If there's one thing which has fascinated mathematicians over the last 100 years or so, it's the shape of blancmange. The shape comes from a mould, of course; blancmange is a kind of wobbly milk jelly. The mould sets the dessert into a rounded shape, with a regular yet aesthetically-pleasing bumpy top. If you cut one in half, its cross section is interesting to study - all those bumps with smaller bumps on top of them. Lovely.
Okay, let's be honest. It's not the blancmange which interests mathematicians per se, it's a mathematical curve which happens to look exactly like one, and in a moment of uncharacteristic levity, the world of mathematics nicknamed this the blancmange function - in fact it was the English mathematician David Tall who coined the term in the 1980s. You can construct one by drawing a series of zigzags, known as sawtooth functions, and combining them, as we shall see.
How Do You Draw A Blancmange?
First, find a piece of graph paper and draw a single sawtooth which goes right across your piece of paper. So, your line will start at zero on the left hand side, then rise at 45 degrees towards the middle of the paper. When it reaches the middle it will suddenly start to fall at 45 degrees, reaching zero at the right-hand side of the paper. A triangular mountain-shape.
Now draw another pattern, this time with two mountains fitting across the page, but each one is half the height of the big one you first drew. Here's the tricky bit - you have to add the heights of these sawteeth to the first single sawtooth you drew. If you get this right, you get a kind of plateau shape.
Now draw another sawtooth, but this time with four mountains. Each will be one quarter the size of the original sawtooth. Add the heights of this shape to the plateau, and your composite shape now has two peaks.
Keep doing this, each time doubling the number of mountains and halving their size, and adding the heights of the shape to the others. The composite shape will eventually start to look a lot like a wobbly milk jelly dessert.
So Who Do We Have To Thank For This?
It was discovered by the Japanese mathematician Teiji Takagi in 1903. The shape is also known as the Takagi curve1.
One interesting thing about the shape is that you can see mini-blancmanges within the bumps. Upon these mini-blancmanges there are smaller micro-blancmanges, which in turn have even smaller nano-blancmanges, and so on. This property is called self-similarity, and it's fundamental to the set of mathematical functions which are collectively known as fractals. Computers have been used to generate some fascinating and complex quasi-organic patterns from these; the images are often enhanced with bright colours to make stunning artwork.
But Why Do Mathematicians Find This Interesting?
The blancmange has one other very unusual property, but in order to explain it, we must first talk a little bit about two mathematical concepts - continuity and differentiability. Some of the following may only be of interest to mathematicians, so feel free to skip a bit if it's not your thing.
Quite simply, a continuous function is one which has no gaps in it when you draw it. A straight line on a piece of paper goes from one side of the paper to the other without any gaps. This is the same for many simple curves and some trigonometric functions, like a sine wave. Our sawtooth function is continuous too - a zig-zag which you can draw without taking your pencil off the paper.
Not all functions are continuous, though. You could have more complex curves which disappear off to infinity at the top or bottom of your paper, or shapes which suddenly step up or down at some point.
The blancmange function is continuous. There are some complicated ways to prove it, but in the end it's made up from lots of sawtooth functions which are themselves continuous, and when you add continuous functions together, they remain continuous. So far so good.
Mathematicians love slopes. If you imagine a mathematical curve as some sort of landscape of rolling hills and valleys, then at any point these will have a gradient. It can be steep or narrow, and it can slope upwards or downwards. The most simple curve is a straight line, and this has a constant slope everywhere. Other curves have slopes which change, but which you can calculate. When mathematicians have a curve, then they like to be able to work out what the slope of it is at any point, and they call this process differentiation.
Now, you might think that any continuous function would have a slope. After all, you know it has no gaps, you can draw it on a piece of paper, and you know how high it is at any point. Well, not quite, as some functions are not smooth - they suddenly change direction, like the sawtooth function does at the top or bottom of one of its peaks. What is the slope at the top of a mountain? You could rest a plank of wood on the sides of a mountain and know what the slope was, but if you rested it on the peak it could point in one of a number of directions.
Mathematicians are happy to state that a function is differentiable - ie, they can calculate the slope - at all places where it is continuous and smooth, but not at the points where it suddenly changes direction.
Up to the 19th Century it was thought that all continuous functions had to be differentiable at most points. However, in 1834 the Bohemian priest and mathematician Bernhard Bolzano discovered a continuous function which wasn't differentiable anywhere. The German Karl Weierstrass discovered another in 1872. Our blancmange function, discovered by Takagi, is a third.
Because there are infinitely many sawteeth making up the final blancmange, there are infinitely many tiny mountain peaks to add to the shape, each one being a point which doesn't have a slope, and so you can't differentiate a blancmange anywhere. Mathematicians call these pathological functions. Their discovery stunned the mathematical world, to the extent that in 1893 the Frenchman Charles Hermite wrote:
I recoil in fear and loathing from that deplorable evil: continuous functions with no derivatives.
Perhaps if he had seen the blancmange he would have been a little sweeter.
The last word should go to the man who coined the name. David Tall's explanation for its strange behaviour was that the blancmange is nowhere differentiable because it wobbles too much!
Another Interesting Curve
You might like to try drawing another curve from combining sawtooth functions, but at each stage multiplying the number of sawteeth by two, but dividing their height by four. You should end up with a special curve called a parabola - it's the curve you get when you cut through a cone at an angle which is parallel to one of the cone's sloping sides. This particular 'conic section' has many interesting properties and applications. For one thing, it's the shape of the reflector you would find in a torch, as it reflects the bulb's light in a parallel beam.
For anyone who has read this far expecting to find a recipe for blancmange, then help is at hand from the Cook It Simply website. Other online recipe sites are available. Don't forget to buy a mould - one with a nice bumpy shape.