The Golden Section and

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:

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:

This gives two values for x, (-l-V5)/2 and (V5-l)/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 0618 of the way up and 0618 of the way down it. Also mark in the vertical lines which are 0618 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 (0618 of the way down and up the frame and 0618 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 imagebased on Fibonacci spirals and rectangles the Web Museumpages 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 Musicweb resources from Mathematics Archivesthat is worth looking at.

Links to major sources of Art on the Web:

Top9.com's List of the top art sources on the webis an excellent place for links to good art sources on the web. Highly recommended!

The Metropolitan Museum of Artin New York houses more than 2 million works of art.

The Fine Arts Museums of San Francisco sitehas 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

Michelangelois 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 Davidsculpture, 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) itselffrom which town Leonardo is named, of course.

There are many sketches and paintings of Leonardo's at The WebMuseum, Paristoo.

The work of modern artists using the Golden Section

Billie Ruth Sudduthis 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 Schoolhas a good online introductionto her work (January 1999).

K ees van Prooijenof 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 Spiralsusing Visual Basic (an example is shown here on the right).

1-61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 ..More..

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