Second-Rate Minds

How would you calculate ? I remember being sent as a boy of ten into the schoolyard to find circular objects to measure. Our attempts to wrap a tape around a dustbin lid clearly did not represent the optimal method.

Perhaps you know a few digits of . Starting from scratch, how would you find more? We know is irrational, so it cannot be expressed as the ratio of two numbers. Worse, is transcendental, so it is not the solution of any polynomial equation with integer coefficients.

There is a passage of the Bible (1 Kings 7:23) which refers to “a molten sea” which is “ten cubits from the one brim to the other” and “a line of thirty cubits did compass it about”. This figure of did not represent the state of the art, with more accurate values being recorded much earlier in Egypt and Mesopotamia.

A famous approximation is attributed to Archimedes (287-212 BC), who worked using only geometric methods. Archimedes took a circle and inscribed within it a polygon. The perimeter of the polygon, which is contained within the circle, is smaller than the circumference of the circle itself. Then he superscribed a polygon outside the circle. The perimeter of this second, larger polygon is larger than the circumference of the circle itself. If you think about increasing the number of sides on a polygon you will notice that the more sides a polygon has, the closer it gets to the shape of a circle. By increasing the number of sides on his two polygons and considering their perimeters, Archimedes could define two series, one increasing and one decreasing, both of which have in their limit.

Archimedes took this as far as polygons of 96 sides and came up with an upper limit for of and a lower limit of . Because these coincide in the first three digits, we can conclude that begins

Others, following the method of Archimedes or developing it independently, extended to polygons with greater numbers of sides. This method culminated with Ludolph Van Ceulen who, around 1600, produced an approximation for to 35 decimal places. Those 35 digits were inscribed on Van Ceulen’s tombstone.

More modern approximations involve calculating the terms of an infinite series involving , which can be rearranged to find an approximation for . For example, the following was discovered by James Gregory:

Using such a formula, one can generate to as many decimal places as necessary by taking more terms. However, this series converges very slowly. For example, 10,000 terms are needed to get just four decimal places. John Machin found a series that converges much more quickly to :

which uses a more general James Gregory result:

Machin used this formula to calculate one hundred digits of in 1706, the year William Jones introduced the modern use of the Greek letter π.

Computers enter the story in 1947 when D. F. Ferguson of the Royal Naval College in England calculated to 710 digits with a desk calculator. In 1950 George W. Reitwiesner published more than 2000 digits of in the journal Mathematics of Computation. These digits were calculated on the ENIAC computer at the suggestion of John von Neumann, who was interested to obtain a statistical measure of the randomness of the distribution of the digits (whether digits occur equally often in remains an open question).  Reitwiesner and his team used the same formula as Machin and ran ENIAC for around 70 hours, including the time taken to handle the punch cards.

More recently, series are used which operate on similar principles but converge even more quickly. The latest computer record used the 1989 Chudnovsky algorithm, which is based on the series:

In 2011 Shigeru Kondo set this record on a custom-built home computer using y-cruncher, a program written by Alexander Yee, to calculate to ten trillion digits.

The light bulb puzzle presents you with three switches, one of which controls a light bulb inside a closed room. You are permitted to flip switches as much as you like, then you must open the door and say which switch controls the light bulb.

You don’t seem to have enough information. You can flip one switch and open the door. If the light is on then you have found your switch. However, if the light is off you can’t tell which of the other two switches controls the bulb.

The solution is to first flip a switch on for ten minutes, then off again. Now flip a second switch on and open the door. Leaving the light bulb on for ten minutes would allow it to heat up. Then, if the light is on or off and hot, you know which of these two switches was the cause. If the bulb is off and cold, you can identify the third switch.

This puzzle annoys me, but I was not sure why.

I heard the following as a “grammar test”: which is correct, “the yolk of an egg is white” or “the yolk of an egg are white”? The answer was given as: “don’t be silly, the yolk of an egg is yellow”. This annoyed me because it was set up as a grammar test so I took the incorrect yolk colour to be a mistake in the statement of the problem that was safe to ignore, restraining myself to the grammatical aspect.

Here is another puzzle, a sorites. Sorites were included by Lewis Carroll in his book Symbolic Logic (1897). Your task is to arrange the following logical statements into a chain to draw a conclusion.

1. babies are illogical;
2. illogical persons are despised;
3. no one is despised who can manage a crocodile.

In this case, since babies are illogical, illogical persons are despised and despised people cannot manage crocodiles, we conclude: “babies cannot manage crocodiles”.

I have presented this to someone who said that the result is obvious since of course a baby cannot manage a crocodile. However, there is no requirement for the statements in a sorites to be true. Are illogical persons, including babies, really despised? This is a logic puzzle in which the question of whether a baby could manage a crocodile in reality is not relevant.

