“Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.”
― Bertrand Russell, A History of Western Philosophy

In 428 or 427 BCE, either in Athens or perhaps on the nearby island of Aegina, Aristocles, son of Ariston, of the deme of Collytus was born. You probably know him better by his street name: “Plato” (given to him because of his big-assed, humongous forehead). Ancient bloggers suggested that his mom, Perictione, was a virgin and that Apollo was the baby-daddy. They said that on the day he was born, the bees of Mount Hymettus landed on his tiny newborn lips and dripped honey into his mouth.

Whether or not Plato enjoyed such a Disney-level birth experience, he did indeed live an amazing life. He grew up in a very rich and politically powerful family. This didn’t protect him from the draft, however. As a teenager he fought along with his fellow Athenians on the wrong side of history, crossing swords against the Spartans in the Peloponnesian War.

Sometime in his twenties his forehead grew another inch or two when his mind was blown by the experience of meeting the great pederast philosopher Socrates. Almost immediately, Plato became “woke” to philosophy and proceeded to study under Socrates for eight years until Socrates pissed off some Athenian politicians and was forced to kill himself with a cup of hemlock tea.

Socrates’ death crushed Plato. He sadly wandered around Greece and the Mediterranean for over a decade, visiting wise men, priests, and prophets until (as some legends say) he was captured by pirates and sold into slavery, only to be purchased and released by a friend (this tale was somewhat ironic as Plato owned five slaves himself at the time of his death).

Sometime after these tragic adventures, Plato established the first Western university (the Academy) and proceeded to educate some of the greatest (and richest) minds of Ancient Greece, including Aristotle (who subsequently educated Alexander the Great who went on to conquer most of the 4th century BC Western world). Throughout the remainder of his lifetime, Plato continued to teach and developed many thought-provoking ideas that would become the foundations of modern Western philosophy, as well as many that weren’t so good.

By now, you’re probably wondering what-the-heck does Plato have to do with anything on Truman’s blog? Well, today, with your generous patience, I’d like to share a couple of short (if not poorly studied) musings on a few of Plato’s ideas that I believe abstractly connect to, and may somewhat inform our vague understanding of the fractal world. 

Plato’s Theory of Forms

The serendipitous modern marriage of inexpensive desktop computers and complicated mathematics has birthed an intriguing window into the weird world of fractals. Today, we’re privileged to enjoy a virtually endless variety of visually entertaining images and videos of two- and three-dimensional fractal forms sculpted from the colorful quantum foam of countless recursive digital decisions. The resulting repetitious elegance of these fractal landscapes and objects suggests worlds beyond our innate understandings of beauty and reality.

On the topic of reality, Plato had a relatively offbeat perspective about this world in which we exist. He believed that reality is composed of two realms: a real world of tangible things, and a world of intangible and eternal ideal “Forms”. In Plato’s theory, everything that we see, touch, eat, drink, or cuddle up next to is just an imperfect copy of a true, ideal Form. These Forms are the archetypes of reality and therefore are superior to the copies of the world we experience.

Plato believed that these ideal Forms exist beyond space and time and that everything within our reality is an imperfect instance that “participates in” the greater Form. In other words, somewhere out there, beyond time and space, there exists the Form of a perfect version of this blog post. This, unfortunately, is just an imperfect copy (that’s trying really, really hard to participate in the Form of a perfect blog post). 🙂

So how does this relate to fractals? Let’s look at the theory itself, and then at an argument against it.

In and of itself, Plato’s theory implies an infinity of imperfect potential copies existing because of the very existence of any unique Form. If say for example, there exists an ideal “Circle” Form, then accordingly there must exist countless potential imperfect variations of “Circle”, right? To this end, as we explore the endless tapestry of any fractal image, we see repeated variations of the same shapes and forms, each uniquely warped into different versions of the one originating shape. As Plato would say, each fractal shape “participates in” the greater Form of the fractal. This definitive self-similarity of fractals presents a paradox to the perfection of Platonic Forms, as problematically the ideal fractal Form (by definition) is self-possessed of its imperfect copies (therefore how can it be perfect?)!

This paradox allows a good segue into the most popular formal argument against Plato’s Theory of Forms: the Third Man Argument. The logic of the argument can be illustrated like this:

Plato’s theory says that if a set of things have a quality of “Round”, such as ball, egg, planet, or my big belly, there must exist an ideal Form “Round” by virtue of which we comprehend these things as “Round”. The Third Man Argument says that because “Round” is like ball, egg, planet, and my big belly, “Round” therefore resembles ball, egg, planet, and my big belly. Therefore, “Round”, ball, egg, planet, and my big belly are similar and require the existence of another ideal Form to participate in (perhaps “Rounderness”?).

I know it sounds really convoluted, but the basic concept of the argument is that Forms can be seen as the same as their copies, and therefore require new Forms for this new group. Through this logic, an infinite growth of new Forms are created, each composed of the ideal Forms that came before it. This argument is fractal in its inherent self-similarity, as each new Form possesses all prior Forms, but is unique in its own perfection.

