Art and Math: The Golden Ratio

Without mathematics there is no art. So said the Italian mathematician Luca Pacioli, who published an entire book on the subject in 1509. Entitled “De divina proportione” (Divine Proportion), Pacioli’s work was illustrated by Leonardo da Vinci.

Down through the ages, the “divine proportion” has gone by many names, like the Golden Mean or the Golden Section. Today it is most commonly referred to as the Golden Ratio. So let’s first talk about the mathematics of it.

Begin with a line segment and cut it into two pieces of length 1 and x, where 1 is less than x. Let the ratio of x to 1 be equal to the ratio of the length of the entire segment to the x. Mathematically, we describe this relationship as x/1 = (1+x)/x, where 1 < x.

Now we just have an algebraic equation involving a fraction. In order to get rid of the fraction, we’re going to multiply the equation by x to get x^2 = 1+x, or x^2 – x – 1 =0, where x^2 is read x-squared. When we apply the quadratic formula, we find that the positive root of x equals [1 + SQRT(5)]/2 ~ 1.618, where SQRT(5) is the square root of 5. This is the Golden Ratio, sometimes referred to by the Greek letter PHI.

So what? The “so what” is that the Golden Ratio turns up in mathematics, nature and art. Let’s start with mathematics.

The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, …. The sequence is generated by taking the two previous entries and adding them to arrive at the new number in the sequence. So 1+1 = 2, 1+2 = 3, 2+3 = 5 and so on and so on. So in general, the next element of the sequence a(n) looks like a(n-2) + a(n-1). Now here’s the interesting part. The further you go along in the Fibonacci sequence, the closer the ratio of a(n)/a(n-1) approximates the Golden Ratio PHI.

Now we’ll show the relationship with nature. We’re going to use the Fibonacci sequence to do a geometric construction. Begin with two squares of side 1 and draw them adjacently. Now take the side of this drawing which has the total width of 2, and draw a square of side 2. Now take the side of this drawing which has the total length of 3, and draw a square of side 3. See the Fibonacci sequence here? In the next iteration, you’re going to draw a square of side length 5. Now join the diagonals of the squares in such a way that you draw a curve that looks like a spiral curve, like a nautilus shell. If you can’t visualize this, look at a construction online like http://jwilson.coe.uga.edu/emt669/Student.Folders/Frietag.Mark/Homepage/Goldenratio/goldenratio.html. So we find that the Golden Ratio appears in nature, in shells, flowers, leaf patterns, and just about everywhere else, once you start looking for it.

The Golden Ratio also appears in art. Classical painting styles emphasize proportions of the part to the whole. The Golden Ratio appears in the work of many painters, including da Vinci (obviously), Michelangelo, Raphael, Rembrandt, and much later in the paintings of Salvador Dali, Georges Seurat and Piet Mondrian.

In architecture, early examples proportioned according to the Golden Ratio are the Great Pyramid of Giza and the Parthenon. Usage of the Golden Ratio is also evident in Gothic cathedrals like Notre Dame and Chartres from medieval times. It also appears in India’s Taj Mahal (1653), and the Eiffel Tower in Paris (1889). In the 20th century, the Golden Ratio appeared in the architecture of such greats as Mies van der Rohe and Le Corbusier.

Do nature and art always subscribe to the Golden Ratio? No. Nature doesn’t not always follow the rules. Natural variations occur. Not every nautilus shell is of divine measure. Nor do humans always follow the Golden Ratio. Rather, it is a guideline for understanding the mysteries of our world, and imitating its natural beauty. As Picasso said, “Learn the rules like a pro, so you can break them like an artist.”