For a while now I’ve been fascinated by the story of Claude Shannon, the pioneer of information theory and the originator of many fundamental concepts now used in all modern manipulation and transmission of data. Being sent a copy of this biography to review was a great chance to find out more about his work and life.

**A Mind At Play: How Claude Shannon Invented the Information Age
**

The authors, who describe themselves as biographers and writers foremost, have taught themselves the mathematics they need to explain Shannon’s work, and weave in some excellent and succinct explanations of the concepts amongst a fascinating human story. From his early years as an enthusiastic maker and tinkerer, through his various university courses and his placement at Bell Labs, to his later years at MIT and retirement, Shannon’s life is chronicled in detail, with a spread of well-chosen photographs to accompany the story.

Claude Shannon is described as the father of information theory – his seminal 1948 paper outlined concepts including the fundamental nature of binary numbers (coining the word ‘bit’, a binary digit), information density, communication channels, and the theoretical Shannon Limit of how quickly digital information can theoretically be transmitted in a noisy channel. These ideas predated even simple computing machines, and Shannon’s work was perfectly timed to provide a foundation for those creating early computers.

The story gives a real sense of how Shannon was well placed to create the mathematics he did – with a sharp intellect that was torn between his love of abstract mathematical theory and his fondness for hands-on inventing and engineering, he had just the right mindset to see what communication theory would become and how it could be made rigorous in a mathematical framework.

It’s also fascinating to learn about Shannon’s other passions in life – nothing he did before or since comes close to the major impact his work on information theory had, but it was far from his only passion. Other areas of mathematics and engineering, as well as pastimes including juggling, stock market predictions, and building robots all fell to his mighty intellect and he brought huge joy to the people around him with his stories and ideas.

The book is well written and lovingly put together (and has a frankly beautiful cover in the hardback edition). It was enjoyable to read, and full of interesting facts and stories. I didn’t realise until reading this book that a wooden box I have at home, which has a switch on top that when flipped, engages a robotic arm that pops out and flips the switch back again, is a modern incarnation of an invention of Shannon’s – he called it ‘the ultimate machine’, one which switches itself off. Knowing this was his original creation, and the joy I find in it, gives me a real sense of connection to this brilliant mathematician whose work changed the world for all of us.

A Mind At Play: How Claude Shannon Invented the Information Age by Jimmy Soni and Rob Goodman is published by Simon and Schuster.

]]>Here’s the final set of photos and video clips from the last week, and for the data fiends among you, a sneaky look at my spreadsheet of runs. With a graph, as requested by Hannah Fry.

Day 23 (aka @sportrelief day): stupid bloody GPS watch failed to log my distance, but I fixed it using the screen grab from my route planning map (which I won't share because it gives away where I live). A good pace! #pikmdotrun pic.twitter.com/6YWz03EeEl

— Katie Steckles (@stecks) March 23, 2018

Next up was this stroke of genius:

Day 25: had the genius idea of getting dropped of 3.14km away from the pub, where we're having a Sunday roast #winning #pikmdotrun pic.twitter.com/kndBgPY2gB

— Katie Steckles (@stecks) March 25, 2018

Augh! Just realised my tweet from yesterday didn't send! Day 27: ran out in the countryside near my nice hotel, missed the turning, had to just run 1.57km away and turn round. #pikmdotrun pic.twitter.com/62pzulsbUZ

— Katie Steckles (@stecks) March 28, 2018

Day 30: Final gym run (will be outdoors tomorrow). Accompanied by @aPaulTaylor & Waterbot. https://t.co/dtKU5Rj0qH pic.twitter.com/rm8AGaJqDQ

— Katie Steckles (@stecks) March 30, 2018

This final run video shows clips and photos from a bunch of the days, plus my triumphant final approach on Day 31:

There’s still time to chuck a final £3.14 on the pile at pikm.run, if you haven’t already. Thanks again to everyone! Now I’m going for a sit down.

]]>If I can make it to £1000 before the end of the month, I’ll be pretty pleased! Donate at pikm.run, or see below for my daily sweaty photos/videos/instagram posts.

Day 9: yep, I'm still doing this #pikmdotrun pic.twitter.com/MpFQvEImgr

— Katie Steckles (@stecks) March 9, 2018

On the 9th, I thought I’d make use of the mathematical properties of π to do a slightly silly one, and made a video:

Day 11: gym again. If you missed it yesterday, here's my video from Day 10: https://t.co/rdiPCJ534N pic.twitter.com/knyzH4L6f9

— Katie Steckles (@stecks) March 11, 2018

Day 13: still going. Thanks to everyone who's supported so far! https://t.co/dtKU5Rj0qH pic.twitter.com/p5qUUNncSD

— Katie Steckles (@stecks) March 13, 2018

I also managed to get in a few more joint runs with running companions:

Day 15: now officially kinda halfway! Buddy gym run again with @elsie_m_ #pikmdotrun pic.twitter.com/VhcVIsghll

