The following comprises a set of slides on "Golden Section in Art and Architecture".
In the commentary, it is presumed that the user is familiar with the geometry of phi.
[If not, visit Fascinating Flat Facts about Phi for a wealth of information.]
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The ancient Egyptians were the first to use mathematics in art. It seems almost certain that they ascribed magical properties to the golden section (golden ratio, divine proportion, phi) and used in the design of their great pyramids.
pyramids, Giza
If we take a cross section of the Great Pyramid, we get a right triangle, the so-called Egyptian Triangle. The ratio of the slant height of the pyramid (hypotenuse of the triangle) to the distance from ground center (half the base dimension) is 1.61804 ... which differs from phi by only one unit in the fifth decimal place. If we let the base dimension be 2 units, then the sides of the right triangle are in the proportion 1:sqrt(phi):phi and the pyramid has a height of sqrt(phi). Visit the web page Cairo and the Pyramids of Giza for more details.
Pythagoras (560-480 BC), the Greek geometer, was especially interested in the golden section, and proved that it was the basis for the proportions of the human figure. He showed that the human body is built with each part in a definite golden proportion to all the other parts.
Pythagoras detail, from School of Athens by Raphael
Pythagoras' discoveries of the proportions of the human figure had a tremendous effect on Greek art. Every part of their major buildings, down to the smallest detail of decoration, was constructed upon this proportion.
Acropolis, Athens
The Parthenon was perhaps the best example of a mathematical approach to art.
Parthenon, Acropolis, Athens
Once its ruined triangular pediment is restored, ...
the ancient temple fits almost precisely into a golden rectangle.
Further classic subdivisions of the rectangle align perfectly with major architectural features of the structure.

Mathematicians had the contribution of the Greeks in mind when they christened the ratio "phi" in tribute to the great Phidias, who used the proportion frequently in his sculpture.
Porch of Maidens, Acropolis, Athens
The Medieval builders of churches and cathedrals approached the design of their buildings in much the same way as the Greeks. A good geometric structure was their aim. Inside and out, their buildings were intricate constructions based on the golden section.
Chartres Cathedral
Rose Window, Chartres Cathedral
But whilst in architecture there was this very great interest in geometry, artists seemed to have lost all interest in the golden section and in mathematics as a whole. In the 16th Century, Luca Pacioli (1445-1514), geometer and friend of the great Renaissance painters, rediscovered the "golden secret". His publication devoted to the number phi, Divina Proportione, was illustrated by no less an artist than ...
Fra' Luca Pacioli (attributed to Jacopo de Barbari)
Leonardo da Vinci (1451-1519). Leonardo had for a long time displayed an ardent interest in the mathematics of art and nature.
He had earlier, like Pythagoras, made a close study of the human figure and had shown how all its different parts were related by the golden section.
Study of Human Proportions According to Vitruvious
Study of Facial Proportions
Leonardo's unfinished canvas Saint Jerome shows the great scholar with a lion lying at his feet.
A golden rectangle fits so neatly around the central figure that it is often said the artist deliberately painted the figure to conform to those proportions. Knowing Leonardo's love of "geometrical recreations" as he described them, this is quite likely.
Notice how the classic subdivision of the rectangle lines up with St. Jerome's extended arm.

The golden rectangles in Da Vinci's Mona Lisa abound. Visit the web page Mona Lisa Applet to add golden rectangles interactively to his famous masterpiece.
Hacking back to classical themes and techniques for their inspiration, artists of the Renaissance like Michelangelo (1475-1564) and Raphael (1483-1530) once more began to construct their compositions on the golden ratio. The proportions of Michelangelo's David conform to the golden ratio from the location of the navel with respect to the height to the placement of the joints in the fingers. Visit the web page Florence and Human Proportion for more details.
Michelangelo's Holy Family ...
is notable for its positioning of the principal figures in alignment with a pentagram or golden star.

Raphael's Crucifixion ...
is another well-known example. The principal figures outline a golden triangle ...
which can be used to locate one of its underlying pentagrams.

This self-portrait by Rembrandt (1606-1669) ...
is an example of triangular composition - holding together an intricate subject within three straight lines. The different lengths of the sides add a little variety.
A perpendicular line from the apex of the triangle to the base would cut the base in golden section.

The English romantic artistic Joseph Mallord William Turner (1775-1851) is admired for his use of color and light.
Norham Castle at Sunrise
Of particular interest are the geometric similarities in his various canvases, with their obvious golden subdivisions.

Rain, Steam and Speed
The Fighting Temeraire
Slavers Throwing Overboard the Dead and Dying
The more recent search for a grammar of art inevitably led to the use of the golden section in abstract art. La Parade, painted in the characteristic multi-dotted style of the French neo-impressionist Seurat (1859-1891), contains numerous examples of golden proportions.
According to one art expert, Seurat "attacked every canvas by the golden section". His Bathers ...
has obvious golden subdivisions.
Three golden figures have been added here. Can you find more?

The Sacrament of the Last Supper by Salvador Dali (1904-1989) is painted inside a golden rectangle. Golden proportions were used for positioning the figures. Part of an enormous dodecahedron floats above the table. The polyhedron consists of 12 regular pentagons and has fundamental golden connections.
The 20th Century architect Le Corbusier (1887-1965) developed a scale of proportions which he called Le Modulor, based on a human body whose height is divided in golden section commencing at the navel.
The same proportion is to be seen in his modern flats. Le Corbusier felt that human life was "comforted" by mathematics.
Today architects are still frequently using the golden section in their work. In the United Nations building, the ratio of width of the building compared with the height of every ten floors is golden.
The CN Tower in Toronto, the tallest tower and freestanding structure in the world, contains the golden section in its design. The ratio of its total height of 553.33 meters to the height of the observation deck at 342 meters is 1.618.
We close with this quotation by Luca Pacioli.

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