Why is the golden ratio important (golden ratio explained)

The “golden ratio” is something most people have heard of… but why is the golden ratio important? This number arises in art, design, biology and other, more unexpected fields.

The golden ratio appears as a byproduct of certain situations. These certain situations include anything that produces numbers in the Fibonacci sequence. This happens a lot in nature, so keep reading to see why.

When using the golden ratio, the resulting proportions are pleasing to the eye because ancestrally we see the healthy organisms and processes in nature follow more exactly to these proportions than diseased, perturbed or mutated organisms and processes. Shortly, we’ll see why the golden ratio is important in so many different fields.

Is the golden ratio important historically?

Pentagons have a lot to do with the golden ratio, which you can read more about here. From first principles of geometry, the ratio was originally emphasized in studying regular pentagons over 2400 years ago. Euclid and Plato (300s BC) expounded on the golden ratio found in geometry. After multiple Greek and Roman scholars introduced, described, and analyzed the golden ratio, it was readily applied to art and computing. In art, the golden ratio makes paintings and sculptures more realistic. Is it a coincidence that in the centuries following Euclid there was a boom of realism in European art?

Of course Europe isn’t entirely the beginning of the story, just the beginning of the mathematical formulation of the golden ratio. Even in middle east we don’t see it written as early as Euclid. But in Africa we see the self-similar principle of the golden ratio used in building.

Much later, scientists like Fibonacci (1200s) and Kepler (1500s) were able to add more to the story. And the story of the golden ratio cropping up in various fields is one that continues to this day’s research. Statistical software and machine learning algorithms sometimes use the this ratio to mimic biology to provide more efficient algorithm and more realistic statistical models.

Any time you have pentagonal symmetry the golden ratio appears. Undeniably, that’s why we see it both in solid-state critical structures and quantum condensates (at the extreme micro level) and the structure of galaxies (extreme macro level). Furthermore, dodecahedrons are a 3D form with pentagonal faces. Many structures in nature form dodecahedral shapes to optimize for space packing and other advantages. Shungite, for example, has dodecahedral fullerenes in its structure.

[The dodecahedron is the shape] “which the god used for embroidering the constellations on the whole heaven”

Plato, Timeaus

Scientists that do cutting edge research on emergence theory seem to always try to slide the golden ratio into their equations to see if it makes things “work”, and sometimes it does! (Kinda.) Even in peer-reviewed research. There was a popular expression famous physicist Paul Davies used in deriving the change in specific heat of black holes, in which the golden ratio arose. However, many people noticed this wasn’t a meaningfully significant quantity in terms of physical properties. (Paul C. W. Davies, Thermodynamic phase transitions of Kerr-Newman black holes in de Sitter space, Classical and Quantum Gravity 6 (1989), 1909–1914.)

Any new or existing field with numbers… the golden ratio seemingly fills in the gaps somewhere. The golden ratio is definitely integral to close-packing problems and morphology of emergent structures in organisms. But people unarguably trying to put it where it’s not, showing renaissance art overlain with the common rectangle sections which are a terrible approximation… so what’s the ACTUAL proven significance of the golden ratio?

Some people like to find close approximations of the golden ratio like a treasure hunt. Numbers considered “close enough” are even found in the bible, in the instructions to build the ark of the covenant.

The importance of the Golden Ratio is continually being added upon and as well as called into question. Some examples of the golden ratio in art can be greatly exaggerated. We’ll talk about the deliberate versus incidental use of the golden ratio, but first let’s get familiar with what its value is.

What is φ?

The most overt and fundamental appearance of the golden ratio happens when analyzing a pentagram star. In setting up a ratio of any “long to short” lines in the picture, the result, after working out algebraically and with geometric proofs, give [1 + √(5)] / 2. Since √(5) is irrational, this value is irrational as well. The proof can be given using high school level algebra and geometry, but takes a while to demonstrate, steps-wise.

But this ratio has a feature more general, extending outside the pentagram concept. The ratio of the large part to the whole is the same as the small part to the large part. This is an extremely important concept in the study of fractals, called “self-similarity.”

The most important concept to internalize if you want to understand and recognize the golden ratio is this “the part to whole is the same ratio as the part to the smaller part.” Keep reading to see this explicitly applied in golden spirals & golden sections.

This “self-similar” concept can be applied when something needs to feel raw and natural in art and design. For example, in Feng Shi, the grid overlain on the room (called a bagua), can be overlain onto the whole house and the whole property, as well as individual areas of a bigger room. The more levels and layers and considered, the stronger the results of organic energy flow will be achieved in the house (i.e. qi).

How are Fibonacci sequence and the Golden Ratio related?

You’ll see some people using “Fibonacci sequence” and “golden ratio” interchangably, so allow me to iron it out. With regard to the Fibonacci sequence, a sequence is an ordered list of numbers. There is a rule that allows you to generate the next number. Fibonacci developed the formulation of the Fibonacci sequence while postulating on various phenomena in macro-biology, particularly predicting the number of rabbits in a population.

Fibonacci Sequence’s Rabbits

The situation is very specific when applied to rabbits, and only roughly approximates reality, but here are the rules. You start with one pair of rabbits that breed and create a new pair of rabbits each month. Once a new pair of rabbits is born, a month must pass before they can reproduce.

The Fibonacci sequence has to be seeded with the first numbers (the first pair of rabbits). You can not start it from nothingness. It must start with two 1’s. The next number adds the two ones to get 2, next adds the 1 and the 2 to get 3, the 3 and the 2 to get 5, and so on. Fibonacci sequence is not the same thing as the golden ratio, as it is a list of ordered numbers by which you can always find the next and “keep going.” The golden ratio is a NUMBER, referred to by the symbol φ, a ratio of one thing divided by the other.