— Katie Steckles (@stecks) March 15, 2018

Day 17: logistically complex. Managed to run πkm fully inside the B'ham NEC, as I've been working at @BigBangFair today. GPS watch gave up after 2.04km (no signal). #pikmdotrun pic.twitter.com/eebrhKjUgh

— Katie Steckles (@stecks) March 17, 2018

Day 19: staying in the gym due to the cold weather, on the world's shiniest treadmill #pikmdotrun pic.twitter.com/WO9xaPKaSV

— Katie Steckles (@stecks) March 19, 2018

I was also given an amazing gift by maths/knitting fan Linda Pollard, who came to see me perform at a show. She’s written up the knit of these magnificent π/sum gloves on a Ravelry page. I took the opportunity to test out their warmness on my next outdoor run:

This mild cry for help resulted in plenty of nice replies on Twitter, which has been a real boost – including Aperiodichum Colin Beveridge, who pointed out that my total is around $\pi^4$, a pleasing coincidence.

Day 21: Gym again (but went to a different gym for variety). Found it hard today. Encouragement please. #pikmdotrun pic.twitter.com/lIvuC7yBUZ

— Katie Steckles (@stecks) March 21, 2018

And of course, today’s effort:

Running continues. Watch this space for a final wrap-up and fundraising total at the end of the month.

Katie’s fundraising page at Sport Relief

More information about Sport Relief

This month I'm doing a completely irrational sponsored run for Sport Relief, aiming to raise £100π by running πkm per day, every day in March. I'm one week in, and here's the story so far.

Given the first few days of my challenge coincided with one of the most ridiculous periods of cold weather we've seen in a while, I wasn't quite willing to brave the outdoors yet, so my first few days were done on a treadmill in the gym.

On the fourth day, the weather broke, so I took advantage of the fact that my parents live near a lake whose perimeter works out to almost exactly πkm. I dragged my dad round it – and even made a short video:

On day 5, I dragged another of my family members for an outdoor run:

And finally, today I managed my first solo outdoor run, literally running an errand to fetch some workshop materials from the Museum of Science and Industry across town. By careful route choice, I managed to hit my target within sight of the front doors:

The best news of all is that on day four, I managed to hit my fundraising target! I reached £314.15 on 4th March, and promptly decided to extend the challenge by upping the target. My new goal is** £3141.59**, which I'm almost certainly not going to reach, but it gives me something to shoot for. I've got the rest of the month!

You can make a contribution (all to the brilliant cause that is BBC Sport Relief), and continue to follow along with my social media updates/proof, by heading to my fundraising page at pikm.run.

]]>

SIR – As one obsessed with prime numbers, I note that we have gone from 2017 (a prime) into 2018 (two times a prime), which will be followed by 2019 (three times a prime). I believe this sequence has only happened three other times in the past 1,129 years.

Keith Burgess-Clements

Maidstone, Kent

It’s lovely that newspapers will print this kind of letter, and a quick check verifies that Mr Burgess-Clements is indeed correct that these three numbers have the properties described:

- $2017$ prime, $2018 = 2 \times 1009$, $2019 = 3 \times 673$

His follow-up statement, that this sequence has only happened three other times in the last 1,129 years, takes a little more checking. But only a little – as we have a resident CL-P, who describes the necessary calculation as ‘a trivial amount of Python code’, and quickly came up with the following list:

- $13$ prime, $14 = 2 \times 7$, $15 = 3 \times 5$
- $37$ prime, $38 = 2 \times 19$, $39 = 3 \times 13$
- $157$ prime, $158 = 2 \times 79$, $159 = 3 \times 53$
- $541$ prime, $542 = 2 \times 271$, $543 = 3 \times 181$
- $877$ prime, $878 = 2 \times 439$, $879 = 3 \times 293$
- $1201$ prime, $1202 = 2 \times 601$, $1203 = 3 \times 401$
- $1381$ prime, $1382 = 2 \times 691$, $1383 = 3 \times 461$
- $1621$ prime, $1622 = 2 \times 811$, $1623 = 3 \times 541$

*Here’s that Python code, in case you’re curious. It uses Sage’s Primes() function.*

pr = set([p for p in range(2018) if p in Primes()])

double = set(2*p-1 for p in pr)

triple = set(3*p-2 for p in pr)

years = sorted(pr & double & triple)

This is a full list of all the cases below 2017, and hints at some nice more interesting patterns – $13$ and $541$ both occur as the prime year and the prime factor that's a third of another year. But we don't have time to dig into that now – we have to check if a person in the newspaper was wrong!

Keith's claim that this has happened thrice in the last 1,129 years is indeed correct – $2018 – 1129 = 889$, and three sets of years have occurred since then. I suspect this may have been a typo though, as if he'd said "the last 1,139 years", that would have included the tail end of the set starting in $877$. Maybe he was looking for the most impressive length of time with the fewest occurrences, to illustrate how rare it is (in which case 1139 would be your best bet, and probably what he meant). We favour "only four times since 1000AD" which still sounds pretty good.