For me, saying “don’t be silly, the yolk is yellow” is like saying “of course babies can’t manage crocodiles”. You have been placed in the constrained reality of the puzzle and it is irritating to allow extra information from outside of the puzzle. In the light bulb puzzle we have switches that cause lights to illuminate. The fact they also cause the bulb to heat up is a fact from the real world intruding on the world of the puzzle. You might equally well say “flip each of the switches and look at the crack under the door to see if any light comes out” or “why are there three switches, anyway?”

Is there a way to fix the puzzle to remove the irritation? Mentioning in the statement of the puzzle that the bulbs heat up would give the game away, while not mentioning it situates the solution outside the constrained reality of the puzzle. Consequently, this cannot work as a viable puzzle.

That is why the hot light bulb annoys me.

Diffusion is a physical process that is an essential component of innumerable phenomena in nature, including when your lungs transfer oxygen into and carbon dioxide out of your blood. Now a fundamental principle behind diffusion has at last been experimentally verified.

An ergodic dynamical system is one for which the average behaviour of a single component over time is the same as the average for all components at a single time. This allows researchers to assume for some systems that measurement of a given variable on one particle – such as the distance covered by a particle in a given time interval – should yield the same average value as a single measurement of the same variable on a collection of particles, so that each particle behaves as a small version of the system as a whole.

It is generally accepted that the ergodic hypothesis is applicable to the dynamics of diffusive processes. Diffusion describes the spread of particles through random motion under the influence of thermal energy. In virtually all chemical reactions, diffusion is responsible for bringing reactants sufficiently close together to enable them to react.

However, as Professor Jörg Kärger at Leipzig University points out, “although diffusive processes have been investigated for the past 150 years, the principle of ergodicity has not yet been experimentally verified”. This is because so far it has only been possible to quantify diffusive processes by measuring many particles simultaneously.

A collaborative effort by Professor Kärger’s group at Leipzig University and Professor Christoph Bräuchle’s team at LMU Munich looked to measure the diffusive behaviour of a system of particles and the trajectories of single molecules in the same system. The LMU researchers were able to track individual molecules, while the Leipzig group could study the collective behaviour of the whole system.

The problem for the teams to overcome was that successful application of the two methods requires apparently conflicting conditions. The Leipzig group’s method needed high concentrations of molecules with large diffusion coefficients, while the LMU team’s method works best with extremely dilute solutions of species with small diffusion coefficients.

Using particular organic dyes with high fluorescence yields in combination with porous silicate glasses containing networks of nanometer-sized channels in which the dye molecules can diffuse, the researchers were able to create a system on which both techniques could be performed.

Comparing data, the two teams found that the diffusion coefficients obtained by the two techniques agreed with each other, providing the first experimental confirmation of the ergodic hypothesis in this context.

This work is important because sometimes a mathematical theory can make accurate predictions about the real world up to a point but not actually reflect the real world context. A nice example is from cosmology in the late 16^th Century, when mathematical models of the universe with the Earth or the Sun at the centre both produced viable predictions of astronomical events like eclipses. Further experimental data was needed to demonstrate that one reflected physical reality more accurately than the other. In this way, experimental checking of fundamental principles, such as the current work at the Large Hadron Collider, is important whether or not it backs current theory.

The next step for this research will be to look at systems that do not conform to the ergodic principle and determine the reasons for this. “The diffusion of nanoparticles in cells looks like an interesting example,” says Bräuchle, “and for us the important thing is to find out why the ergodic theorem doesn’t hold in this case.”

Notes:
Based on press release: ‘Glowing beacons reveal hidden order in dynamical systems – Experimental confirmation of a fundamental physical theorem‘, 19 October 2011, Ludwig-Maximilians-Universitaet Muenchen (LMU).
Research published as Feil, F., Naumov, S., Michaelis, J., Valiullin, R., Enke, D., Kärger, J. and Bräuchle, C. (2011), Single-Particle and Ensemble Diffusivities—Test of Ergodicity. Angewandte Chemie.

In recent years I’ve felt quite glad I never read G.H. Hardy’s massively influential book A Mathematician’s Apology (1940). I’d heard of this by the time I arrived at university but never got around to borrowing the copy I saw in the Hall library. Recently I decided to take a look and found much to dislike.

First, there’s his view of public communication of mathematics. Sometimes it seems all advice on public engagement starts with a quote from the Apology; as though a mathematician must acknowledge, like Hardy in his introduction, the improper nature of their endeavour.

