I'm still thinking how sense of magnitude gets confounded when we look at certain shapes. Look, for example, at these pictures of mountains and how it’s difficult to tell how big the mountain is when you only have the surrounding rocks to compare to. The effect is shown more starkly in a previous post which showed Base Camp of Everest.

This is a classical example of a property called self-similarity. As you zoom in or zoom out, the complexity of the rock’s shape seems to be preserved- the shape of craggy cliff is copied again at different orders of magnitude.

I was talking to a friend of mine about this a few weeks ago, and he pointed another example of another convergence at two different orders of magnitude- this picture of dried mud compared with the zoomed in pictured of dried blood.

What’s interesting here is that in the blood and the mud, the cracked surfaces look almost indistinguishable. My friend pointed out that both would have begun as viscous liquids and the physics of their drying probably overlap and follow the same differential equations. This comment made me think about the concept of the attractor, and how the look of the cracked surface (that also looks like a birds-eye-view of a ravine) may fulfill the definition of being an attractor of both drying blood and mud.

So let’s talk about attractors. They are properties of a system that compel it to move in a certain direction, no matter what the starting conditions of the system are. A kinda-cool-but-not-really-cool example of an attractor is the point at the bottom of a damped pendulum system where the pendulum eventually stops after all its energy is dissipated. An actually-really-cool example of an attractor is the Sierpinski Gasket, a fractal that is wholly described by something called an Iterated Function System (or IFS). An IFS, basically, is a set of instructions that tell you how to manipulate an image over and over and over again, for an infinite number of times. Let me give an example. Let’s start with a cat as our initial condition (can you tell I drew it myself?):

Now, my IFS is a set of instructions that says something like, “Take your cat, make three copies, shrink the copies down to a quarter of the original size, and arrange the copies in the shape of a right triangle” Like so:

Now the IFS says to do the same set of operations again-

and again

and again

onto infinity

The shape we get at the end is this very interesting fractal shape called the Sierpinski Gasket. The fascinating thing is that there really is no trace of the underlying cat we started with at all. Really, we could begin with any starting image, whether it be a cat, or a house, or a pile of dirt, and by applying the IFS to it, we would end up again at the Sierpinski Gasket. Like an incredible infatuation, we can’t help end up at the Gasket, no matter how dissimilar our base image is from the final fractal. That’s why the fractal is the attractor of the IFS!

The Gasket also exhibits striking self-similarity. Zoom in on any part of the Gasket, and the entire gasket will appear again in the tiny part. You could never actually tell how deep you were in the fractal, because each level looks the same as the zoomed in or zoomed out picture. Order of magnitude is practically impossible to tell in a fractal.

It’s interesting to think about the fractal nature of Nature. We know that self-similarity is a large part of what makes the size of mountains difficult to judge. It also seems that no matter whether you begin with blood or mud, whether it’s at the scale of a field or on a microscope slide, their drying properties will attract them to the shape of a cracked, dendritic web.

Rather than talk about science in this post, I’m going to talk a little bit about the Zhuangzi, a Daoist text written by a Chinese philosopher (also named Zhuangzi) who lived around 400 BC. For those who don’t know a lot about Daoism, it’s a popular philosophy/religion that began in China that emphasizes the play of opposites, the unity and multiplicity of the natural world, and a fundamentally mystical ontology. Dao means ‘way,’ and within the religion it represents both a kind of ethics as well as a mystical life source from which the world springs from. I won’t go into an in-depth discussion about the religion (You can always read about it on wikipedia) but maybe some parts of it can be intuitively understood in the famous symbol from the religion, the yin and yang.

The black arm and white arm are always pushing on each other, but equally balanced, and each contains a bit of the other within itself. Equally as helpful are the first two lines from the main text of the religion, the Dao De Jing, which go:

“The Dao that can be described is not the absolute Dao; the name that can be given is not the absolute name.”

which has a wonderful self-undermining quality in which the whole text pretty much begins by saying that everything that follows is not the true way/Dao and any attempt to name or describe the way/Dao will not succeed. Dao, then, is forever an incommunicable and mystical concept.

The Zhuangzi is one of the other major texts of the religion besides the Dao De Jing, and is notable because it’s often funny, absurd, and full of word play and talking animals- a unique thing among religious books. One of the main themes in the book is actually about relative magnitudes of space. The opening line of the book beings with a parable:

“In the dark sea of the north there is a fish; it is named the Kun. The Kun is so huge no one knows how many thousand li he measures. Changing, it becomes a bird; it is named the Peng, so huge no one knows how many thousand li he measures. Aroused, it soars aloft, its wings like clouds hung from the sky. As the sea shifts, it turns to set its course toward the dark sea of the south, the Pool of Heaven.”

From the translation’s footnotes, Kun means “roe,” or fish-egg, the tiniest little thing. A “li” is a unit of measure which is about a third of a mile. So already, magnitudes of scale are being mixed and reinterpreted. Since we are humans, we naturally compare everything relative to our own size. Zhaungzi begins immediately by trying to muddle this and get us to think beyond our anthropocentric view, or at least see it in a new light. Things like mountains are big because it takes days or weeks for us to climb up one, while an anthill is small because we can step over one with barely any effort. But to an aphid, the anthill would seem large, and the mountain probably unimaginable.

As a corollary, it’s interesting to think about how with the invention of cars, we can now go up mountains much quicker than we can hike one, and how this probably has changed how big mountains feel to us. Same with the distance between cities. When someone asks you “how far away is Philadelphia from DC?” people often give a response like “Oh, it’s about 2.5 hours away.” This is a very interesting way of responding, as it allows the distance between cities to shrink proportionally to the speed of the transportation that connects the two places.

