list of books and articles on the golden section and music with abstracts, some of which is in German.
• 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.
• Casey Mongoven is a composer who has used Fibonacci numbers and golden sections in his own musical compositions. You can hear them and read more on his web site. Casey has an impressively large collection of pieces, most of them a few seconds only in length but they are fascinating to listen to and very different from conventional music. The pitches of his notes are often based on powers of Phi and their order is fixed by a number sequence, such as the Fibonacci numbers, or R(n) - the number of Fibonacci representations of n or on many other sequences that are described here on my Fibonacci site.
  His scores too are images that illustrate many of the series you will have seen here. You can experiment for yourself with the Fibonacci Sequence Visualiser that was designed specifically for Casey's works.
• Ted Froberg explains how he used the Fibonacci numbers "mod 7" (that is the remainders when we divide each Fibonacci number by 7) to make a "theme" which he then harmonizes and has made into a Fibonacci waltz.

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.

Who was Fibonacci?

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