I remember being told by an established communicator of mathematics about the importance of maintaining a strong research record to quell critics who might accuse him of undertaking public engagement work only because he has ‘lost his touch’ for research. In the introduction to the Apology, Hardy says that writing about mathematics,

is a confession of weakness, for which I may rightly be scorned or pitied by younger and more vigorous mathematicians. I write about mathematics because… I have no longer the freshness of mind, the energy, or the patience to carry on effectively with my proper job.

You may be aware that this blog takes its name from the Apology‘s opening paragraph. Hardy regards communication of mathematics as work for ‘Second-Rate Minds‘. We quote Hardy with irony, because we do not agree with him.

I believe there is great importance in communicating mathematics as widely as possible. I think it is important that children are encouraged to enjoy mathematics so that they might take further interest in the subject. Equally important is the view of mathematics held by the general public. Despite Hardy’s disdain for applications, mathematics nevertheless pervades the modern world and benefits from society valuing its role.

I feel sorry that a passionate and able communicator may not be valued by mathematicians, and that such abilities might be tempered by concerns about the scorn of one’s fellows. I happily accept that outreach is not for everyone; indeed society will benefit more from most mathematicians getting on with their research. Still, I don’t think we should look on those with a passion for communicating their subject as having failed.

I’ve chosen to pick on Hardy for his views on public communication of mathematics, though there’s much else I find fault with in his Apology. He expresses several opinions which are these days widely held and which I dislike.

Hardy is a proponent of the unfortunate cult of youth, writing, “if a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or himself”. Hardy hates teaching, although loves lecturing to “extremely able classes”.

Perhaps most famously, there’s his view on the “utility” of mathematics. It is easy to pick on a number theorist writing in strong terms against mathematics that can be applied to the real world, particularly to the business of war, at a time when modern cryptography was being nurtured at Bletchley Park. I have seen people take this argument to an extreme, somewhat unfairly since Hardy also writes against the “misconception” that “pure mathematicians glory in the uselessness of their work, and make it a boast that it has no practical applications”.

Having criticised these views, there is much in the Apology that is thought provoking and it is certainly worth a read. I like some of what Hardy says about the nature of mathematics and there is an interesting and frank account of what led him to the subject as a boy. Still, I am not sad that I missed reading the book in a more formative state of my development.

Afterword: Although I didn’t realise it at the time I sent the first draft of this text to Samuel for editing the morning after Sarah Shepherd died. A few days later when I received Samuel’s comments I had heard the news. I haven’t changed the substance of the piece since then, just tidied up a couple of useful points Samuel made about the flow and grammar of what I had written, yet the topic is curiously close to my involvement with Sarah’s work. I think Sarah would have agreed about the importance of promoting mathematics and the place of applied mathematics. She wrote passionately about the value of mathematics applied to the real world and the importance of communicating this to children and the general public. Through her iSquared Magazine she encouraged many young researchers to take their first steps in public engagement by writing about their research area for a popular audience. Consequently I dedicate this essay to Sarah Shepherd.

The puzzle grid

Here is a puzzle I enjoy giving to students. Given a 4×4 grid of sixteen dots, draw six straight lines that form a continuous path passing through all of the dots. Here, continuous means you must be able to draw over your six lines in one go without taking your pen off the paper.
This task is easy to complete with seven lines and impossible with five. Six is where the interesting puzzle lies.
I have given this to undergraduate students up and down the UK, and I have noticed a common response that makes me enjoy this puzzle and the lesson it teaches.
Many students start playing with the puzzle, drawing lines on grids right away. This is good because playing is how you start to understand puzzles and by showing their thought processes I can look out for when they have misunderstood the problem. In my experience, this is a problem often misunderstood.
One approach I see quite often is to draw lines within the grid always moving to the next adjacent dot – either up, down, left, right or diagonally. I ask the student, why aren’t you drawing outside the grid? “What,” the student typically replies, “you’re allowed to go outside the grid?”

Going outside of the grid on a strange diagonal

Then I ask, why are you only moving to adjacent dots, why not move at strange diagonals and jump around the grid in a more irregular fashion? Again, the response: “is that allowed?”
The statement of the problem was: “draw six straight lines that form a continuous path passing through all of the dots”. This says nothing to limit how you can move through the dots or where on the page you can draw your lines.
In fact, I know of fourteen solutions to this problem and each of them involves a strange diagonal and leaving the confines of the grid. By imagining these additional conditions the student creates a new problem – one that cannot be solved.
Good puzzles lie on a thin line between easy and impossible, and the particular wording in the statement of the problem is likely to be very carefully chosen to give you enough to solve it without being too easy. A would-be puzzle solver must check the wording to be sure they are attempting the puzzle that has actually been set and not some imagined, impossible version.