Flipping Classrooms and Hegartymaths

17th October, 2013

For many years, maths lessons have run in roughly the same way: the teacher stands at the blackboard, giving a mini-lecture on some mathematical topic or technique, introducing the idea, outlining the theory, and then running through an example or two. The students would sit patiently (or not), taking notes (or not), so that at the end of the day they will be able to tackle some exercises on the same subject as homework… or not.

As the years have rolled by, the blackboard may have been replaced by a whiteboard, and then by a smartboard, but otherwise the formula has remained by and large the same: theory in the classroom followed by exercises for homework.

The idea of “flipping” the classroom, is to reverse this process. The insight is that in the age of youtube, today’s students can perfectly well take in the lectury bit at home — with the added advantage of being able to pause, rewind, and rewatch in their own time. This leaves lesson-time free for practice, providing the teacher with more time to go around talking to students individually or in groups, meaning more opportunities to help those who are struggling or to supply extra challenges to those who need stretching.

I have no experience of this method myself — all the same it immediately appeals to me as a way in which technology may really be able to improve the teaching and learning experience, rather than just adding bells and whistles. One person who is convinced by this new approach is Colin Hegarty, an old friend of mine from university, who I’m delighted to say has returned to the world of maths after a few years in finance, and is now expertly flipping classrooms in North London.

Even if you don’t have the good fortune to be one of Mr Hegarty’s students, you can still peruse the 611 (!) videos that he and his colleague Brian Arnold have created for this purpose, all of which are freely available on Hegartymaths, or on youtube. Judging by the 800,000 odd views their videos have gathered, it’s not only their own pupils who are benefiting from these experiments in flipping.

As a sample I’m embedding one in which Colin talks through the Chinese postman problem:

Interviewed by Kevin Houston

7th October, 2013

You can read my interview with Kevin Houston (or should that be Kevin’s interview with me?) on his blog.

Constructible Numbers

13th September, 2013

This blog-post is an extract from my book Maths in 100 Key Breakthroughs

Newton by William Blake (1795)

Constructible Numbers

A sure route to mathematical fame is to resolve a problem that has stood open for centuries, defying the greatest minds of previous generations. In 1837, Pierre Wantzel’s seminal analysis of constructible numbers was enough to settle not just one, but an entire slew of the most famous problems in the subject, namely those relating to ruler-and-compass constructions.

As with so much in the history of mathematics, the topic had its origins in the empire of ancient Greece. The geometers of that period were interested not only in contemplating shapes in the abstract, but also in creating them physically. Initially, this was for artistic and architectural purposes, but later for the sheer challenge it posed. In time, mathematicians came to understand that the obstacles theyencountered in these ruler-and-compass constructions brought with them a great deal of mathematical insight. Nowhere was this more true than in the ancient enigma of squaring the circle, and what that revealed about the number $$\pi$$.

Classical problems

Greek geometers decided on a set of simple rules for building shapes, using only the simplest possible tools: a ruler and pair of compasses. The ruler is unmarked, so it can only be used for drawing straight lines, not for measuring length (therefore these are sometimes called straight-edge-and-compass constructions). The compass is used to draw circles, but it may only be set to a length that has already been constructed.

Today’s schoolchildren still learn how to use these devices to divide a segment of straight line into two equal halves and to bisect a given angle. These were two of the very first ruler-and-compass constructions. A more sophisticated technique allows a line to be trisected, that is, divided into three equal parts. What of trisecting an angle, though? Various approximate methods were discovered, which were accurate enough for most practical purposes, but no one could find a method which worked exactly. This proved a mystery, and gave the first hint that there was real depth beneath this question. But what does it mean if one task can be carried out by ruler and compass and another cannot?

The most famous of the ruler-and-compass problems, and indeed one of the most celebrated questions in mathematics, is that of squaring the circle. The question is this: given a circle, is it possible to create, by ruler and compass, a square which has exactly the same area? At the heart of this question lies the number $$\pi$$ (see page 54). The problem ultimately reduces to this: given a line 1 unit long, is it possible to construct by ruler and compass another line exactly $$\pi$$ units long?

Another classical problem was that of doubling the cube. This problem had its origins in a legend from around 430 BC. To overcome a terrible plague, the citizens of the island of Delos sought help from the Oracle of Apollo. They were instructed to build a new altar exactly twice the size as the original. At first they thought it should be easy: it could be done by doubling the length of each side. But that process leads to the volume of the altar increasing by a factor of 8 (since that is the number of smaller cubes that can fit inside the new one). To produce a cube whose volume is double that of the original, the sides need to be increased by a factor of $$\sqrt[3]{2}$$ (that is the cube root of 2, just as 2 is itself the cube root of 8). The question of doubling the cube therefore reduces to this: given a line segment 1 unit long, is it possible to construct another exactly $$\sqrt[3]{2}$$ units long?

