Mathematical Monsters

The Machines of Mathematics

I’m going to digress briefly – and I promise to keep it brief – into the language of mathematics. This is important, because with this language the idea of feedback is much easier to understand. I promise to keep it short and simple; I think you’ll find it quite painless, and maybe even fun.

Mathematicians invent machines all the time – it’s part of their work. But most people don’t know that, because mathematicians don’t call them machines; they call them “functions”. Mathematicians are really good at creating language that is accurate and meaningful for them, even though it’s confusing for the rest of us. A function in math is just a machine that turns one thing into something else.

Think of making coffee. If you’re really into coffee, you grind your own beans. So when you get up in the morning and want to start your day with a great pot of coffee, you put the beans into the coffee grinder, press the button, and after a bit of time and a bit of noise, ground coffee comes out. The coffee grinder is a machine that turns the beans into coffee grounds. Then you put the grounds into a coffee maker, and out comes coffee – the coffee maker is a machine that turns the grounds into coffee. Mathematicians would call the coffee grinder and the coffee maker “functions.”

In English we would describe the machines for making coffee as:

The coffee grinder turns the coffee beans into coffee grounds.

The coffee maker turns the coffee grounds into coffee.

Mathematicians like to keep their sentences very concise and short, in order to say a lot with as few symbols as possible – they love efficiency. So they typically name their machines with a single letter, often f or g, though they can use any symbol they want. If you were a mathematician you might call the coffee grinder “g” and the coffee maker “m”, and write:

The g turns the coffee beans into coffee grounds.

The m turns the coffee grounds into coffee.

That’s a little shorter, but mathematicians want to spend as little time as possible writing stuff down, so they also name the things that go into their machines with single letters, often x, or y, or z, though again they could use any symbol they want. And when something has been changed by a machine, they indicate that with a symbol that shows it’s been changed – e.g., a subscript or superscript. So they might call the coffee beans “x”, the ground coffee x1 and the coffee itself x2. (x1 is read as “x sub 1” and x2 as “x sub 2”.) Then you could write:

The g turns the x into x1 (in English: the coffee grinder turns the coffee beans into coffee grounds)

The m turns the x1 into x2 (in English: the coffee maker turns the coffee grounds into coffee)

This is getting shorter and more concise (and also harder to understand), but mathematicians still aren’t satisfied; they really want to get rid of anything that has to be spelled out, so they throw away the word the altogether, they With the word “turns” with parentheses and the word “into” with an equal sign. So they would write the above as:

g(x) = x1 (in mathematics you would say “g of x equals x sub 1”; in English: the coffee grinder turns the coffee beans into coffee grounds)

m(x1) = x2 (in mathematics you would say “m of x sub 1 equals x sub 2”; in English: the coffee maker turns the coffee grounds into coffee)

Now the statements are about as short and concise – and unintelligible to non-mathematicians – as possible. But if you understood the language of math you would know exactly what this meant, and you could write about it and talk about it easily.

One of the things that makes mathematics fun is that the machines (functions) that mathematicians build are abstract, so they are built as fast as you can think of them – you don’t have to take the time to physically build anything; as soon as you have the idea, you have the machine. And if you tell someone about the idea, then they have the machine too. They can think of a way to change the machine, and as soon as they tell you about it, you have the changed machines as well. And the machines can do anything you can imagine, because they are not constrained by the laws of physics. So, for example, mathematicians make up “reverse” machines – ones that undo what other machines have done. You could make up a machine that turns coffee into coffee grounds, and another that turns coffee grounds into coffee beans.

Mathematicians call reverse machines “inverse functions”. They use a negative 1 superscript (-1) to show this. Remember our plain English version of the coffee grinder? It was:

The coffee grinder turns the coffee beans into coffee grounds.

