The word “quincunx” refers to a pattern of four objects arranged in a square with a
fifth in the centre, like the spots on a dice. Although it’s an obscure word today, it has resonances and survivals in subjects as diverse as garden design, architecture and various aspects of religious and mystical symbolism. It’s also a name for a remarkable device designed by Sir Francis Galton in 1894 to demonstrate the “normal distribution”, a statistical object that had gone from obscurity to ubiquity during the preceding century. Galton’s board, as it’s also sometimes known, can still teach us interesting things about randomness and order, and takes us from high finance to quantum computing and back to deep philosophical questions about the nature of uncertainty itself.
|Galton’s board is illustrated on the left. A collection of little balls is poured into the top, which funnels them down to an opening (marked “removable stop”). They then fall down towards the compartments at the bottom, but in the way is a lattice of pins driven into the board — the pattern of these pins is what gives the quincunx its name.
The balls bounce “randomly” (more on this in a minute) off the pins, which makes it effectively impossible to predict which compartment a particular ball will end up in. Yet if a lot of balls are sent down the funnel, they form a predictable pattern: a bell-shaped curve that has a hump in the middle and tails off on either side.
This curve is not yet a normal distribution — it’s something called a binomial distribution, which is essentially the lumpy version of the one we’re after. But imagine making the balls smaller and smaller, adding more and more pins and compartments to match. As you did this, your quincunx would produce a result that was ever less lumpy, and that looked more and more like a normal distribution. In fact, the definition of the normal distribution is it’s what you’d get if the balls were infinitely tiny and there were an infinite number of compartments for them to fall into.
The key idea is that when a ball hits a pin, whether it goes left or right depends very sensitively on exactly how it hit the pin, just as which face a dice lands on when thrown depends on exactly how you threw it. It’s so sensitive, in fact, that in many cases you really can’t tell which way it’s going to go even though our understanding of physics tells us there’s nothing truly indeterminate about it: it’s just that it’s really hard to observe the situation closely enough to make a good prediction. In fact, in principle there should always be cases where we can’t do this, no matter how sophisticated our measuring devices might be.
Here’s a really nice double-sided quincunx in action:
and here’s a computer simulation of a quincunx that associates a musical note with each horizontal position, so that each ball’s path is translated into a sort of melody:
With both of these, you only get a very rough binomial distribution. I decided to make my own version using Processing, the language we use on our programming course here at Central Saint Martins (the code is on the course’s Moodle site). In my version the ball falls from left to right, and bounces off a pin at every pixel, so the effect is much more pronounced. The red region on the right shows the actual binomial distribution, and the green curve is a smoothed-out version that looks very like a normal distribution:
Making the paths of the “balls” look like lightening was a bit whimsical, I admit, but the paths taken by actual lightening bolts do indeed follow this same principle, at least under some fairly general assumptions; which way the spark jumps depends on very fine differences in local electric charge.
As the pins get closer together, these paths become closer to Brownian motion, another concept that results from letting things “go to infinity”. There’s a technical discussion of this, and its relationship with both finance and physics, in this video and the subsequent parts.