The magical mathematical beauty of the Mandelbrot set

A special friend, Rob Yost, sent me the following message, and I wanted to share it with you. It's something to ponder and marvel about.

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First of all, just go to google, search for mandelbrot set images, and scroll through the myriad of incredible pictures that will result.

Do it.

Really.

Amazing, huh.

But where do they come from?

Imagine a photo of the Earth taken from space. You can see the whole Earth in a few square inches of paper. But to really understand the interaction of mass and energy that's going on at the interface between the Earth and the space around it, you have to zoom in... Zoom in close enough to see the plants and the animals and the rocks and the trees that are all happening just at the surface. Zoom in on the interior of the Earth, and it's all dark and boring. Zoom in on an area of space just a few thousand feet above the surface, and it's all just boring space... but all the action happens in those few feet where the Earth meets the sky--and that's how it is in a Mandelbrot set as well. As you zoom in closer and closer, it quickly becomes clear that you will never be able to comprehend everything going on within those few square inches you hold in your hand.

So you zoom in close enough to see the surface of the Earth, and you can kind of understand the life that's happening there, but if you really want to understand life itself, you have to zoom in even closer, to the structure of the organisms, to the systems and organs that make them up, and even to the level of the cells which make THEM up. But to understand how the cells work, you have to zoom in even closer, to the level of molecules and atoms, and within them, to the level of electrons and protons and neutrons, and within them, to the gluons and quarks, and within them, to... ???

Or think about the length of a coastline. It seems easy to measure a coastline in miles when you view it from a satellite or even from an airplane. But get closer, and what appears from a distance to be a roughly straight line becomes more and more infinitely curved--and therefore longer--the closer you get to it. Eventually you get down to the water curving around each individual grain of sand, such that the actual length of the coastline--the interface between water and land--becomes almost infinitely long. And so it is with the "coastline" between the Mandelbrot set and the space around it.

I will try not to bore you with too much technical detail, but the Mandelbrot set is a graph of a mathematical equation. You see all those pictures of what looks like a big turtle or maybe a beetle pointing to the left? That is the whole of the Mandelbrot set--the picture of the Earth from space. You see how inside it is dark and boring, like the Earth, and the space around it, though showing some color variations, is largely just boring space, like the Earth? All of the other pictures you saw on google come from that fine line of interface between the "earth" and the "sky" of the Mandelbrot set. Zooming closer and closer to that area of interface reveals an ever-deepening level of detail that is even more rich and infinite than the Earth itself.

To demonstrate this, I will show you a few youtube videos that take you on a "zoom journey" into the infinite complexity of the Mandelbrot set. You will see that there are some structures which share a basic similarity of form, like a certain "species" of plant on Earth, yet every individual expression of that "species" is unique, much like life on Earth. The idea that you are continuing to zoom inward can almost get lost in the psychedelic show that is captured on the videos.

Keep in mind that back when I first learned about Mandelbrot sets in college, like 25 years ago, we didn't have computers powerful enough to achieve nearly this level of zoom. In fact, as you approach the end of some of these videos, you will be looking at such an infinitesimal level of detail that it would take a computer screen spanning the width of our solar system, our galaxy, and even the entire universe itself (!) to show the entire Mandelbrot set at once, at these extreme levels of detail.

Just watch, and wonder...





Now watch--and listen to--this one, and think about how the Mandelbrot sets represent us as being made in God's image, each of us a feather, but only an expression of the Great Feather, each with our own individual aura of decorations, but still connected to the whole.



Along the same lines...



And one more for good measure...



Are you starting to understand why the Mandelbrot set has been called the most complex object in mathematics?

Ok, so I told you that the Mandelbrot set is a graph of a mathematical function. Has your mind started to wonder how many endless hallways lined with chalkboards it must take to contain a mathematical function of such incredible detail, powerful enough to account for this entire universe filled with sparrows and beetles and algae and bacteria?

Well--and this is the part that is so profound that it brings tears to my eyes--the mathematical function responsible for the infinite detail of the Mandelbrot set can be handwritten on the back of a postage stamp:

Z(n+1)=Z(n)^2+c

That's it. I won't bore you with details (you can look it up on wikipedia if you want them), but all of this complexity springs from the fount of that simple little calculation. I have to wonder if our entire universe itself doesn't spring from a similarly simple idea in the mind of God as well.

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Thank you, Rob!