A Hénon Map Inspired by an Artwork Inspired by a Hénon map

As part of my maths in museums work I was talking to someone at the National Galleries of Scotland. The world of art lies a little outside my comfort zone, which is partly why I was interested in talking to them: if I want to encourage other people to venture outside their comfort zones in the direction of mathematics I should probably practice what I preach.

Anyway, just outside the Modern One gallery there is a sculpted land feature created by artist Charles Jencks called Landform which you can see to the top-right of the embedded satellite view of the area below:

The sweeping curves and other-worldly, natural-but-not-quite feel to this formation reminded me of something, but I couldn't immediately remember what. Somewhere in the back of my mind, Jeff Goldblum popped up:

MALCOLM:    Dr. Grant, Dr. Sattler -- you've heard of Chaos Theory?

SATTLER:       No.

MALCOLM:    No? Non-linear equations? Strange attractors?

SATTLER:       Shrugs

- Jurassic Park (1993)

Thanks Dr Malcolm^[1], that's it! I was thinking of a strange attractor, specifically a Hénon map.

Explaining strange attractors with any degree of accuracy inside what is supposed to be a relatively quick blog post is going to be difficult, so let's try this: imagine a process that generates some form of data - some aspect of the weather, maybe. You plot this data on a graph, but it seems to be randomly distributed: the points you plot don't seem to have any relationship with each other, except you know they all came from this same process. However, as you plot more and more of the points produced by this process it does seem that they are actually following some sort of grand plan. They don't form a neat line or curve, but they do appear to have been shepherded into a particular region, or to be avoiding certain areas of the graph. As you plot more and more points, a clearer shape emerges, but it remains a fuzzy one: there seem to be points, curves or regions that the data points are attracted to, but not so much that they stick to them, instead being happy enough hanging around in the general vicinity.

This point, curve or shape is an "attractor". If the attractor is fractal then we call it a "strange attractor"^[2].

The Hénon map is a relatively simple chaotic system with a strange attractor, devised (or discovered?) by Michel Hénon following his attendance at a seminar by Yves Pomeau.

Essentially, the process starts with an initial point in the coordinate plane which is plotted and then used as the input to a function for which the output is the second point, which is itself plotted and then used to generate the third point in the same way. And so on, as far as your patience and computing power allows.

I had a go at doing this in a spreadsheet, starting with the point (0,0) and applying the function  described on the Hénon map Wikipedia page:

For a coordinate \((x_n,y_n)\), the next coordinate is \((1-1.4{x_n}^2+y_n,0.3x_n)\).

That means, in English, that to get the next point's \(x\) coordinate you need to square the \(x\) coordinate of the previous point, multiply that by 1.4, then subtract this from \(1\), and add on the value of the previous point's \(y\) coordinate. To get the next point's \(y\) coordinate, you need to multiply the previous point's \(x\) coordinate by \(0.3\).^[3]

Starting with the point \((0,0)\), the first five coordinates plotted were:

\[\begin{array} {|r|r|}\hline x_1 & y_1 \\ \hline 0 & 0 \\ \hline 1 & 0 \\ \hline -0.4 & 0.3 \\ \hline 1.076 & -0.12 \\ \hline -0.7408864 & 0.3228 \\ \hline  \end{array}\] 

The chart below shows these five coordinate pairs plotted together:

Unimpressive, right? Looks like my mum's first go at Duck Hunt.

Here's what happens if we plot the first 50 points:

And now the first 500 points:

Let's skip to 5,000 points:

There's very definitely some structure in there, and it's looking a bit more like Jencks' land sculpture. These look a lot like they'd become solid curves if we kept on plotting coordinates, but if we zoom in...

This shows a portion of the right-hand side of the previous graph, with the horizontal grey line across the middle indicating 0.2 on the y-axis (I've made the crosses that represent the points smaller too). It can be seen that a certain fuzziness exists at this scale. In fact, if I made the crosses indicating the points even smaller and plotted a load more, we'd see a similar theme to what we see in the chart as a whole: sweeping curves doubling back on themselves. Those thick, dark curves are actually a number of apparent curves running close to each other. Zooming in further we'd see that each of those curves is similarly actually made up of a number of apparent curves, and zooming in still further... Yes, it's a fractal!

Here's a cool Hénon map zoom video that somebody else made, which should help to get the idea across more clearly than I've probably achieved in the last paragraph:

This is just one example of the kind of mathematical rabbit hole that you can fall down that starts with a museum or gallery object, which is itself one reason why I enjoy museums and galleries as sources of mathematical inspiration and exploration.

And if you're thinking something along the lines of "yeah, you've just picked a squiggle and said it looks a bit like art," Charles Jencks, the artist behind Landform, is known for taking inspiration from aspects of contemporary physics, and actually wrote about strange attractors (see Part Two, chapter VIII).

Footnotes:

1. If you're not a Jurassic Park fan the last few lines won't mean much to you, but don't worry: it's over now. (back)
2. I think this is what separates "strange attractors" from "attractors", but I'll admit to not having formally studied any of this: please correct me if you know more about this than I do, and suggest an as-pithy-as-possible alternative. (back)
3. Please note, if you're an "I don't see the point of algebra" sort of person, that this paragraph (in English prose) says exactly the same thing is the previous one (which utilises algebra): the only difference is that the one using algebra is clearer, much more succinct, and a heck of a lot quicker to type. (back)