“What makes a single number so captivating that it has persisted in our imagination for more than two thousand years?” -Gary B, Meisner The Golden Ratio: The Divine Beauty of Mathematics
We’re going to dive into what was hands down my favorite thing to learn about in art history class: The Golden Ratio.
I know just enough about The Golden Ratio to be equally in awe of it and completely baffled by it. It’s one of those concepts that lights up your brain when you discover, but is hard to grasp unless you are mathematically inclined. The more you learn about it, the more you will see Golden Ratio proportions everywhere.
Luckily for us, there are a handful of people who understand the concept clearly and can explain it to the rest of us cotton-headed ninny muggins. I will be focusing mostly on the Golden Ratio in art and architecture here, but I can’t help but touch on how it pops up in nature as well. Get your geek pants on, fellas!
What the What is the Golden Ratio?
The Golden Ratio is an irrational number, approximately 1.618, that is prevalent in nature, art, architecture, and design. (Other names for it are golden mean, golden section, Phi (in mathematics), divine section, golden number, Fibonacci sequence.) Actually, the Fibonacci sequence is super-closely related to the Golden Ratio, but the not exactly the same. This is one example of math confusion.
The Golden Ratio Rectangle
Visually, it is a rectangle, that when cut into a square, results in the remaining rectangle being the same proportion as the original rectangle. I absolutely cannot explain the math behind it, so go here to understand it in a more practical way.
Try this: If you feel like drawing a rectangle using the Golden Ratio, here’s how you do it:
- Draw a square.
- Draw a dot halfway across the bottom line.
- Draw a line from that dot to either opposite corner.
- Drop that line down so it overlaps the bottom line of the square.
- Wherever the end of it lands is where you can draw a line up and over to make your rectangle.
- Notice that the new rectangle you just drew can be separated into the same proportions as the large rectangle. So can every subsequent little rectangle you draw!
Take it a step further by breaking down the new rectangle into smaller golden rectangles and then drawing a spiral using lines going from one corner to the opposite corner in each square of the golden rectangle. Huh? Here, take a look at this:
I also felt the need to draw a page of phi symbols, and please notice I used a lovely golden color in keeping with the theme.
Who Discovered the Golden Ratio?
I did! In the early 1600’s. JK.
According to this site, Euclid explained the Golden ratio formula in his book The Elements, even though he didn’t name it anything. It isn’t clear where he might have learned of the idea.
The number phi was named (in the 1900’s) for the Greek sculptor and mathematician Phidias who lived from 500 BC – 432 BC.
The Golden Ratio in Nature
This is so, so cool, and will make you like nature, even if you abhorred nature previously. Nature is RIDDLED with The Golden Ratio, in plants, faces, microscopic whatevers.
The Golden Ratio tends to show up in nature in many flower seed formations (sunflower), seashell patterns (nautilus), and even galaxies! Read this article on examples of the Golden Ratio in nature to blow your mind.
Your Face is a Golden Ratio
Keep that comeback handy the next time you’re at a loss for words. It’s likely the recipient will have no idea what you’re talking about, and you can giggle at your secret joke.
Supposedly if you look at a human face straight on, the closer to ideal beauty it is, the more measurements you can take all over it that work out to be the Golden Ratio.
Here are some examples of the Golden Mean found on the perfect face:
- Divide the height of your face (from the top of your head to the bottom of your chin) by the width of your face at the widest point, and you will get somewhere right around phi. (1.618)
- The distance between your eyes is about the same width as one eye.
- The measurement from the hairline to right between the eyes, and from between the eyes to the tip of the nose, and from the tip of the nose to the bottom of the chin should all be close to equal to achieve perfect beauty.
I measured this good-lookin’ dude and found that his 3 vertical measurements don’t match, and his face height divided by his face width came out to be roughly 1.72 inches. We had to wing it because of his hair do obviously, but it does work out to be close.
Try this: Print out a photo of your face and measure it to see how perfect you are. I haven’t done this yet, but am assuming my giant forehead will throw everything off. 🙂
Remember my Faces are Mathy post? You can find more face measurements there that help when drawing faces.
The Golden Ratio in Art
Okay, here we are at the good stuff. Since the Golden Ratio was discovered, it’s not a surprise that it has been widely adopted by artists, designers and architects to determine the most visually pleasing proportions to make their creations.
There are many ways artists have used the golden ratio in art; using the golden rectangle itself to determine the composition of the artwork, using the path of the spiral in the golden rectangle, and even placing important subject matter at measured points inside the rectangle.
The Golden Ratio in Painting
Leonardo da Vinci
Probably the most famous artist to put it to use is Leonardo da Vinci. Da Vinci illustrated a book written by Luca Pacioli in the late 15th century all about the Divine Proportion. He used the measurement in many of his paintings, including Mona Lisa and The Last Supper. This post shows instances of The Golden Ratio in some of his works.
Try This: I had fun playing with a transparent png of the Golden ratio rectangle over Mona Lisa. Here is the png if you want to download it and try it, too:
Here is Mona Lisa:
I opened her up in PicMonkey, then added the Fibonacci spiral as an ‘Add your own’ graphic and moved it, resized it, rotated it in all sorts of ways.
