The End of the Line

Jun 11 · 10 min read

An essay of reflections

This is a story of the search for a rare animal

… a line, in the wild.

The amazing thing about physics that should stop all of us in our tracks and make our jaws drag along the floor for a while is this:

Real world events can be described by simple mathematics.

That is, it is truly wondrous that something as complicated as the real world sometimes is describable by something as simple as addition, multiplication, subtraction, and division. When we begin to realize how many other choices there are to describe anything, this simplicity is stunning.

Let’s break that down a little.

First of all, to be more exact, the sentence should say:

Some real world events or phenomena can sometimes be somewhat modeled by somewhat simple mathematics.

The challenge for non-physicists is that many ideas lie just slightly outside of our everyday intuition. They are built of concepts written in the language of mathematics, daunting because they exist in a mental space that is particularly formal and idealized. If you’ve ever seen a ‘clean room’ — where workers must wear white smocks, and disinfect before entering — math, physics and some other sciences are like that. When you enter, you have to leave many everyday intuitive ways of thinking behind.

Or do you? The story of leaving intuition behind is the popular story about math and physics. In fact, intuition can be very useful, but it must be developed in this particular ‘clean room’ environment. Sometimes this ideal environment is likened to the summit of a mountain, such as the Himalaya, an environment seen as pure and clean of worldly imperfections. This ideal world is populated by rarely seen and beautiful snow leopards.

The first line was perhaps sighted 300-400 B.C.

Let us note that it is not possible to find a line in our everyday world. We can think of many ‘almost-lines’, but can always think of a reason why they aren’t, in fact, lines, in our intuitive understanding of a line as a straight geometric figure:

  • the horizon — it bends due to the curvature of the earth
  • a string with a weight on it — there are little imperfections in the string
  • where a wall meets the floor — also not perfectly straight
  • a line drawn on a piece of paper — same reason
  • etc. many other examples we can think of have similar problems

We start to understand that a line is something that exists in our mind.

The jaw-dropping statement we saw at the start of this article begins to be revealed.

We start to see that the line belongs to the end of the sentence, ‘mathematics’, that idealized, Himalayan clean-room. But wait, we can apply our intuition to it, we ‘know’ and can almost ‘feel’ whether something is a line. Hmm.

And it becomes apparent that our list of ‘almost-lines’ belongs to the left-hand side of the first sentence.

As an aside, I’ll just say that the ‘clean-room’ or mountain-top perspective is an aesthetic. That is, not everyone thinks it is beautiful, and I’m not trying to focus on some kind of judgement about what is good or better, but instead to view it like visiting a museum and appreciate it, and also put it in context. Similarly, the too-worldly opposite of the ‘clean-room’, the swamp or ‘grunge’ is not for everyone either. This is also an aesthetic. Most of us live or appreciate something in between. An interesting thing is that in the case of mathematics, physics, or other sciences or engineering, this aesthetic and its accompanying idealized language have been able to yield tools that actually have had a direct physical effect on the world. (for example, an airplane, or unfortunately, weapons — whereas art and its language, for example, operate on the world through human interpretation — this is a long ongoing discussion!)

So, for something like 2500 years or more, humans have been day-dreaming about lines (and circles, squares, etc.). We know what they are, even though we never actually see them.

And we have tried to make them. We started by building buildings using plum-lines, making yarn, string, clothing, bricks or beams, wells, roads, spears, arrows, and many other things.

When we think about it, a line seems to basically mean ‘distance’, since it is indeed the shortest distance between two points.

In fact, a mathematician or physicist would say ‘it had to be something’!

And as time went on and we realized many abstract things can be described by lines. The path of an object we drop, or a beam of light. (hold your horses about Einstein) We also saw that even curves, if you zoom in enough, are basically lines. So, smooth curves are basically also lines.

(To see this: start with a very gentle curve and zoom in, it will look more and more like a line. Then zoom out, and the gentle curve will look like sharp curve…!)

Then, as we entered the industrial revolution, we got better at making things in lines and tried to make everything that way. Even the economics of large-scale industry relies on lines and thinking about lines, since curves are just lines. This has been an incredibly useful idea.

In fact, the idea that curves are basically just lines has dominated a great deal of our science and thinking through the 20th century. This idea has various names, mostly around the name ‘Calculus’. Again, the basic idea is that a great deal of behavior of physical things can be described by curves, and that, if we zoom in enough, curves look like lines.

One everyday example of zooming into curves to see lines is the horizon. Oddly enough, the horizon is actually not very far away, only about 2.9 miles (4.7 km), you can walk there in little over an hour! But the earth is very big! It has a circumference of approximately 24,901 miles (40,075 km). So, if we walk to a point on the horizon, we walk just over 1/10000 or 0.01% of the way around the world. Unfortunately, we modern Western humans don’t deal with this kind of zooming or scaling in our everyday lives. This is outside our intuition. (why? more on that later)

That is, most of the time, we deal with things on a local scale. Roughly speaking, this means the distance and time of most things we do is pretty similar. For example, it takes me a bit of time to go to work or home, but not 13.5 months (the same as the 1/10000 horizon-earth scale). Similar for eating, having a conversation, etc. Even sleep takes a big chunk of time, but nothing like 1 year for every day of being awake!

