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 in architecture, art and music.

Contents of this page

Things to do

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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 golden proportion (1 : 1.618 which is the same as 0.618 : 1). It occurs naturally in some of the proportions of the Five Platonic Solids (as we have already seen). A construction for the golden section point is found in Euclid's Elements. The golden rectangle is supposed to appear in many of the proportions of that famous ancient 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!) at Nashville which calls itself "The Athens of South USA".)

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

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

Links

There is a wonderful collection of pictures of the Parthenon and the Acropolis at Indiana University's web site. Dr Ann M Nicgorski of the Department of Art and Art History at Williamette University in the USA has a large collection of links to Parthenon pictures with many details of the building. David Silverman's page on the Parthenon has lots of information. Look at the plan of the Parthenon. The dividing partition in the inner temple seems to be on the golden section both of the main temple and the inner temple. Apart from that, I cannot see any other clear golden sections - can you? Allan T Kohl's Art Images for College Teaching has a lot of images on ancient art and architecture.

Modern Architecture

The Eden Project's new Education Building

California Polytechnic Engineering Plaza

As a guiding element, we selected the Fibonacci series spiral, or golden mean, as the representation of engineering knowledge.

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 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. 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

A M B

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.

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".



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?



Graham Sutherland's (1903-1980) huge tapestry of Christ The King behind the altar in Coventry Cathedral here in a picture taken by Rob Orland. It seems to have been designed with some clear golden sections as I've shown on Rob's picture: Show golden sections on the 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.

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? 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):

Links to major sources of Art on the Web:

The work of modern artists using the Golden Section

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.

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Stress, Meter and Sanskrit Poetry

Twinkle twinkle little star

How I wonder what you are.

Up above the world, so high

Like a diamond in the sky

...

S

s

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:

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

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

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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

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

Did Mozart use the Golden mean?

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?

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.

Bartók, Debussy, Schubert, Bach and Satie

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

The Golden String as Music

1

10

101

10110

10110101

1011010110110

101101011011010110101

...

Other Fibonacci and Phi related music

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

TV Stations in Halifax, Canada

Turku Power Station, Finland

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 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

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?





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): 12.674=

12h 40.453m 1.76513=

1h 45.908m 2.85604=

2h 51.362m 3.94695=

3h 56.817m 5.03786=

5h 2.271m 6.12876=

6h 7.726m 7.21967=

7h 13.180m 8.31058=

8h 18.635m 9.40149=

9h 24.089m 10.4924=

10h 29.544m 11.5833=

11h 34.999m It is much easier to compute the times if :At the following times the hands form aof 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 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

Things to do What other logos can you find that are golden rectangles? 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. Email me with any answers to these questions and I'll try to include them on this page.

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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°

References and Links on the golden section in Music and Art

Key: a book an article in a magazine or

a paper in an academic journal a website

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