Salvador Dali’s painting, The Sacrament of the Last Supper (1955) is widely cited as utilizing the Golden Ratio, and it’s pretty cool to break it down in these terms:
- The room is a dodecahedron, which is related to a golden rectangle in a mathematical way I absolutely cannot understand.
- The entire painting is a golden rectangle.
- The table and 2 disciples next to Christ are positioned perfectly at the sections of the golden rectangle.
Mondrian is said to have used The Golden Ratio in his abstract paintings, but when I take the trusty golden rectangle to some of them, nothing ever quite lines up perfectly. Apparently some of his paintings line up better with the proportions than others, so maybe he wasn’t too picky about precise measurements.
Interestingly, my obsession with placing the golden rectangle over everything I can get my hands on led me to placing it over this photo of Yves Saint Laurent models wearing Mondrian dresses in front of a Mondrian painting. Look how beautifully that works out. 🙂
Look at his paintings with a new sense of proportion, eh? Only some of his paintings work out this way, so I’m not sure if this was intentional or not on his part.
The Golden Ratio in Photography
The rule of thirds is commonly taught to beginning photographers as a way to set up the composition of their photos. It is based on the idea that your photo can be separated into a grid of 9 equal spaces, and the focal points of the image should line up with one or more spots where the lines intersect.
The phi grid is used in photography as another way to break up the image, following the 1:1.618 ratio. The lines are laid out in a grid, but they are not spaced evenly as in the rule of thirds grid.
Both of the grids are used to help the viewer’s eye move around the photograph, and to create a more interesting composition instead of just plopping your subject dead center. It causes a sort of off-center balance that is more interesting for us to look at.
Here is Wikipedia’s gif showing a photo cropped using the rule of thirds and not using it:
Here’s the photo on the right cropped down further after I overlaid a phi grid on it. Which do you prefer? Either? Download your own Golden Ratio templates here.
Henri Cartier-Bresson’s work are great examples of The Golden Ratio in photography. And by great, I mean it’s really fun to overlay his photos with the golden rectangle spiral. So satisfying.
The Golden Ratio in Architecture
The Parthenon has long been one of the most cited architecture examples of The Golden Ratio, but lately is discredited by some. Supposedly the space between its columns were the Golden Ratio proportions. I guess maybe we have to chalk that up to coincidence now, since no one can prove the ancient Greeks were overly inspired by this idea.
The Great Pyramid of Egypt is rumored to have Golden Ratio proportions, but some people have discredited this as well.
Architecture from The Middle Ages and The Renaissance show the Golden Ratio in use in churches and cathedrals, as the Golden Ratio grew in popularity. The Modena Cathedral in Italy, Notre Dame, and Chartres Cathedral to name a few, are said to have been designed using Golden Ratio proportions.
The architect Le Corbusier developed his own system of measurement based on The Golden Ratio. This was called The Modulor system, and he made a big impact on many architects of the time, and since. Who knows how many things have been designed based on Le Corbusier’s system.
He designed his Unite d’Habitation in Marseille and UN building in NY using Golden Ratio proportions. In the United Nations building, the width of the building compared with the height of every ten floors is the Golden Ratio.
Here’s a thoughtful article on the Golden Ratio in architecture.
More Golden Ratio Resources
Watch this 1959 short Disney film called Donald in Mathmagic Land for a special appearance by the Golden Ratio:
And this amazing video on Fibonacci/Golden Ratio that you seriously will not be able to stop staring at.
Books on the Golden Ratio
The Golden Ratio: The Divine Beauty of MathematicsThe Golden Ratio: The Story of PHI, the World’s Most Astonishing NumberThe Golden Section: Nature’s Greatest Secret (Wooden Books)The Golden Ratio Coloring Book: And Other Mathematical Patterns Inspired by Nature and ArtThe Power of Limits: Proportional Harmonies in Nature, Art, and ArchitectureBlockhead: The Life of Fibonacci
Also check out this list of Fibonacci kid books from What Do We Do All Day.
Other Cool Golden Ratio Stuff
Contemporary Examples of Artists Using The Golden Ratio
The mystery and satisfying geometry of the Golden Ratio make it a compelling idea to explore in art-making even today. I’m curious to try my hand at making something based on all this fun mathy stuff.
Here are some current artworks I found that refer to The Golden Ratio to varying degrees.
The Golden Mean– Nautilus shell photograph by Javiera Estrada
D3GMV Photography by Douglas McIntosh
BF-10-4/4 Sculpture by Benoist Van Borren
Compass II Collage by Curtis Olson
The Golden Ratio in Art is Majorly Controversial
Given all these examples in nature and artwork of The Golden Ratio, it’s easy to be tempted to start trying to force our own art into these rules. As much as I love learning about this, and seeing how visual artists have interpreted this wonder of geometry, I feel like it’s still more important to tuck the information in your brain to let it seep out intuitively when you make work.
This is just me loving the magic in art-making. The part of us that births it without too much planning and overthinking, even though we hold onto the rules and practice and techniques we’ve accumulated over the years.
As I’ve touched on, some of the previously unchallenged accounts of the Golden ratio in art and architecture have come up for debate. There are plenty of people who completely discount the Golden Ratio as a total scam.
What do you think? I want to know your thoughts on this fascinating topic.