Anyway, perspective is a big topic in the Zhuangzi. He often writes from the perspective of other animals and plants to get readers to stop making assumptions about the nature of the world that aren’t necessarily true. Big things aren’t necessarily big, it depends on what you’re referencing size with. To every concept of length, there is a concept of a ruler. Likewise, a feeling of time is coupled to the kind of clock you’re using. Old things aren’t necessarily old, and beautiful things aren’t necessarily beautiful. Same with almost everything else. The only thing that seems to be free from this relativism according to the Daoist is the Dao itself, which is conveniently never defined.

Below is an excerpt of the beginning page of the Zhuangzi, continuing from the excerpt I gave above. It’s a fun book to pick up whenever you want to be forced to think from a different perspective.

Little understanding cannot come up to great understanding; the short-lived cannot come up to the long-lived. How can we know this is so? The morning mushroom can understand nothing of the alternation of night and day; the summer cicada can understand nothing of the progress of the seasons. Such are the short-lived. South of Chu one finds a lizard called the Dimspirit which counts five hundred years as one spring and five hundred years as one autumn. In high antiquity there grew a great rose that counted eight thousand years as one spring and eight thousand years as one autumn. Such are the long-lived – yet today Pengzu is the best known exemplar of longevity, whom crowds of men wish to equal. How pitiful!

There is a funny scene from the cartoon Adventure Time where there is “Powers of Ten” zoom to such an extreme degree that the zoom ends up returning to where it began, leading the protagonist to yell “Everything small is just a small version of something big!”. Thinking about the physics of the situation, are similar structures and objects really repeated and copied at different orders of magnitude? Let’s start by considering the fundamental interactions that make up all the objects we can see.

There are four forces that we know of in the universe:

1. The electromagnetic force, which is what you think about when you see a compass needle move, or see lightning, or feel a statically-charged ballon raise the hair off your head.
2. The strong force, which is what keeps the protons and neutrons in the atomic nuclei together.
3. The weak force, which is responsible for radioactive decay, and finally
4. gravity, which keeps us firmly on the Earth’s surface.

Both the weak and strong force only really act on atomic scales and do not extend past the diameter of a nucleus. We don’t see their interactions in our day to day life. The strong force doesn’t have a lot to do with keeping the bricks together in a building or making steel and stone as hard as they are, and isn’t noticeable unless you’re looking at things as small as nuclei. Gravity, on the other hand, has an infinite range of distance, and we definitely notice how apples fall to the Earth and how the Earth orbits the Sun, but gravity is so weak that we need things as big as the Earth and Sun before we begin to see its effect. The gravitational force between my desk lamp and my pencils is completely unnoticeable.

So really, the interactions that we most readily grapple with as human-sized beings are those caused by electromagnetism. The pencil does not fall through the desk because of electromagnetic forces from the atoms in the pencil repelling the atoms in the desk. The pressure I feel on my back and bottom when I sit down in a chair is a similar kind of repulsion. The light I use to see the world is electromagnetic radiation interacting and reflecting off the charged particles of the objects around me. Taste and smell, which are chemical reactions, are exchanges of ions and electrons which are all primarily governed by electromagnetism. My brain and muscles are predominantly electrical impulses and signals.

Planets and stars all really look the same, and that’s predominately the fault of gravity. Atoms are practically/theoretically indistinguishable from each other. But animals, plants, and minerals- their wide difference in form, color, shape, texture and motion is all a product of electromagnetism. What variety! It’s amazing how one force can create such diversity. And yet, even in all this diversity, how do things change as you zoom in or zoom out?

Look at this picture of a mountain. How big are the rocks in the foreground, if you had to guess? How high? 4 feet high? 400 feet high? A mile high?

Do you see those colors close to the base of the picture? Those yellow, blue, and red spots? It looks like trash on the ground, right? Let’s look closer.

It’s actually base camp of Everest. (Here is the website where you can really play/zoom with the image).

What looked like scattered trash is actually the living structures of quite a few hundred people. And those rocks that looked like boulders that might rise up 4 feet or so, are actually cliffs that rise up a few 100. Our eye ( without context) can’t seem to recognize how big something with a ‘natural’ shape (like the shape of a rock) is. So nature really does copy and repeat its patterns at multiple scales of size. It’s a lot like a fractal, in which you can zoom in quite a bit and then return to a copy of the thing you started out with.

So in a sense, sometimes big things are just copies of small things, but it depends on how small or how big you go! If you go too small, like the size of an atom, you won’t see these shapes repeated again. Once you get to the size of a nucleus, the electromagnetic force stops being the only player in the way things are structured, and you need to start taking into account the strong force. Same thing for when you go too big. Gravity plays a much larger role on the scale of planets and stars and will have a similar effect on changing how things are structured.

But for sizes that aren’t as big as planets or small as atoms, why do shapes repeat at different scales? Why can’t I tell if the rock is 5 feet tall or 500 feet tall? It’s a fascinating question! I like to think it has to do with the fact that it’s a product of how the force of electromagnetism likes to align molecules in regularly repeating patters, whether in a crystal or other such structure, and that structure can simply keep on repeating and growing and repeating to any size. Whether that size is a mile or a meter, the underlying, organizing structure is the same, and so it looks the same.

But even with the tendency to repeat the same pattern at different scales, how is it that electromagnetism can allow such diversity of phenomenon? The fact that electromagnetism can create forests and trees and animals, as well as being the source of how we look and feel (consciousness is electricity, if you reduce it enough) while on the other hand gravity only seems adept at making very big spheres— is there something special about the mathematical nature of electromagnetism that gives it the ability to be so myriad?