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Piet Mondriaan painting

Boston City Hall

Van Koch snowflake algorithm

Real Fern leaf

Fern leaf recreation from fractal algorithm

Beauty

*By Eduardo Gutiérrez Prieto*

Biologists and neuroscientists already have a good answer explaining human-to-human attraction. Our genes push us to find a healthy partner with high chances of survival when we mate, so our descendants will inherit those traits and will thus be able to pass on even more copies of our genes. The fitness of a candidate can be somehow guessed from its appearance, therefore, what we call a beautiful face, is nothing else but the accumulation of features that makes our brains think that person is fit for survival. And if you don’t believe it, just look at how the societal standards of beauty have changed over the years. In medieval times, the top models depicted by famous painters were most times overweight for nowadays standards and they all had very pale skins. These features were indicative of very high chances of survival, as being fat and pale meant easy access to food without the need to labour the fields. However, when looking at the current western standard of beauty, they are basically opposite. What is rare and indicative of high socio-economical status, and thus of beauty, now is the capacity to eat expensive nutritious and low-caloric food and being tanned as a consequence of being able to spend time outdoors.

It is hard to extrapolate this reasoning to justify why we find beauty on the infinite branching of a tree, or on the perfectly ordered geometry of a snowflake. The reason surely lies in our biological complexity but it is not clear how. Indeed, the closer scientific branch to explain it is the one that explores complexity at its peak, Chaos and its study of fractals.

Before going into the mathematical complexity of chaos, we can perform a much more down to Earth approach by looking into the evolution of art, the field that throughout history has studied the perception of beauty the most. Different painting styles have evolved throughout the years, starting even before the appearance of Homo Sapiens with schematic representation of hunting scenes on cave walls. The painting styles were refined through ancient history aiming to perform an ever more accurate representation of reality, finding its summit at the mid XVIII century with the naturalism, or realism, style. However, it is around this time that photography was popularized. It thus made no sense for artists to aim for an accurate representation of reality. This is when, and why, the abstraction of painting started to occur. Abstract artists’ goal was to destil those elements that are indispensable to evoke human emotions and awaken the feelings of beauty.

Going down the road of abstraction we find the Dutch painter Piet Mondrin and his celebrated drawings of colored squares with which he aimed to capture some essence of reality and transmit beauty. His artistic talent to abstract the world into a bunch of straight lines and colors is undeniable. However, despite being celebrated, his paintings are not widely understood. Showing his canvas to a group of untrained laymen who have never heard of Piet Mondrin, some may find the drawings appealing but most will probably not think they are precisely beautiful and feel a jolt of joy. The same applies to one very special architectural style: Brutalism. Brutalism is characterized by the massive use of concrete in very straight geometries, the application of the simplification of shapes to structures. It is the style of the *Aula building*, one of TU Delft’s prides. Despite how technical and hard it is to design in this style, it is crystal clear that a big sector of society doesn’t like them. There is even a social movement to demolish the Boston City Hall building for how horrendous and disruptive it seems to most people.

Piet Mondriaan painting

Boston City Hall

Looking into art we see then that the road to abstraction of nature into simple Euclidean geometries has not led to identifying the key elements that make humans find beauty in something. In order to find answers we may better look now into nature, where there is little doubt that true beauty is found. Very few people would say they do not find pretty the infinitely branching of trees in a forest, the rough skyline of mountainous chains or the delicate geometry of a snowflake. These apparently very complicated geometries seem to widely induce a sensation of beauty when staring at them. They are surely far more recognized as beautiful than colored squares or massive concrete blocks.

For a person without any background in mathematics, the common observation on the branching of a tree, the delicacy of a snowflake or the myriad peaks of a mountain may share nothing in common. A trained mathematician, however, can see a clear pattern. They are all highly fractal geometries.

Does this mean that we find beauty on fractals? Maybe. If you ask your mathematician friend to explain to you why Mathematics is beautiful, she will likely show you the Mandelbrot set. The never-ending, never-repeating spiraling down of the set is simply mesmerizing and few will find it unappealing. Indeed, Thomas J. Watson, as an IBM researcher, financed the expensive calculation of an ever-finer structure of the set by selling to the public the beautiful images he was able to calculate.