If you had a reverse machine that turned grounds into beans, you would write:

The reverse coffee grinder turns the coffee grounds into coffee beans

Using the inverse notation of mathematics you would write:

The coffee grinder-1 turns the coffee grounds into coffee beans (you would read this as “The inverse coffee grinder turns the coffee grounds into coffee beans)

Remember that the mathematical notation for the coffee grinder was:

g(x) = x1 (in English: the coffee grinder turns the coffee beans into coffee grounds)

The mathematical notation for the inverse coffee grinder would be:

g-1(x1) = x (in mathematics speak you would say “g inverse of x1 equals x”)

There is another reason, beyond efficiency, that mathematicians created this shorthand notation, and it’s actually quite useful. Since their machines are abstract, they don’t have to worry about what they put into them – they could put cardboard into their grinder, and get ground up cardboard out of it. This is handy because it means that you can work with the concepts without being constrained by reality. The concept here is a “grinder”, but in the real world we have to specify what the grinder is intended to grind, so we buy the right kind and don’t put the wrong stuff into it. In the world of mathematics we can just think of the concept “grinder” and think of it as working with anything. So by using this short, generic notation, when they write:

g(x) = x1

they can With the “x” with anything they want. They could say g(coffee) and know they’d get ground coffee out, or g(cardboard) and know they’d get ground cardboard out, and so on. Their notation allows them to think of the essential concepts, without being distracted by physical details that aren’t important to the concept. The downside to their notation is, you have to know the language, and you have to be good at thinking about pure concepts without having them related to anything physical – a challenge for many of us. Fortunately there are people who are good at that, because without mathematics we would never have been able to create all of the wonderful inventions that make our lives so rich.

Of course the machines that mathematicians create don’t work on coffee beans, or cardboard, or any other physical thing. They work on numbers; they take numbers and turn them into other numbers. So, for example, a mathematical machine might take a number and double it – i.e., multiply it by 2. If that foundation was called “f” it would look like this:

f(x) = 2x (in English: take whatever number you start with, and multiply it by two.

Mathematical notation works well for mathematicians, because they speak the language of mathematics, but for those of us who don’t speak that language, their writing looks very complicated and challenging – giving the impression you’d have to be really really smart to be able to make sense of it. (I think that’s another reason they do it.)

The ability to focus on, and think and communicate about, essential concepts, free of the constraints of the physical world, is very powerful and will be useful as we explore the notions of feedback and systems thinking. But I’ll limit it as much as possible, I’ll keep it simple, and I will strive to explain what the language means when I do use it. Lines, Planes and Cubes

In addition to functions, which deal with numbers, it’s important to understand a few things about the branch of mathematics called geometry. Geometry deals with shapes, just like algebra deals with numbers. Shapes area things like lines, planes and cubes.

A straight line looks like this:


and a curved line looks like this:

curved line

Here’s a line with both flat and curved parts:


In addition to segments that are straight or curved, a line can have points that are neither flat nor curved – like the sharp points where straight and curved segments meet in this figure:


In mathematics, lines are “one dimensional” – they have no width, only length. In a sense, lines are the idea of length. Imagine a thread stretched from left to right, so it’s perfectly straight. The thread has some width and height, even if it’s a very fine thread. Now imagine the thread’s width and height shrinking – whatever thickness your imaginary thread had when you thought of it, think of it as half as thick, while it’s length stays the same. Now imagine it shrinking again, and again, and again, until it has no thickness at all – it’s just a length. That’s the mathematical idea of a line. Of course you can’t really visualize it, because it has no thickness, but you can still think of it.

Lines can be curved – all of the pictures above are lines – but from a mathematical perspective they have no width. So they can only be measured in terms of length, not in terms of width or height. Since these imaginary lines have no width, no matter how many of them you put side by side on a flat surface, they would never take up any space on a flat surface – they are one dimensional, so they can never occupy any space in two dimensions. You can measure the length of a line, but it will never cover any area.

A surface – like a piece of paper – is two dimensional. It has both length and width. A surface looks like this:


But a surface has no height. A plane is the idea of an area. Imagine a piece of paper laying flat on your desk. You know that if you put several hundred of them on top of each other they will create a stack that might be an inch or two tall. But now imagine that piece of paper has half the thickness it originally had – your stack would be half as tall. Now imagine it half again as thick, and half again, and so on, until it has no thickness at all. At that point your stack has no height. That’s the mathematical idea of a plane, it extends in two dimensions but not three. Each of its edges is a line, and you can measure their length. By multiplying the length and width you can measure it’s area. But it has no volume.

