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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
5
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
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).- Here are three more impressive photographs that you can use for your
own investigation (part of the New
York Skyscrapers WWW pages).
- The Secretariat building from the visitors entrance (photo by Lawrence A Martin)
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
An excellent
source of architecture images is the University
of Wisconsin's Library of Art History images - well worth checking
out! It has many images of the Parthenon, pictures of its friezes and other
details. Use their searcher
selecting the Period Ancient Greece: Classical and the Site Athens.
Note: the images cannot be copied or even made into links, only viewed
on their page!
Also see University
of Michigan, June
Komisar's page of architectural links. She points to the Great Building
Collection which has some excellent photo images on their Parthenon
page. Do check this out as they have a FREE 3D viewer to download and
lots
of buildings in 3D to view. You can take your own virtual walk through
the Parthenon!
There is a link
to some nice pictures of Greek temples etc at http://tony.ai/KW/golden.html.
The golden section
in The Kings Tomb in Egypt.
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The Golden Section and Art
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 becomesPacioli'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.
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 1This gives two values for x, (-1-
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).
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
- ( 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
- 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.
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.
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.
Ned May has generated
some
beautiful pictures based on Fibonacci Spirals using Visual Basic (an
example is shown here on the right).
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Fibonacci and Poetry
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Fibonacci and Music
Various composers have used the Fibonacci numbers when composing music - more details in Garland's book.
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.
Beethoven's Fifth
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:
Bartók, Debussy, Schubert, Bach and Satie
-
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;
-
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. After its first publication in 1986, this book is now (February 2000) back in print.
-
See also Roy
Howat's Web site for more information.
-
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 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).
-
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 organisation.
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!
WHERE TO NOW???
This is the last page on
There is a
very useful set of mathematical links to Art
and Music web resources from Mathematics
Archives that is worth looking at.
1·61803 39887 49894
84820 45868 34365 63811 77203 09179 80576 ..More..
A Controversial Issue
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!
Important article that point out the weaknesses in parts of "the golden-section
is the most pleasing shape" theory:
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.
George Markowsky's
Misconceptions
about the Golden ratio in The College Mathematics Journal Vol
23, January 1992, pages 2-19. How to Find
the "Golden Number" without really trying Roger Fischler, Fibonacci
Quarterly, 1981, Vol 19, pp 406 - 410 The Fibonacci
Drawing Board Design of the Great Pyramid of Gizeh Col. R S Beard in
Fibonacci
Quarterly vol 6, 1968, pages 85 - 87;
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, An example
of Fibonacci Numbers used to Generate Rhythmic Values in Modern Music
Links to other Music Web sites
Gamelan music
Other music
Gamelan New
music Some CDs
Art
The
Fibonacci
Sequence
A Mathematical History
of the Golden Section ISBN 0486400077. Education through
Art (3rd edition) H Read, The New Landscape
in Art and Science G Kepes H E Huntley's, The
Divine Proportion: A study in mathematical beauty, C. F. Linn, The
Golden Mean: Mathematics and the Fine Arts, Gyorgy
Doczi, The
Power of Limits: Proportional Harmonies in Nature, Art, and Architecture M. Boles, The Golden
Relationship: Art, Math, Nature, 2nd ed.,
Who
was Fibonacci?
Fibonacci
Home Page
More Applications of the Fibonacci Numbers and Phi. The next topics...
Fibonacci,
Phi and Lucas numbers Formulae
Links
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