# The Magic of Math: Artists and the Golden Ratio

*Sometimes it is impossible to identify what makes a masterpiece so compelling. Is it the color palette or the careful modeling? The compelling nature of the subject matter, perhaps? While many works draw us in without a clear reason, the captivating quality of some paintings can be tied to the use of the “Golden Ratio”. Known by many other names and variations – including the golden section, the divine proportion, or the rule of thirds – the Golden Ratio has been used for centuries by both architects and artists as a means to enliven their work with a certain ‘je ne sais quoi’. *

For those curious about this captivating mathematical magic that has woven through art for generations, this article is a must read. We’ll explore the use of the Golden Ratio by offering a bit of its origins and showcasing its use across art history.

## The Golden Ratio Explained

Though the term itself was first coined by mathematician Martin Ohm in 1815, the golden ratio had been well understood in mathematical terms for generations prior. Numerically speaking, the golden ratio is tied to φ, or phi, an irrational number that is calculated using the following equation:

Φ = (1 + √5)/2

The result of this calculation can be rounded to 1.618 (its decimal can continue to almost infinite ends).

Thus, the specific proportional relationship of the golden ratio can be represented as 1:1.1618. Mathematicians have used this relationship in a variety of ways. For example, Leonardo da Pisa’s 13th-century numerical sequence, the Fibonacci sequence, a sequence of increasing numbers where each number is equal to the sum of the preceding two numbers, relies upon this ratio.

How, though, to put this ratio into practice from a design perspective? The easiest means to visualize this relationship is to begin with a square, which by its nature has four equal sides (which one can label “X”). Next, one needs to expand this square into a rectangle, which can be accomplished by taking this initial length and using it in the following equation to calculate the rectangle’s adjacent long edge as follows:

(X+Y)/X = X/Y = Φ

So, as an example, one can imagine that X is 1. This means that:

(1+Y)/1 = 1/Y= 1.618

So, in this example, the first, shorter edge (X) would be 1, while the next, longer edge (Y) would be 1.618. Completing the remaining two sides creates the golden rectangle. With this rectangle created, one can repeatedly use the same calculation to create infinitely expanding rectangles. In addition, one can then also draft an arc diagonally from corner to corner that, when the golden rectangle is replicated continuously, forms an endless golden spiral.

## The Golden Ratio in Art

Evidence suggests that architects used the golden ratio far back into antiquity – the Great Pyramid of Egypt and the Parthenon perched on the Athenian acropolis both stand as evidence. For artists, this golden ratio similarly proved central to the pursuit of harmony in painting specifically through the use of golden rectangle and golden spiral. This harmony was in part considered divine, given the fluid consistency of proportions that seemed to have only been heaven-sent, but it also further tied art to nature. The arc, for example, resembles much of the forms from the natural world, from the curve of a nautilus shell to the bloom of a flower. It is perhaps not a surprise, then, that closer look can reveal that this same curve forms the foundation of many artists’ paintings over time.

## 16th- and 17th-Century Golden Ratio Revolutionaries

Sixteenth-century Europe witnessed some of the more clear-cut examples of the golden ratio in action. Italian artists such as Leonardo da Vinci, Michelangelo, and Raphael all demonstrated the use of this central proportional relationship in their paintings. In his famed *Last Supper* (1495-1498), for example, Leonardo employed aspects of the golden ratio to build a harmonious panorama of the fateful meal that revolves seamlessly around the central figure of Christ.

Other artists of Leonardo’s generation also picked up on the power of such divine proportions. For instance, Raphael relied upon the famed ratio for his devotional image of the *Madonna del Cardellino (Madonna of the Goldfinch)* (1505-1506), where the tender embrace between the Christ child, the young Saint John the Baptist, and the Madonna gains emotional intensity given the golden ratio relationship that unites their positions within the composition. Similarly, Michelangelo’s *Creation of Adam* (1508-1512), the centerpiece of the Sistine Chapel ceiling, features a reclining figure of Adam whose curved form follows the golden arc and further guides our eye to the figure of God the Father that looms over him.

The Baroque painters of the 17th-century also found inspiration in the golden ratio. Dutch painter Rembrandt van Rijn used it as the core unifying element in his captivating *Anatomy Lesson of Doctor Nicolaes Tulp* (1632); meanwhile, Diego Velasquez was rumored to use the golden ratio in paintings such as *Christ Crucified* (1632), where the downturned head of Christ and the gentle arc of his dangling arms reflect an artful use of golden proportions.

## 19th- and 20th-Century Golden Ratio Nuance

As time progressed, artists continued to apply the ideas of the golden ratio to their work as a means to heighten its visual impact. A prime example can be found in the prints of Katsushika Hokusai’s famed series, “Thirty-Six Views of Mount Fuji.” Of this print cycle, the most acclaimed is arguably *Great Wave off Kanagawa* (1831), which reveals colossal crashing waves about to engulf a diminutive fishing vessel set against the looming peak of Mount Fuji in the distance. These monumental foamy crests conveniently follow the continuous arc conjured from the golden ratio.

Even as modern art pushed forward into new approaches and techniques, the golden ratio remained a core tool. Famed Surrealist Salvador Dalí relied upon the proportional system in *The Sacrament of the Last Supper* (1995) so extensively that he designed the entire painting within the space of a dodecahedron derived from the golden ratio. The result is a neatly symmetrical painting that seems to resonate with an almost divine harmony.

Perhaps less spiritual in nature but equally mathematically derived, Dutch painter Piet Mondrian’s De Stijl works resonated with an uncanny appreciation for the potential of the golden ratio. Gaining acclaim for his geometric designs that favored simplified palettes, like *Composition in Red, Yellow, Blue and Black* (1930), Mondrian recalled the principles of the golden ratio in its neat, proportional division of compositional space.

## Going for (the) Gold(den Ratio)

As this quick tour of art history showcases, artists have turned to the neat proportions provided by the golden ratio to conjure some of history’s most captivating paintings. What is also striking in this selection is that, despite the evolution of painterly techniques and styles, the golden ratio resonates nevertheless as a means of creating a satisfying, harmonious visual relationship of form and space. Given this, the next time a work of art proves irresistible, look closely to investigate whether this magical mathematical relationship might be at play.