And a three dimensional object – a box, or a ball, or any real object – has length, width and height. With a box, say, you can measure its length, it’s width, and its height. Each of its edges is a line, so you can measure their lengths; each of its sides is a plane, so you can measure their areas; and the entire box is a cube, so you can measure its volume. A cube looks like this:


So it makes sense to think of dimensions in terms of whole numbers – 1, 2 and 3. Mathematicians even make up things that have more dimensions, like 4 or 5 or more, but there is always a whole number of dimensions – or so they believed for a long time. They’re pretty hard to imagine, but in mathematics they are useful. Important points about the monsters

They diminished our sense of certainty about that field that was supposed to be the bastion of certainty. One more chink was taken out of the armor of certainty.

The machines all used feedback to create the monsters.

They were fundamentally very simple machines – they didn’t do anything very complex – yet they gave rise to these bizarre, and in some cases extraordinarily complex, structures.

Nature loves efficiency, so she is bound to discover anything that create complex but orderly systems with the greatest economy possible. Back to the Monsters

So now that you understand the notion of a function (i.e., a mathematical machine), and of lines, planes and cubes, we can get back to the interesting things – the monsters, and feedback.

There are certain basic beliefs that mathematicians held on to for centuries – and with good reason, because these principles gave them a sense of order, a sense that the world – at least the world of mathematics – could be understood.

Every Sharp Point on a Line is Connected by a Straight or Curved Segment

For example, they believed that any line you could draw had to have at least some areas that were either flat, or smooth curves. To understand this, consider the following. Recall that earlier I showed you that a line can consist of straight segments, curved segments, and sharp points where different segments meet at an angle. It’s obvious, if you think about it, that no line could consist only of sharp points – there would always have to be a segment between any two sharp points that connected them, that was either a straight line or a curve. After all, what makes a spot pointy is that two different straight or curved sections come together at an angle. For example, this line has lots of sharp points, but they are always connected by a either a straight or curved section:


One of the beliefs that mathematicians held dear was that no line could consist only of sharp points. This seems obvious – that any continuous line (one without any gaps in it) would have to have sections that were either straight or curved between any pointy spots.

And yet, in 1904 Helge von Koch, a Swedish mathematician, introduced what is now called the “Koch Snowflake.” The Koch Snowflake is produced by a machine that Koch invented that starts with a simple triangle, and goes through a very simple process. Here’s what Koch’s machine does:

Start with an equilateral triangle (an equilateral triangle is a triangle with all three sides the same length), like this one:


Divide each straight line of the triangle (i.e., the sides) into three equal parts, and With the middle segment with another equilateral triangle, whose sides are 1/3 the size of the original:


Now remove the bottom of the triangles you just added (the lines shown in green above) and you get the first result of Koch’s machine:

koch machine

Now repeat this – take the image above and feed it back into the machine, which Withs each straight line with a triangle whose sides are 1/3 the size of the one just used. Feeding the image above back into the machine produces this:

koch machine

And if you feed this image back into the machine, replacing each straight line with a triangle whose side are 1/3 the size of the one used in this step, you get:

koch machine

You can see where this is going. If you run this machine an infinite number of times, all of the straight lines disappear and all you are left with is the points. There are no straight or curved segments, even though it is a continuous, unbroken line going around the snowflake. It is impossible to imagine what this would look like, but Koch didn’t care about that – he was concerned with demonstrating that you could, mathematically, create such a structure, thus violating one of the things that mathematicians felt quite certain about.

A Box Has Volume

Another belief mathematicians held strongly was that if you made a box, it would have to enclose some measurable volume. For example, a box that’s 1 foot on each side would hold one cubic foot of space. Obviously, any box that has measurable sides must contain some space inside the box. And any collection of boxes would therefore also have to contain some space inside them.