Wantzel’s deconstruction

Working in the turbulent setting of France in the early 19th century, Pierre Wantzel turned these ancient questions over in his mind. He recognized that the form of many ruler-and-compass questions is the same. The key to them was this: given a line 1 unit long, which other lengths can be constructed? And which cannot? If a line of length $$x$$ can be constructed, then Wantzel deemed $$x$$ a constructible number. Setting aside the geometrical origins of these problems, he devoted himself to studying the algebra of constructible numbers. Some things were obvious: for example, if $$a$$ and $$b$$ are constructible, then so must be $$a + b$$, $$a – b$$, $$a \times b$$, and $$a \div b$$. But these operations do not exhaust the range of constructible numbers; Wantzel realized that it is also possible to construct square roots, such as $$\sqrt{a}$$.

His great triumph came in 1837, when he showed that everything constructible by ruler and compass must boil down to some combination of addition, subtraction, multiplication, division and square roots. Since $$\sqrt[3]{2}$$ is a cube root, and cannot be obtained via these algebraic operations, it followed immediately that the Delians’ ambition to double the cube was unattainable. A similar line of thought revealed the impossibility of trisecting an angle.

As for the greatest problem of all, squaring the circle, the final piece didn’t fall into place until 1882, when Ferdinand von Lindemann proved that $$\pi$$ is a transcendental number (see page 197). Then Wantzel’s work immediately implied the non-constructibility of $$\pi$$, and the impossibility of squaring the circle was finally established.

Maths in 100 Breakthroughs

12th September, 2013

I’m pleased to present a new book: Maths in 100 Key Breakthroughs is published by Quercus and is now available as a softback or e-book. You can buy it from the publisher here, or in the usual other places.

As the title suggests, its hundred chapters, ordered chronologically, each deal with a major mathematical development (e.g. Aristotle’s analysis of logical syllogisms circa 350BC, the discovery of transcendental numbers in 1844, and the creation of Weaire-Phelan foam in 1993). My hope is that it should be accessible, attractive, and entertaining to people with little or no background in the subject – jargon and technical notation are kept to a minimum, and each chapter is accompanied by a beautiful full-page colour illustration.

My major concern was to avoid wrenching these breakthroughs out of context and artificially presenting them as stand-alone events. After all, mathematicians typically make advances by contemplating the insights of previous generations and answering questions posed by earlier thinkers. Without Kelvin’s conjecture (and perhaps without the work of Pappus and Thomas Hales on related geometrical questions) the discovery of Weaire-Phelan foam would have been less exciting. Equally, it often takes time and further insight for the significance of a breakthrough to become apparent: it was some years after their initial discovery that the deep importance of transcendental numbers was recognised.

So I hope that the book not only presents some wonderful discoveries, but also tells the back-stories, gives some sense of what the characters involved thought they were up to, and discuss why their work matters to us today.

The revenge of the Perko pair

14th August, 2013

A few weeks ago, I was excited to receive correspondence from a certain Kenneth Perko. The tale of the Perko pair is a wonderful mathematical story, and one I have told on numerous occasions, so let me do so again:

The Perko Pair I: the story so far

In the late 19th century, mathematicians and physicists started producing tables of knots. The idea here is that some knots are genuinely different from each other, while others can be deformed to match each other without cutting or gluing, making them essentially the same thing from a topologist’s point of view. The trouble is whether or not two knots are really the same is extremely hard to tell on first sight…. as we shall see.

The tables begin with the unknot, the only knot with no crossings. Then comes the trefoil or overhand knot, which (so long as we don’t distinguish between it and its mirror image) is the only knot with three crossings. There’s similarly one knot with four crossings, then two with five, and so on. In the late 19th century Peter Guthrie Tait and Charles Little got as far as listing the knots with 10 crossings.

Knot table up to 7 crossings, from Wikipedia

However the early knot tabulators (unsurprisingly) made a few errors, and it was not until 1976 that Dale Rolfsen put together a comprehensive list of the knots with up to 10 crossings, based on earlier efforts by John Conway, in turn building on the work of James Alexander in the 1920s. Of the ten-crossing knots they counted 166 separate varieties. (Today’s mathematicians have made it as far as 16 crossings.)

