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.
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, (-1-5)/2 and (5-1)/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).
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".
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.
Various composers have used the Fibonacci numbers when composing music - more details in Garland's book.
Baginsky's method of constructing violins is also based on golden sections.
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.
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.
References and Links on the golden section in Music and Art
an article in a magazine or
a paper in an academic journal
Links to other Music Web sites
More Applications of the Fibonacci Numbers and Phi.
The next topics...
Fibonacci, Phi and Lucas numbers Formulae
Links and References
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