In 1926 Karl Menger, an Austrian mathematician who eventually emigrated to the United States and taught mathematics at Notre Dame and Illinois Institute of Technology, wrote a paper describing a machine that produces something known as the “Menger Sponge.” Like Koch’s machine, it starts with a simple figure – in this case a cube:


Menger’s machine takes a cube that is put into it, divides each side of the cube into nine equal squares, which turns the big cube into a set of 27 smaller cubes. The machine then removes the middle cube from each side, and the cube at the center, leaving a set of 20 smaller cubes – or a large cube with square holes drilled through the middle of each side, depending on how you choose to look at it. It looks like this:


Now when you put this construction back into Menger’s machine, it takes each individual cube and repeats the process, producing this:


Now you repeat with this construction, which results in this:


It looks a bit like a hotel in a science fiction horror film, one where you might enter but never leave, or try to enter but keep finding yourself forever outside.

As with the Koch Snowflake, you repeat this process an infinite number of times, and you end up with a collection of boxes that, taken together, have an infinite surface area – that is, if you took all of their sides and laid them out next to each other, they would stretch to infinity in all directions – and yet no volume whatsoever, because you’ve removed all the space inside the boxes.

This seems crazy, and you certainly can’t visualize it, but the mathematical result is a set of boxes that contain no space. Once again the sense of certainty that mathematicians loved was challenged and found wanting.

Dimensions Come In Whole Numbers

Yet another belief that mathematicians held dear was that dimensions only exist in whole numbers. A line is one dimensional, a surface two dimensional, and a cube three dimensional. But the idea of a fractional dimension – an object that was, say, 2.5 dimensional, would be absurd, and mathematics should not allow it.

But it turns out that when you study the question of dimension carefully, it is nowhere near as straightforward as it seems. And in the 1960’s Benoit Mandelbrot, a brilliant multi-disciplinary scientist, demonstrated that in fact there are many objects, both in nature and in mathematics, that exhibit fractional dimensions. For example, the fractal dimension of the Koch Snowflake is about 1.26, and the fractal dimension of the Menger Sponge is about 2.73.

I’ve just described three beliefs mathematicians held for centuries – beliefs that make perfect sense to us, so much sense that if someone told us that any of these beliefs were wrong, we’d be likely to think that person might be somewhat wrong in the head.

With the little bit of math that I’ve shown you, there are some fun and fascinating ways to explore the power of feedback. The Chaos game is one …

What all of these have in common is that they are self-referential – that is, the product of each step provides the raw material for the next step. It turns out that anytime something becomes self-referential, the situation is pregnant with complexity unpredictability. As you will see shortly, this has far-reaching implications for how human beings live, work, and interact with each other.

It would be easy to write these things off as the musings of nerdy mathematicians who had nothing better to do with their time than think up wild ideas. In fact that is exactly how many of the greatest mathematicians initially reacted to these mathematical machines that were being designed. And yet, throughout the 20th century, as scientists encountered more and more aspects of nature that simply didn’t yield to more traditional techniques, they found themselves turning the fascinating emerging field of chaos theory and fractals, which explores how these monsters actually work. And what they found was that long before human beings showed up, nature had also discovered these monsters, and put them to good use.

The mathematics of fractals enables extraordinarily efficient techniques for solving difficult problems, and nature takes full advantage of them – e.g., to name a few: the growth pattern of trees; the relationship between an animals physical size and it’s life span; the distribution of human cities of different sizes; and the distribution of earthquakes of different magnitudes all appear to be governed by fractal mathematics. Nature uses fractal mathematics in the design of lungs in animals, to achieve the maximum possible surface area in the smallest amount of space, so that a very compact lung can absorb large amounts of oxygen quickly.

And, now that scientists are respecting and studying this mathematics instead of writing it off, they are finding more and more uses for it. To name a couple: printer manufacturers use fractal mathematics to improve the quality of the images their printers can make, and cell phone companies use fractal mathematics to improve the efficiency with which cell phone signals can be distributed across their towers. And geneticists are beginning to understand that the dynamics of feedback are deeply woven into the mechanisms of genetics – how we inherit attributes from our parents; how genes turn each other on and off; how some genes alter the structure of other genes; what determines whether a gene that predisposes you to cancer gets expressed or remains dormant. The more we come to understand the dynamics of feedback, the more profoundly we see them shaping all aspects of our lives.