The surprise was that Rolfsen and Conway had also made a mistake! Even armed with 20th century techniques of algebraic topology, a duplication had slipped through. In 1973[1] Kenneth Perko had been studying a 19th century table of Little, and had realised that two ten-crossing knots (subsequently labelled by Rolfsen as 10161 and 10162) were actually the same thing in disguise.

This story has a very clear moral: telling knots apart is difficult. Really difficult. Even after 75 years of brainpower, mathematicians were still coming badly unstuck, and who knows how long it might have taken without Perko’s alertness. And of course, this just for knots with a paltry ten crossings. Imagine trying to decipher knots with thousands…

Now for the sequel:

The Perko Pair II: Perko strikes again

The above story has been told countless times, and is usually accompanied by a picture of the offending pair of knots, something like this:

Warning: Not the Perko pair!

The trouble is that this picture is itself wrong!

This error has infected, most likely among many other places, Wolfram Mathworld and Mathematics 1001 by myself. And guess who pointed this error out…

The explanation of the mistake is that two updated versions of Rolfsen’s 1976 table are now in circulation. In both, 10162 has been deleted as it should be. But one version (occurring for instance in the latest editions of his book) keeps his original numbering up to 10166 with a space between 10161 and 10163. In the other, see here for example, Rolfsen’s 10163 has been renumbered as 10162, and 10164 as 10163, etc., thus counting up to 10165.

So the knot numbered 10162 in the picture above was actually Rolfsen’s original 10163, and thus not equivalent to 10161! For the avoidance of doubt, here is the real Perko pair, with drawings provided by the man himself. Accept no imitations!

While we’re at it, let’s also put paid to the to the idea that Ken is an ‘amateur’ mathematician. Before going on to a career in law, he says “at Princeton I studied under the world’s top knot theory topologists (Fox, Milnor, Neuwirth, Stallings, Trotter and Tucker)”. I apologise for suggesting otherwise. At least I didn’t pronounce him dead unlike certain other popular maths authors… Also, he adds “That stuff about rope on the living room floor is pure internet nonsense. I did it with diagrams on a yellow legal pad. Ropes wouldn’t work anyway since the two knots are non-isotopic mirror images of each other.”

[1] Note the corrected date, usually given as 1974.

Eric Jaligot (1972-2013)

28th July, 2013

I have just heard the very sad news that Eric Jaligot has died.

I did not know Eric well, although our paths crossed several times and we once collaborated in a piece of work [pdf] on the model theory of groups, an area in which Eric was a leading expert. He was knowledgeable, thoughtful, kind and generous with his time, and often to be seen with a wry smile on his face. I’m sure he will be sorely missed by his many friends within the model-theory community and beyond.

A tribute page is hosted at the Institut Camille Jordan in Lyon, where Eric was based.

A nautical problem

10th July, 2013

Here’s a little probability exercise. My wife and I are having some things shipped to UK from Japan, on board the container ship MOL Comfort. On 17th June, the ship broke clean in half. (Click on the pictures for more details.)

Breaking in Half

The crew escaped, and for the next 10 days, the two halves of the ship drifted apart in the Arabian Ocean, each laden with containers.

Broken in Half

On 27th June, the rear half sank with all its cargo, in 4km of water.

The aft sinks

Meanwhile the front half began to be towed (backwards) towards Oman.

The fore under tow

Since 6th July, the front section has been on fire.

The fore on fire

The salvage company/coastguard estimate that approximately 90% of the front half’s cargo has already been burnt. Further, because of the possible presence of dangerous chemicals among the cargo, along with large quantities of oil within the hull, they fear the risk of explosion.

Exercise
Estimate the probability that our stuff is:

1. at the bottom of the ocean
2. on fire
3. unharmed

Extension
Assuming that our stuff has survived so far, estimate the probability that it is:

1. going to blow up
2. going to sink
3. going to reach us safe and sound

Update: did you guess right?

The fore half continued burning until its balance was so out of kilter that it sank. It didn’t explode.

The End of the MOL Comfort, 10th July 2013

Carnival of Mathematics 100

3rd July, 2013

Welcome to the 100th edition of this Carnival of Maths! Yes, that’s the 13 + 23 + 33 + 43th edition.