There’s fractal flavoring in both the theory and the argument against the theory. Ultimately, however, the reality of every instance of a fractal form is informed by the ideal perfection of the mathematic formula that creates it. Of course, this raises the question of “is the formula a Form, or just a generator of imperfection? “ And “does a fractal even exist without a formula to draw it?” That’s a bit too philosophical for my tiny forehead, so let’s proceed onward.

Plato’s Allegory of the Cave

If you’re unfamiliar with Plato’s allegory, it goes something like this: imagine that there exists an immense underground cave somewhere (say beneath Disneyland) wherein a group of very unfortunate kidnap victims have been imprisoned and chained for the entirety of their lives. They have never experienced anything except the stale emptiness of the cave. They can’t look left, right, up, or down because they’re bound by their legs and by their necks in such a manner that they can only stare straight ahead at the empty wall in front of them. Kinda sucks, huh? Can you imagine the depth of depravity required to build and maintain such an inhuman scenario? I can’t.

Continuing onward, not far behind the captives a fire blazes and illuminates the cave with a powerful, if not eerie glow. Upon the empty wall in front of them, they can see their own wavering shadows.

Halfway between the captives and the fire there lies a raised perpendicular walkway across which, for some unknown and bizarre reason, strange people walk back and forth periodically. Attached to the side of the walkway is a wall that is just tall enough to hide the people as they walk back and forth. Like puppeteers, these strange people make their way from one end of the walkway to the other, carrying assorted wooden and stone objects over their heads. The objects are shaped like statues and animals, and like the chained inhabitants themselves, the shapes of the various objects create blurry, moving shadows on their wall.

As the shadows pass across the wall, the captives give them names. As the strange people on the walkway speak and their voices echo through the cavern, the captives believe the voices belong to the shadows. The echoes and shadows are the only reality the captives know.

So, having presented this bizarre scenario, Plato rubbed his humongous forehead and went on to ask a lot of interesting philosophical questions speculating what would happen if one of the chained inhabitants were to be released and disabused of their shadowy understanding of reality. How would they react upon seeing what existed beyond the shadows? How would the other prisoners react when the newly enlightened inhabitant tried to share their new knowledge and understanding of reality? Unfortunately, for the purposes of this thought exercise, I’m going to leave you hanging and end the allegory here (feel free to Google “Plato’s Allegory of the Cave” if you want to catch the rest of the story).

Once again, what does this have to do with fractals? Well, I believe that a connection can be drawn between the plight of Plato’s captives and the modern visualization of 3-dimensional fractals.

First, let’s look upon our mental wall at the recent past. In 1979, Polish-born French mathematician Benoit B. Mandelbrot discovered the first, and perhaps most famous fractal, the Mandelbrot Set. Mandelbrot’s iconic fractal has been described as the “Thumbprint of God” because of its infinite detail and chaotically beautiful color patterns. This intricately weird, bug-shaped 2-dimesional diagram spawned an entire generation of fractal enthusiasts, who, chained to their VGA screens, explored its mysterious depths with abandon.

Digging deeper into the mysteries of this flat, endless world, programmers developed new formulas that generated new varieties of bizarre, yet equally repetitious shapes, all consistently projected upon the flat wall of our collective reality. Tip-toeing across the virtual walkway of mathematic discovery, they brought in the non-spatial dimensions of time, color, and perspective to warp and add variety to what we saw, but in the end, we still simply experienced fractals as wonderfully weird 2-dimensional visualizations. To us, the uninformed viewer, the 2-dimensional fractal was the only reality we understood because that was all that we saw.

Time zipped by, computers got faster, smaller and cheaper, and in 2009 Daniel White and Paul Nylander discovered the first three-dimensional manifestation of the Mandelbrot Set, the Mandelbulb. This was the event that broke the chains off of our collective entrapment and enabled us to “see the world beyond the shadows”.

As we turned our heads and were disabused of the flat limitations of the Mandelbrot Set and its other 2-dimensional shadow puppets, we came to understand that 2-dimensional fractals are merely cross-sections of 3-dimensional fractal entities. As we step closer to flame, we begin to grasp that 3-dimensional fractals are also cross-sections of 4-dimensional forms. And as we step further toward the exit of the cave are we not still chained to the 4K screen of our collective 3D fractal understanding, waiting for the arrival of new technology that will release us and disabuse us of our current sense of fractal reality, and envisage us with wonders unknown?

Conclusion

Thank you for your patience.

I think Plato was intriguing, and in turn, would have been quite intrigued by the modern world of fractals. I can even visualize him standing on the steps of a Starbucks in his ruffled toga, furrowing his big-assed forehead as he sips his latte and waxes philosophic about the fundamentally fractal nature of humanity – that by our self-similar reproductive nature, we may be the ultimate organically temporal fractal.

Hope you enjoy the videos.