Beyond the Golden Ratio

Beyond the Golden Ratio

What do a sunflower, The School of Athens by Raphael, and the Apple logo have in common? At first glance, it seems they’re completely unrelated. However, looking closer, each has been linked to the golden ratio, a mathematical proportion that’s shaped both design and scientific inquiry alike. In revealing connections between numbers, nature, and creativity, the golden ratio continues to prove that some of the most profound ideas are found where science and art meet!

A number frequently appearing in both art and nature, the golden ratio is often denoted by the Greek letter phi ($$φ$$). In mathematics, two numbers a and b are in the golden ratio if their ratio is the same as the ratio of their sum to the larger number. The golden ratio is defined as the unique positive solution to the equation $$φ^2=φ+1$$, derived from $$\frac{1}{φ} = \frac{φ}{1+φ}$$. Its value is $$\frac{1+\sqrt{5}}{2}$$, an irrational number.

In other words, the golden ratio is a ratio of $$1:1.618$$. More concretely, if we had two pieces of string (where the length of one string was 1.618 times the other), the sum of the lengths of both strings would be 1.618 times the length of the longer string! This may all seem very abstract and random, but as we will see, the ratio is a very useful design tool for both artists and plants alike.

What’s interesting is how the ratio can be written in terms of itself! This can be expanded into an infinite continued fraction.

Additionally, there’s a unique relationship between the golden ratio and the Fibonacci sequence (0, 1, 1, 2, 3, 5, …), a mathematical sequence where each number is the sum of the two preceding ones! When we take two successive numbers in this sequence, their ratio tends to 1.618—see the below example calculations. $$φ$$ is the limit of the sequence of these ratios!

While pure maths can be exciting in and of itself, let’s move on to some of the many fascinating appearances of the golden ratio that can be found in nature! You can find it in as varied and far-reaching examples as pinecones, plant growth, and nautilus shells!

In particular, sunflower seed spirals tend to adhere closely to the golden ratio.

Sunflower heads display a set of clockwise spirals and another set of counterclockwise spirals. The number of spirals in each direction are often consecutive Fibonacci numbers, maximizing the efficiency of the seed packing [1].

Additionally, the angular spacing between successive seeds is close to the golden angle! If the angle were a rational fraction of 360, the seeds would align in radial lines, creating gaps. The golden angle, the “most irrational” number, prevents this alignment from occurring and allows for optimal distribution.

In geometry, the golden angle is the smaller of the two angles created when a circle is sectioned according to the golden ratio. The division produces two arcs where the ratio of the shorter arc to the longer arc is equal to the ratio of the longer arc to the circle's circumference.

This pattern in plants is most interesting. For plants with alternate leaves, the leaves often grow in a spiral manner. Since the angle between two adjacent leaves is around 137.5 degrees, the golden angle, this allows for each leaf to get the maximum amount of sunlight without being shadowed by the leaves above, as no leaf is directly above another.

Found in architecture and various paintings throughout centuries, the golden ratio has been a constant subject of fascination for creatives. The proof lies in the name itself——”golden”. It has even been dubbed the “divine proportion” by Luca Pacioli, who believed it to reflect the divine simplicity, omnipresence, and incomprehensibility of God. His book on the subject, Divina Proportione, was illustrated by Leonardo da Vinci.

Built in ancient Athens, the Parthenon has been analyzed as fitting within golden rectangles——where the side lengths are in the golden ratio. The iconic Notre Dame’s horizontal layers and west facade, alongside the Taj Mahal’s symmetry and scaling, also have been linked to the ratio. Additionally, The Sacrament of the Last Supper by Dali explicitly uses a golden rectangle as the canvas, while The Great Wave depicts a wave——a naturally-occurring spiral.

Clearly, painters and architects alike have been inspired by this “golden” number in ages past. But is its appeal truly universal? There’s been continuous debate on this topic.

Salera et al.’s study on preferences for the ratio provides evidence of an implicit preference for the golden ratio––the study’s participants responded faster and more accurately when golden ratio patterns were paired with positive words, leading to a positive D–score. This preference occurred regardless of how the golden ratio patterns were designed. However, when it came to explicit preference, the participants rated golden ratio patterns only slightly more pleasant than random patterns. When participants were asked to create proportions (for instance, dividing lines or money), they did not naturally use the golden ratio. Thus, people may not consciously apply the ratio in everyday decisions, but may associate the ratio’s patterns with beauty without knowing why [2].