I gather a party is planned for the 101st issue (that being a prime number of course). All the same, in our bimanual pentadactyl species, one hundred is a number laden with cultural significance. Making it to a hundred can earn you a letter from the Queen or can even see your name on the honours board at Lords. Although I can’t offer a prize so grand, let me at least express hearty thanks and congratulations on behalf of all readers, contributors, and hosts to Alon Levy who got this thing started back in 2007, [added later: and to Mike Croucher at Walking Randomly who ran it 2009-2012], as well as to Katie Steckles, Peter Rowlett, and Christian Perfect at the Aperiodical who between them have done a stirling job of keeping the ball rolling. Now on with this month’s delights…

• Let’s start with the biggest mathematical news story at the moment. Back in May, Yitang Zhang announced a proof of the following great breakthrough: there are infinitely many pairs of primes which are at most 70,000,000 (= H) apart. Terry Tao reports on subsequent huge success:

This project is still ongoing, but we have made significant progress; currently, we have confirmed that [the same thing] holds for H as low as 12,006, and provisionally for H as low as 6,966 subject to certain lengthy arguments being checked.

• Sticking with the number theory, if you want to express 5 as a sum of positive integers there are 7 ways to do it: 1+1+1+1+1, 2+1+1+1, 2+2+1, 3+1+1, 3+2, 4+1, 5.

So we say that the partition number of 5 is 7, or P(5)=7. Partition numbers are a fascinating and deep branch of number theory. In this post, Adam Goucher delves into the world of partition numbers, unearthing some astonishing discoveries of Ramanujan, and what is surely this month’s most terrifying formula:

• And some more number theory! What is the probability that two randomly chosen whole numbers are coprime (i.e. the only number that divides them both is 1)? David Cushing leads us to the answer, introducing some classic number theoretic objects along the way.

• On the subject of the number 100, via Seek Echo (via Diogenes Laertius via Apollodorus), we hear that a certain classical mathematician may have celebrated his world-famous theorem in grizzly fashion:

When the great Samian sage his noble problem found,
A hundred oxen dyed with their life-blood the ground.

• Hat games are classic genre of mathematical puzzles. Usually the hats come in two colours, and a line of people have to deduce their own hat type. (E.g..) Here, Tanya Khovanova presents a new twist:

A sultan decides to give 100 of his sages a test. The sages will stand in line, one behind the other, so that the last person in the line sees everyone else. The sultan has 101 hats, each of a different color, and the sages know all the colors. The sultan puts all but one of the hats on the sages. The sages can only see the colors of the hats on people in front of them. Then, in any order they want, each sage guesses the color of the hat on his own head. Each hears all previously made guesses, but other than that, the sages cannot speak. They are not allowed to repeat a color that was already announced.

• Lê Nguyên Hoang solves a famous conundrum using a famous theorem:

In this article, we’ll present and solve the classical utilities problem, using one of the most beautiful formulas of mathematics, due to Leonhard Euler. This will have us walking a few steps in the stunning world of algebraic topology!

• Another well-known variety of puzzle is “What comes next…”. Tony Mann has some good examples, but also points to a problem with the whole formulation:

If, at the beginning of next season, I record the number of goals scored by my football team in their first 5 matches as 0, 0, 0, 0 and 0 (which is not unlikely), I might take comfort from the fact that the formula (n-5)(n-4)(n-3)(n-2)(n-1) has successfully predicted the goals in the first five matches and that therefore I can expect my team to score 120 in their sixth match.

• Sticking with the puzzles, here’s a nice geometrical challenge from Futility Closet:

There’s a long way to solve it and a very neat quick way. I did it the long way…

• How old is the oldest person you know? Mr Honner opines that this may not be the right question, and tells us about “the most viewed histogram of all time”. There’s a thoughtful discussion in the comments too. (Are the Pru perhaps capitalising on the availability heuristic?)

• In another blog-post inspired by an advert, Math Goes Pop challenges the man from AT&T to a game of What’s the biggest number you can think of?

• As part of the long-running debate about what mathematical education is, what it should be, what it’s for, and what it should be for, Alice Carey and Stephen Wilson discuss The Faulty Logic of the ‘Math Wars’ in the New York Times Opinionator blog.

• Continuing with the politics of mathematical education, here in the UK Michael Gove wants all children up to the age of nine to know their 12 times table. But in our decimal world, what purpose does this really serve? John McLoone at Wolfram Blog has done some investigations:

The “return on effort” drops very rapidly toward the 10 times table and then barely improves. It seems like a fairly compelling case for stopping our rote learning at 10.