This is also mostly true of the size of things. We might look at our garden rake at 1 foot (.3 meters) distance, but rarely closer, but when it is 3000 m (1.9 miles) away (again, horizon-earth scale), we certainly don’t interact with it, and probably can’t even see it.

Until relatively recently, when we have interacted with things outside of this daily human scale, they have seemed impossible mysteries. Bacterial diseases, for example.

The industrial revolution, large-scale production using powerful new fuel sources, has allowed us to produce a great deal of objects that are several steps closer to the straight line. An example of this is sheet-metal made by hand versus that made once that industry was mechanized (look at old steel ships or the V-2 rocket). And most of our modern industrialized everyday world is built and designed around this ‘religion of the line’, for lack of a better term.

Enter the fractal.

I had a profound eye-opening experience when as a teacher I was required to come up with an activity for students and, being a nerd, I decided upon ‘Fractal Day’.

We learned that fractals are (roughly) objects or shapes that are similar on different scales. A classic example is a tree, which has branching on a large scale, but as you zoom in to those branches, you see smaller branches, all the way down inside the leaves. Another is Romanesco broccoli, which looks like it is made up of many smaller brocolli.

Romanesco broccoli

I was aware of this, but the (literal) eye-opening happened to me when we did the final activity — going outside with cameras and taking pictures of fractals. As we walked among the buildings and trees around the school, looked down at the ground and up at the sky, something hit me. Almost everything in nature was fractal or fractal-like (the scaling or zooming doesn’t go on forever), and almost everything made by humans was not fractal at all.

If you don’t believe me, I suggest you organize a fractal-day of your own. Learn what they are, draw or build a few (the dragon fractal is fun, for example).

You can even do this as a thought experiment if you are trapped in a spaceship. List some things in nature. Most of them will be at least somewhat self-similar on different scales.

  • hand
  • cloud
  • sand
  • waves

Most human-built things are not fractal whatsoever. For example, you will not find little tables inside a table. A bicycle is not made of little bicycles. etc. etc. Perhaps you can think of exceptions to this fractal = nature, non-fractal = human-built rule?

(This is another one of those jaw-dropping moments, in case you were wondering.)

If we go back to the story of lines, we remember that in some way, curves and lines are the same thing. Lines are curves ‘zoomed-in’, such as the horizon, and as we said, a great deal of modern science, industry, and economy is based on the idea of lines.

However, this observation about ‘fractals = nature’ has put a fly in the ointment of the idea that many things can be described by curves which are basically lines. The big story is that, different from smooth curves, when you zoom in to a fractal, you do not see lines.

In fact the more we have studied natural phenomenon, the more we have become aware that this ‘religion’ of lines doesn’t always work, and in fact there are a great, great deal of natural phenomena which behave in a way which is fractal.

In fact, as our industrial development begins to destroy the environment in which it was created, we see that economies and industry cannot scaling indefinitely. Once again, this is a problem of scale.

And in fact, modern mathematics and science has made significant steps beyond the line. Since the 1980s, there have been various areas of study related to this non-linearity of nature, called fractals, chaos, and more recently complex systems.

Let’s zoom out, adjusting our scale to think about historical trends in human development. Again, using the ‘horizon-earth scale’, let’s not go back 20 years or even 2,000 years, but 200,000 years. At that scale, let’s think about daily life. As far as we understand, at that historical scale, there were no lines, neither in the daily life, nor in the mind of humans. (We don’t even have petroglyphs that old, and that’s the age of the oldest human fossils.)

(The old popular adage is “That’s the greatest invention since the wheel”. Well, a simplistic but deep observation is that a wheel is basically useless if you don’t have a road or place flat enough to use it. We are back to curves and lines!)

That is, pre-history is largely a time before lines. They were neither available, nor useful. The forest is full of fractals, the ecological system is full of chaos. The natural world is full of complex networks of interactions. In fact, in this light it is quite amazing that the human neurological or social system is able to deal with a non-fractal world at all, and makes me wonder about deeper causes of stress, depression, metabolic and other systemic illnesses.

And today, we find ourselves in an era where we are gradually both reaching the limit of lines to function in economy, industry, or science, and where we realize the deepest phenomena in nature do not follow them. This was already huge step when Einstein took us beyond straight lines with his descriptions of space-time.

Similarly, the economies of scale of industry are more and more being forced to include environmental factors into their calculations. The earth and its resources are finite, not infinite. Profit does not increase forever, and cost does not continuously decrease.

And the big news of the 21s century, the internet and mobile phone usage, is showing us how these fractal, networked, non-linear structures (social media networks, for example) grow in new systems.

All of this is both exciting and frightening for us humans. We do not have a well-developed math or science of the world beyond lines. In fact, most of the time when scientists say something is non-linear, it means they don’t expect it can be modeled, or it may be very difficult to model, and never really understood satisfactorily. A great deal of this comes from the ‘horizon-earth’ problem of scale. We cannot see the trees and forest at the same time. There are factors at work which are hard to communicate or understand at a human scale.

Luckily, we are all geniuses. Once we realize how incredible it is that we can walk and chew gum, navigate in a jungle, climb rocks, or even appreciate the simplest music, we have to re-calibrate our thinking about what it means to be ‘intelligent’. Our minds are already well-adjusted to the fractal world. The last few hundred years has really been an excursion into a foreign land.

We will do just fine at the end of the line.


Written by


A new PhD student studying self-organization, criticality, learning, and information.