De Bartolo et al. found a slight overall preference for the golden ratio, but this preference did not appear in all categories: for geometric shapes and humanoids, preference did not reach significance. However, a preference was found in all other categories. In particular, there was a more significant preference for stimuli exhibiting the golden ratio in human photographs and paintings [3]. Additionally, eye movements focused more on areas related to the golden ratio. For images that followed the golden ratio, those who rated them as more beautiful spent less time looking at them, suggesting that aesthetically pleasing golden ratio designs may be easier to visually process.

Other studies also indicate the relationship between beauty and the golden ratio isn’t absolute. Green’s review recognizes that there does appear to be a psychological effect associated with the ratio, but the effect is much smaller and less universal than what popular accounts claim [4]. Instead, aesthetic judgment is influenced by various factors, such as symmetry, complexity, familiarity, and cultural experience. Jacobsen et al.’s work further suggests that beauty is evaluated by a brain network underlying evaluative judgements, the same areas involved in social and moral judgment [5]. Thus, perceptions of harmonious proportions may vary upon the context in which they are experienced.

Think about the creation of a painting. A Renaissance artist may have used mathematical proportions to organize a composition, creating balance carefully. However, the impact of an artwork also depends on color, storytelling, cultural significance, and the viewer’s personal interpretation. A painting isn’t beautiful because of numbers alone. Rather, mathematical structure is only one of many tools artists use to guide perception.

The Golden Ratio can be applied to shapes, which have been used to create balance in art and design! Believe it or not, many logos we’re familiar with have used the golden ratio in some form. Using different manifestations of the ratio, such as circles or spirals, we can create more harmonic designs. Alongside this, using the golden rectangle allows for consistency in a design’s proportions across different screen sizes!

You may recall the golden rectangle from the previous section of this article. As a reminder, a golden rectangle is a rectangle with side lengths in the ratio $$φ: 1$$. When a square section is removed or added in, the resulting shape is still called a golden rectangle. We can add or remove squares infinitely.

Taking a deeper look at several well-known logos, the ratio’s influence is clear: the National Geographic logo in particular is just a golden rectangle.

Here’s something else cool: if we draw a quarter circle in each square, from one corner to its opposite corner, we end up with the golden spiral! This spiral maintains the $$φ: 1$$ proportion as it grows outward.

If we draw a full circle in each square, we have circles that follow the $$φ : 1$$ ratio in balanced proportion!

Going back to our logos, the Apple logo consists of perfect Fibonacci circles, while the bite is positioned to create a golden spiral. And the Twitter logo’s a case where golden circles will align perfectly at every curve!

Logos designed with the ratio in mind are balanced and proportionate——despite their simplicity (or maybe because of their simplicity), they are still compelling!

But just using the golden ratio does not magically create a good logo. Designers also use other tools in order to make these logos stand out. Apple’s logo is a clean, uncluttered silhouette, making it easier for us to process and remember. Since it stands on its own without text, it’s able to transcend language barriers and global markets. The Twitter logo’s calm blue, on the other hand, goes back to color symbolism: subconsciously, it’s associated with trust, calmness, and intelligence. National Geographic’s logo plays around with symbolism as well: the iconic yellow rectangle is a window to new science, nature, and discovery!

Many artists use the golden spiral as a composition technique, leading the viewer’s attention to the focal point (to learn more about this technique, check out the article Composition 1). However, the golden ratio is just one of many tools artists use to evoke a reaction or emotion in the viewer!

Taking a look at the above illustration, the golden ratio’s influence is clear, with the bright pink spiral ending at the character’s hand. But the artist also uses value and contrast to draw the viewer’s eye: the hand is at a much brighter value compared to the rest of the piece, naturally standing out against its surroundings. Additionally, the character’s expression, alongside the bright neon colors, provides the viewer with a sense of unease.

My illustration below uses the golden spiral through the butterflies’ movement, leading the viewer’s eye to the dragon in the background. Furthermore, the viewer feels engaged and excited when looking at this scene due to the vibrant, saturated hues, the use of bright lights, and the character’s expression and pose. The dragon, at a lighter value compared to the frontal cars and buildings, is also highlighted as a focal point due to the contrast. Its bright orange hue stands out against the blues of the sky and buildings, due to my choice of complementary colors.

Ultimately, aesthetic appreciation emerges from interactions between science, mathematics, culture, and individual experience. The golden ratio may influence how we perceive balance and harmony, but human creativity extends far beyond any single equation.