• In Turkey we have a more dramatic combination of mathematics and politics, where Ali Nesin has been teaching mathematics amid the current turmoil and teargas in Istanbul. Alexandre Borovik has a photo of Ali speaking in Gezi Park (which adjoins Taksim Square, the centre of the protests).

There’s also news footage here. Ali was previously arrested for the crime of “Education Without Permission” – you may remember the extraordinary image of a blackboard with group-theoretic notes on it behind a police cordon. Once again, we wish Ali well.

• And even more on the theme of the number 100. What would the world look like if it was a village of 100 people? Jim Noble and students of the International School of Toulouse show us.

• Question: what do you get if you combine the ‘u’ species of a Gray Scott reaction diffusion model with the density of a simplified Navier Stokes fluid dynamics model?

I’ve no idea – I can’t even pretend to understand the question. But the answer appears to be solitons which are on fire. Whatever it may mean, this simulation by Simon Gladman is undoubtedly very cool, especially with the banging techno soundtrack.

• Math Tango has a glowing review of a compilation of mathematical articles from the New York Times.

• Of course book reviews are common things, as are film reviews, restaurant reviews… but every week[1], the guys at the Aperiodical have been writing integer sequence reviews. And now they’ve expanded this project into a sort of integer sequence world cup, or as they put it Integer Sequence Review Mêlée Hyper-Battle DX 2000. Be that as it may, the winner of this week’s heat is an absolute corker – well worth taking the time to understand.

• It’s common to look at the average (mean) lifetimes of various things: people, pieces of technology, biological species… In a post at The Endeavour, John D. Cook introduces us to an important variant of this idea:

If something has survived this far, how much longer is it expected to survive? That’s the question answered by mean residual time.

It’s worth clicking through to John’s explanation of an interesting consequence, the Lindy effect: “The longer a technology has been around, the longer it’s likely to stay around.”

• Have you ever calculated the determinant of a 3×3 matrix or bigger? If so, you’ve probably wondered what it is you’re actually doing. Michelle at My Summation provides some insights.

And finally…

• Futility closet introduces us to some new units of measurement:

Since Helen’s face launched a thousand ships, Isaac Asimov proposed that one millihelen was the amount of beauty needed to launch a single ship.

• Which UK city googles for “maths” most often? Mr Gregg has investigated, but you won’t guess the answer.

• What colour uniform would mathematicians wear on the Starship Enterprise? Peter Rowlett has invested a surprising amount of thought into this question.

[1] Most weeks

Flowers of Segregation

2nd July, 2013

As I’ve mentioned in previous posts, I have recently been doing some analysis of Thomas Schelling’s model of racial segregation with Andy Lewis-Pye and George Barmpalias.

I’m delighted to say that a picture coming out of our work, Flowers of Segregation has just been announced as winning the Infographics category of the Royal Society’s Picturing Science Competition.

Click to see it full size, and read here about what it means.

Schelling Segregation (Part 2)

18th June, 2013

I talked in Part 1 about a model of racial segregation devised by Thomas Schelling in 1969 [original pdf]. This post will describe how the model works, in its simplest 1-dimensional incarnation, and what we’ve discovered about it. (I hope this post is also accessible to everyone, but I’ve included the formal statements for the more mathematically-minded reader.)

Schelling’s Model

Suppose that a large number of people, say a hundred thousand (or more generally $$n$$), all live around the edge of a giant circle. This is the city. We’ll imagine that its citizens are of two races: red and blue. Initially we populate the ring randomly, meaning that we go around each house in turn tossing a coin to decide the colour of the resident.

Once all the people are in place, each person will only be concerned with their own immediate neighbourhood: say the 80 people to their left, and the 80 people to their right, making its total size 161. (The number 80 is the called neighbourhood radius, or $$w$$, and as with $$n$$ we can alter it later.)

Now we have to factor in people’s preferences. Let’s imagine that each person is happy so long as they’re in the majority in their neighbourhood, but they’re unhappy if they’re in the minority. So, if some red individual’s neighbourhood contains 75 red people and 86 blues, she’ll be unhappy. But if contains at least 81 red people, including herself, she’s guaranteed to be happy, since the reds are bound to outnumber the blues. We’ll imagine that everyone feels similarly.

The idea is that unhappy people may then move house (while happy people won’t). To model this, at each time-step we pick, at random, a pair of unhappy people of opposite colours and swap them. This will have knock on effects on the neighbours of those two nodes, who may themselves change from being happy to being unhappy, or vice versa. We keep repeating this until we run out of unhappy people of one colour or other.[1]

Ok, those are the rules. And the big question is: what will the ring look like at the end of this process? And the answer is…

As you can see, distinct red and blue regions have developed. If we want to measure the level of segregation, we need to know how big these areas are.

