Leaving the High Road and Walking into the Woods...

 

Leaving the High Road and Walking into the Woods...

A good teacher can explain what is already there in the textbook. But once in a while, a teacher gets a strange thought.

“We all know this topic already. But what if we try looking at it in a different way?”

One such small thought opened the door to a new world in mathematics.

In 1978, a mathematician named John Hubbard was teaching calculus to second-year students in Orsay, France. As part of the course, he was also teaching the famous Newton’s method, a technique used to find the roots of equations.

The basic idea of Newton’s method is quite simple. We first make a guess about where the solution of an equation might be. We then put that guess into a formula to get a better guess. Then we take that new guess and feed it into the same formula again. If we keep repeating this process, step by step, we eventually get closer and closer to an actual root.

For example, if we solve the equation (x^2 = 1) directly, we immediately see that it has two roots: (+1) and (-1). But if we solve the same equation using Newton’s method, we begin with a guessed value and then improve it little by little until we eventually reach one of these two roots. Which one we reach—(+1) or (-1)—depends on where we started.

This is a method that students have been learning for generations.

But Hubbard wanted to do something a little different that year.

“Why should I just work it out on the blackboard? What if the students themselves write a program and run Newton’s method?”

The computers of those days were nothing like the ones we have now. They had less power than even a small phone today. Hubbard’s students used programmable calculators to solve cubic equations—equations with three roots.

And that is where a small question appeared.

In Newton’s method, depending on where we begin, which root do we finally end up reaching?

“At first I thought the answer would be obvious,” Hubbard might have said.

But the more they looked, the less simple it seemed.

If they started in one region, the method converged to the first root. In another region, it went to the second root. In still other places, it went to the third root.

But then another question arose.

What do the boundaries between these three regions look like?

Are they simple and neat, like the borders between three countries on a map?

Hubbard and his students began testing one starting point after another with the computer. They marked which root each point eventually reached.

But the resulting picture did not look like a neat map divided into three simple regions.

It looked messy.

One region pushed into another. Inside that, a third one appeared. If they looked more closely, they saw smaller branches. If they zoomed further in, they found similar patterns again.

At that time, even line printers could not print colourful images. They could only print patterns using symbols like ‘+’ and ‘|’. The first pictures came out in very low resolution and, in Hubbard’s own words, looked like “a mess.” But they did not throw that mess away. They kept looking at it. They kept asking questions.

And that is the important part.

Usually, when a strange result appears in an experiment or a calculation, what is our first reaction?

“Something must have gone wrong.”

And very often, something really has gone wrong!

But some of the great stories in science begin exactly there.

“This may be an error. But… what if it isn’t?”

The strange boundaries that appeared in Hubbard’s study of Newton’s method were connected to the mathematical world of complex dynamics and fractals. Hubbard later said that they quickly realised that the full behaviour of Newton’s method was not as completely understood as people had assumed. Investigating those strange images opened the way to new mathematical research.

There is a great lesson for us in this story.

Most of our education teaches us how to move along a known road.

Read a chapter. Learn a formula. Study an example. Solve ten problems. Write the answer in an exam.

In this way, we learn to travel efficiently on roads that have already been laid down.

But research is not always like that.

Sometimes you notice something strange by the roadside.

A question arises: “What lies over there?”

There is no road there. No signboard. No Google Maps. You do not even know where it will lead.

And yet, you have to leave the road and walk into the woods.

That is what Hubbard did.

He looked again at the long-familiar Newton’s method with the curiosity of a child. “Do we really understand fully how this works?” he asked.

That one question revealed an unknown world hidden behind a familiar lesson.

Great research does not always need a huge laboratory. Sometimes a small doubt is enough.

Doing a calculation once again.

Not throwing away a strange result.

Asking again and again, “Why did this happen?”

And above all, being willing to step away for a while from the path everyone else is taking, and walk into the unknown woods.

Because new paths do not appear on maps. They become paths only after someone has first walked there.