φ as a number

The first few terms: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 …

You can add 144+233 to get the next term, 377 and so on and so forth.

If you take a term in the Fibonacci sequence and divide it by the term before it, you get an approximation. The golden ratio is the limit of what the ratios of consecutive terms in the Fibonacci sequence give. As you divide the two terms farther and farther out, the value of the golden ratio gets more exact. φ is irrational, like pi. So as far out as you go the decimals will never terminate.

• 3/2 = 1.5 (slightly too low)
• 5/3 = 1.66… (slightly too high)
• 8/5 = 1.6 (lower)
• 13/8 = 1.625 (higher)
  … and so on.

The golden ratio, being intricately connected to reproductive processes in nature, and anything with 5-fold symmetry, provides another tool for scientists and statisticians to use. It has been found to be helpful in stock trading and in search algorithms, at least, and often mathematical problems involving chaos (where fractals often crop up) pose a solid opportunity to slide the φ in to see what results it gives.

Golden ratio	The result of the problem “long/short = short/(long-short)” (Formulated from the geometry of a pentagram.)
Golden mean	The convergence value of the Fibonacci sequence, the limit as terms progress (“to infinity”) being equal to the golden ratio
Golden section	A portion cut so it follows self-similarity (“long/short = short/(long-short)”. It could be a section of a circle, a line segment, or something else.
Golden rectangle	A rectangle with the long/short side lengths giving the golden ratio.
Golden spiral	A logarithmic spiral with a growth factor of the golden ratio. the function would look like f(x) = φx

Golden Spiral & Golden Angle

Now let’s tease out the “golden spiral.” Out of all the golden things we could describe, the golden spiral is particularly illuminating. In math, a spiral is best described in terms of angles and radii. In differentiating types of spirals mathematically, spacing between rounds is an important characteristic. The “growth factor” appears in the equation of a spiral to determine how much wider the shape gets with every twist. A golden spiral is simply a logarithmic spiral where the growth factor is φ.

Now you can see how this comes into play for the tentacles picture above. Depending on the example, the golden spiral can be subtle or overt. But it is almost always to be expected in nature. The growth pattern could be plus or minus φ, giving approximately 135 degrees or 222 degrees.

“The position of each new growth is about 222.5 degrees away from the previous one, because it provides, on average, the maximum space for all the shoots. This angle is called the golden angle, and it divides the complete 360 degree circle in the golden section, 0.618033989”

Sinha, in this article

Golden Ratio in Nature

The golden spiral is very obvious in many plants and mollusks. The most blatant place to find it is in the unfurling or flowers or plants like succulents that have clear symmetry. Here are examples you can plainly see.

Mammals	fetus shape; paw shape; cochlea in ear; pineal gland
Mollusks & lower animals	nautalis; radiolarians; starfish
Weather & Space	hurricanes; tornadoes; distances between planetary orbits; spiral galaxies
Chemicals	fullerenes; quantum condensates
Plants	flower petals; leaf, seed and thorn placement around the stem

But even outside these displays of the golden spiral, the property often lies in many different parts of every plant. For example, in pines, we see it in the pinecones, we can count it in the number of pine needles in every bundle, and we can also compute it the the proportions of the branching structure. Nearly every branching plant follows the golden section led branching structure. And, nearly every plant with leaves follows the “rotate-by” regime in the leaves on the stem. See my list of golden ratio in plants examples and my list of golden ratio in animals examples for more examples.

Importance of the Golden Ratio

Why does nature “choose” this ratio? It is just a close-packing problem on a rotating backbone. Nature through genetic evolution wants to optimize certain values. Let’s use a sunflower as an example, in which the golden ratio appears in the pattern of seeds on the head. Nature wants to be able to put as many seeds as it can on that head, but it also can only split off new cells from existing cells. So the golden ratio happens to solve the problem and we get to see nature’s computation of this in the arrangement of structures.

Artists can use this as a starting point to make something that feels realistic. Say I want to draw a tree that has eyeballs on the branches. I can’t find a picture to show what I want to draw because this doesn’t exist, but if I follow the natural order trees do I don’t have to copy a reference to create something that feels realistic. This is exactly what Da Vinci studied with regard to the proportions in the human body, which by the golden ratio are highly self similar just like nearly all other natural phenomena.

We see the golden ratio used in architecture like the pyramids, the parthenon, and nearly all ancient religious sites. While the golden ratio in architecture is the subject of its own article, you may come across works of art and wonder if the golden ratio is involved. There are a number of tools for determining so. It can become important to really measure because in some circumstances, people really stretch the boundaries of what counts to try to claim the golden ratio was used. A golden caliper can be used to measure distances in real life, and softwares such as PhiMatrix can determine the presence of the golden ratio on the computer.

Psychological importance	Represents vitality and abundance Naturally calming, soothing and stable to the mind (because of beauty)
Biological importance	population growth, morphology of structures
Chemical importance	fullerenes, dodecahedrons, quantum condensates
Physical importance	is suspected in gravitation, electrodynamics, statistics in chaos, brownian motion/fluid diffusion
Artistic importance	creates realistic images

So now you’ve at least scratched the surface of the infinite rabbit hole that is the golden ratio, and can see why the golden ratio is important. The golden ratio, fractals, and sacred geometry are all topics I am striving to explain in depth on my blog, so feel free to leave a comment and let me know if you have any questions about the importance of the golden ratio.