A Mathematical Guide to Aesthetics
Have you ever observed a really profound piece of art (or maybe even a movie poster) and wondered why it was so compositionally pleasing? The most likely reason is that the artist intentionally (or unintentionally) employed the use of the Divine Proportion (also known as the golden ratio, the golden section, and part of the Fibonacci series). The Divine Proportion is a little-known phenomenon that suggests that there’s a mathematical equation that’s consistent with the aesthetics of good composition. Yes, it has even been proven that some people with a natural instinct for good composition inadvertently employ certain properties consistent with the golden ratio. Here we’ll examine a brief history of the Divine Proportion and its use over the centuries.
History of Phi
The Divine Proportion is based on the ratio of one object to another as defined by the number 1.618033988749895…(the number is actually quite longer, but you get the idea). For example, if you take the length of a rectangle and divide it by its height and it equals 1.618 (etc.), the rectangle is called a golden rectangle because the lengths of its sides are in the golden ratio.
This number is also often referred to as “Phi” (rhymes with fly), named after Phidias, a Greek sculptor who lived around 490–430 B.C. The ratio was so named because it was believed that Phidias made deliberate and intricate use of the Divine Proportion in much of his work, including the Athena Parthenos in Athens and the Statue of Zeus at Olympia.
Use of the Divine Proportion is all around us, and while knowledge of its existence has been confined to more specialized disciplines, it has slowly gained some mainstream recognition. The extremely popular book and film, The Da Vinci Code, made mention of the Divine Proportion with regard to Leonardo da Vinci’s use of it in several of his paintings. It has been suggested that Leonardo made deliberate use of the Divine Proportion in almost every aspect of his work including such paintings as the Mona Lisa and The Last Supper. Throughout history, however, it has been speculated that the golden ratio is prevalent in every aspect of our universe—from the natural growth pattern of the nautilus seashell to the pyramids of Egypt to the musical compositions of Mozart and Beethoven.
The Divine Proportion is often represented by the golden spiral. This is the tool used by artists and sculptors to achieve remarkably accurate proportion and aesthetic composition. In the golden spiral, the ratio of the length of the side of each square to the length of the side of the next smallest square is in the golden ratio; the rectangles formed by the combination of squares are golden rectangles; and at every quarter-turn, the spiral gets wider by a factor of Phi.
The Golden Spiral
The Golden Spiral
Let’s examine what makes the Divine Proportion so “divine” by illustrating its properties on some pieces of contemporary art. Here are a couple of pieces by photorealistic artist Bert Monroy. Bert’s images are comprised of elements that mimic reality, and despite his technique, which is remarkable, it’s his composition that is so compelling. As you can see here by overlaying the golden spiral over Bert’s art, these images fit very well within the golden ratio.
Now some would probably ask if this was intentional. Only Bert knows for sure, but history has certainly shown that amazing compositions have been put together by both conscious and unconscious awareness of the Divine Proportion. So whether or not the golden ratio was employed deliberately, it can still be used to better understand composition.
Rule of Thirds
The same goes for photography. In this photograph, Scott Kelby employs a compositional technique known as the rule of thirds, which is loosely based on the Divine Proportion. Basically, it’s your camera’s view split into a nine-square grid with the idea that you position your subject at any point where grid lines intersect. As you can see here, this yields a more pleasing composition. The subject of this piece falls at the intersection of two grid lines and is also placed based on the Divine Proportion itself, which clearly defines this as a well-composed image.
Now, of course, I’m not suggesting that every image should be scrutinized in such a way. I’m simply illustrating that design and composition can be measured and perhaps even enhanced by employing this simple scientific device. At the very least, it’s fun to apply it to other people’s art and gain insight to their creative genius.
Building of the Golden Spiral
If you’re ready to take advantage of the Divine Proportion in your designs, here are some easy steps to create your own “approximate” golden spiral in Adobe Illustrator.
The first step is to draw a square with all equal-length sides (hold the Shift key when using the Rectangle tool to constrain the shape to a square). Let’s begin by drawing the largest square first. Tip: Make sure you Fill is set to None so you can see all the shapes as you draw.
Next, from the bottom center of the square draw a circle out until the edge of the circle touches the top corners of the box. You can use a guide to help locate the center of the square, and then hold Shift-Option (PC: Shift-Alt) to draw a perfect circle from the bottom center of the square outward.
On the right side of the square, draw a rectangle from the top-right corner down to the bottom of the first square and out to the edge of the circle. This shape is now 0.618 the size of the main square.
You can now delete the circle. From the top-left corner of the rectangle that you just drew in Step Three, draw a square (hold the Shift key) until you reach the right side of the rectangle.
Continue drawing these squares in a sort of clockwise downward spiral, starting from the bottom right of the last square that you just drew.
To test the proportion accuracy, draw a line from the top-left corner to the bottom-right corner, and then draw a line from the top-right corner to the bottom point of that middle line. These lines should cross at the corner point of each shape that they intersect.
Lastly, starting at the lower-right corner of the smallest square (not the small rectangle), begin a line spiraling out, intersecting the outside corners of each square (as shown here) until it reaches the bottom-left corner of the overall shape.
Now we have a group of shapes that are geometrically representative of the golden ratio. Once you’ve created a complete and accurate golden spiral, you can now save it as a custom shape in Illustrator or Photoshop and use it in your design work as a compositional guide. You can also use it to overlay existing art to understand other artists’ use of composition.