It is clear then that most humans find beauty on fractals. But the question again is, why?. In order to understand it, we need to understand what fractal geometry is.

A fractal geometry is a geometry with a non-integer dimension. A line has dimension 1, a square dimension 2 and a cube dimension 3. A geometry with a dimension 1.5 is somewhere in between a line and a square. That is, it has a finite area, but an infinite length. Think about the island of Ireland. You can draw a circle in which the island is enclosed. The area will be therefore bounded by the area of the circle, it will be smaller and never greater. However, think about the perimeter of its shore. It will depend on the ruler you use to measure it. If your ruler is 1km long, you’ll be filtering out a lot of small features of the coast. Thus, if you instead take a 1m ruler, you’ll be able to capture many more of these features and your measurement will soar. Now imagine you can make that ruler infinitely small, what will be the measurement?. Mathematically, the answer is infinity. That is, the shape of Ireland is fractal with a dimension between 1 and 2, i.e. with finite area and infinite length.

This may seem to be completely irrelevant to our quest for beauty, but believe me, it is not, we are getting there. Stay tuned.

One of the amazing properties of fractals is that, despite their mesmerizing complexity, they can be generated from very simple algorithms. Take an equilateral triangle. Take the central third of the length of every side out, use the gap as the base for a new equilateral triangle. You should obtain a David’s star. Then repeat the action ad infinitum on every side of the geometry, and there you go. You have a snowflake, the so-called Von Koch snowflake. It is simply astonishing how such a complicated shape can be generated from such a simple recipe. What is even more astonishing is that it is possible to easily recreate leaf, branching and other natural shapes by simply adding a probability distribution to the recipe. That is, throwing a dice and using its score to influence the recipe. Imagine you are drawing a line, the score would command you whether to make a turn or not. With those kinds of simple additions, you can recreate naturally occurring geometries.

Van Koch snowflake algorithm

Real Fern leaf

Fern leaf recreation from fractal algorithm

It has been indeed mathematically proven that given a somewhat fractal geometry, it is possible to extract an algorithm to recreate it. The more fractal it is, the simpler and more accurate the algorithm will be. Take for instance the case of a leaf of fern. It is undoubtedly fractal. The algorithm developed by Michael Barnsley to calculate and plot a fern leaf takes less than 30 lines of code! This means that what we see as beautiful complicated geometries is nothing but simplicity.

Let’s now turn our attention to genetics again. Ferns reproduce asexually, meaning they are part of a lineage that appeared even before natural selection discovered the potential of sexual reproduction. This is an indicative of simplicity. As a rule of thumb in biology, the simpler an organism is, the longer ago it probably evolved. For this very reason, it makes a lot of sense for the fern simpler DNA to encode its structure in a very short, highly fractal code. It is an astonishingly efficient way to store the structural geometry of the plant.

Trees are more complex than ferns, and indeed they are not so purely fractal. The algorithms for their structures are more complicated as tree genetics have become more complex with the exploitation of sexual reproduction among other things. Animals are even more complicated than trees, and thus, their body plans are much less fractal. Summarizing, the level of fractality can thus be understood as a level of how complex an organism is. Commonly, the more complex an organism is, the higher it will be in the food chain and the more chances it will be your predator.

It thus makes sense for us, Homo Sapiens, to try to stay away from the more complex organisms. That means that from an evolutionary perspective it is sensible to look for fractality to rest, reducing the levels of adrenaline and increasing the levels of oxytocin and serotonin in our bloodstream. A hormonal state that is linked to feelings of happiness, mindfulness and relaxation. Exactly the kind of feelings we face when we stare at mesmerizing landscapes of incredible beauty.

The theory defended in this article thus suggests that us considering something beautiful, and the attached feeling of happiness we feel, is nothing but an evolutionary strategy that kept us alive when we were roaming in the savanna and foraging in the forests. It allowed us to escape from predators by making us feel joy when surrounded by highly fractal geometries.

Beauty is fractal.