The first rigorous answer to this question was provided in a recent paper by Christina Brandt, Nicole Immorlica, Gautam Kamath, and Robert Kleinberg. They prove, roughly speaking, that the segregated regions are not significantly bigger than the original neighbourhoods.

More technically, their theorem says this: for any $$\epsilon>0$$, for all $$w$$ and all sufficiently large $$n$$, the average length of the segregated regions is $$O(w^2)$$ with probability at least $$1-\epsilon$$. (They conjecture that this can be further tightened to $$O(w)$$.)

Altering the Tolerance

Here’s a new question: what might happen to this picture if we tweak the system to make people more tolerant? Instead of requiring their own colour to be a majority in their neighbourhood, perhaps they’re happy so long as it’s not below 38%, say. Well, this is the picture that emerges if we set the tolerance paramater $$\tau=0.38$$…

This is, I hope you’ll agree, a rather surprising turn of events. After all, one might naively expect that if people are content with a greater level of mixing, this should be reflected in their final arrangement. But what we’re actually faced with are dramatically larger segregated regions.

In more technical terms the segregated regions have jumped from being polynomial (or conjecturally linear) to being exponential relative to the neighbourhood radius.

In even more technical terms, in our paper, Andy Lewis-Pye, George Barmpalias and I prove that in this situation there exists a number $$d>0$$ such that for any $$\epsilon > 0$$ and for all $$n \gg w \gg 0$$ (meaning “for all sufficiently large $$w$$ and all $$n$$ sufficiently large compared to $$w$$”), the probability that a randomly chosen node will end up in a segregated region of length greater than $$e^{\frac{w}{d}}$$ is greater than $$1 – \epsilon$$.

Why should this be? Well, we get a hint if I add in some extra information to the pictures. Here’s the first scenario again, where the tolerance is $$\tau=0.5$$:

The innermost ring is the city’s initial configuration. Outside that are markers for the initially unhappy nodes. Then the body of the ring shows the changes which take place over the course of the run, with distance from the centre proportional to when the change happened. At the outside is the result: the same final configuration we saw before. Now let’s have another look at the case where $$\tau=0.38$$:

In this case, there are visibly fewer initially unhappy nodes. But each of them sets off a domino effect: when they change, the nearby nodes of the same colour become unhappy, and so on. Paradoxically, because almost everyone is initially happy, the resulting firewalls can extend much further before running into each other, which is how the larger segregated regions are formed.

A Threshold Between Segregation and Integration

What if we make people even more tolerant? Suppose $$\tau=0.3$$, which is to say people are happy so long as at least 30% of the neighbourhood are their own colour.

As you can see, there’s not a lot going on here; now everyone is initially happy, and remains that way. In our paper, we show that the threshold value is $$\kappa \approx 0.353092313$$. More precisely, for those who are interested, it is the root of the following equation[2]: $\left( \frac{1}{2}-\kappa \right)^{1-2\kappa} = \left( 1-\kappa \right)^{2-2\kappa}$

For values of $$\tau$$ above this threshold, but below 0.5, the theorem above applies (though the value of $$d$$ will vary). For values below $$\kappa$$, the ring will be static or nearly so. More precisely, we prove that for any for any $$\epsilon > 0$$ and for all $$n \gg w \gg 0$$, the probability that a randomly chosen node changes colour at any point is less than $$\epsilon$$.

This threshold manifests itself very dramatically when running simulations: below it very little happens. Then… boom!

Moving To Total Segregation

Finally, you might wonder what would happen if we make people less tolerant. Perhaps everyone requires their neighbourhood to contain at least 65% of their own colour, meaning $$\tau=0.65$$. In this situation, common-sense suggests that we should see higher levels of segregation, and this time case common-sense gets it right, though the change is again more dramatic than one might have guessed:

(This ring has $$n=10,000, w=20, \tau=0.65$$.)

In fact, the point 0.5 is another crisp threshold: whenever $$\tau$$ exceed 0.5, so long as $$w$$ is large enough, complete global segregation is inevitable sooner or later [3].

[1] Actually what I’ve described is the closed model. There’s an even simpler open model in which we pick at each stage a single unhappy node and change its colour – we might imagine that the unhappy red resident has moved out of the city altogether and been replaced by a blue resident whose moved in from elsewhere. One challenge is to determine to what extent the open model is a reasonable approximation to the closed one.