One final question to answer is, how many of these will there be going forward? The next few will be:

- $2557$ prime, $2558 = 2 \times 1279$, $2559 = 3 \times 853$
- $2857$ prime, $2858 = 2 \times 1429$, $2859 = 3 \times 953$
- $3061$ prime, $3062 = 2 \times 1531$, $3063 = 3 \times 1021$

It's also worth checking when this pattern will continue for four years, so that the fourth year is four times a prime; that's $12721$, which is prime, while $12722 = 2 \times 6361$, $12723 = 3 \times 4241$ and $12724 = 4 \times 3181$.

What's your favourite number fact about 2018 so far? Answers in the comments.

]]>I’ll aim to run **π kilometres** (or as close as I can get, with the measuring instruments I have access to) each day during the month of March. This will either be on the treadmill at my gym – in which case I’ll try to get a photo of the ‘total distance’ readout once I’ve finished – or out in the real world, for which I’ll use some kind of running GPS logging device, to provide proof I’ve done it each day. Some days I’ll run on my own, and others I’ll be accompanied by friends/relatives, who’ll be either running as well or just making supportive noises. At the end of the month, I’ll post an update documenting my progress/success/failure.

**Serious request**: if you know of anywhere in the UK I can reasonably get to where there’s an established circle that’s exactly 1km in diameter, I can try to come and run round the circumference of it. Drop me an email if so.

If you’d like to support my ridiculous plan, you can follow my progress and donate on my fundraising page, or encourage others to do so by visiting pikm.run (I paid £4 for the URL, so now I have to do it). Sport Relief is the even-numbered-years-counterpart of Comic Relief, which together raise money for thousands of projects all over the UK and in the developing world, to help the vulnerable and those in need.

]]>I guess I’m alone in this, since Matt Parker’s ‘guess the mean of the digits in all the entries’ competition received the most entries of any competition meaning he also won a prize.

Bored with ‘guess the mean of all the entries’ style competitions, I decided to come up with something that flips this concept on its head.

My competition was to come up with a function for mapping a set of $N$ integers onto a single value, and whoever’s function takes a pre-determined set of integers I’m thinking of closest to a particular predetermined value I’m thinking of is the winner. I was slightly sloppy in the way I defined the question (I used $N$ to refer to both the set of numbers, and the size of the set) but everyone responded beautifully – with only some people feeling the need to point out my error – in what became a virtual festival of making up your own notation.

I received 32 entries to the competition, not all of which were actually valid entries – several had the bright idea of using ‘the number you’re thinking of’ as a variable in their function, which is 100% cheating. Here are some of my favourites, in increasing order of amazingness, culminating with the winning entry (scroll down to the bottom if you just want to see that).

The set of numbers I was thinking of was in fact The Numbers, from the TV series Lost – in which a seemingly random set of integers play a mysterious yet crucial role in the plot. The numbers are 4, 8, 15, 16, 23 and 42. As well as being one character’s winning lottery numbers, and popping up in various other places, for a whole chunk of season 2 the numbers must be typed into a computer every 108 minutes, or else… something bad will happen.

I actually quite enjoyed it but I have notoriously bad taste in TV and films.

If you haven’t seen Lost, and you have a spare 122 hours of your life you’re happy to not get back, go for it.

My set of numbers is therefore as follows: 4, 8, 15, 16, 23 and 42 (in that order). The target number for the function was chosen to be zero, for reasons I can’t explain (mostly panic).

On to the entries!

Several examples fell into this category – including:

$f(x_0, x_1, \ldots, x_N) = N$

Simply counting the number of numbers (minus one). While we appreciate your lovely and obfuscating zero-indexing, this gives the answer $5$.

$f(x,y) = x^2 – y$

Since this function is only defined on two inputs, we took the first two, giving $ 4^2 – 8 = 8$.

$f(x_1, x_2, \ldots, x_N) = x_N^2 – 1$

Taking the last of the numbers and squaring it, then subtracting one. This gives $42^2 – 1 = 1763$.

Various

statistical functions, including thesquare of the mean, theupper quartile, thelower quartileand the ‘anti-mode‘ (the least common value, and if there’s more than one value equally uncommon, the mean of these – since the set is all distinct, this is just the mean LOL!).

These gave the values $240.25, 19.5, 11.5$ and $18$.

“

The first prime not a factor of any of the numbers”

This is $11$.

“

Return a random element of the set”

Since the target value isn’t in the set, this won’t work but it’s a nice try!

\[f: x_1 \ldots x_N \rightarrow \frac{e^{\min(x_i)} \cdot \pi^{\max(x_i)}}{N}\]

Almost like Euler’s identity in how it effortlessly combines mathematical constants, but sadly returns a result in the region of $7 \times 10^{21}$.

Many entries fell into this category:

$f(x_1, x_1, \ldots, x_N) = \frac{\displaystyle\sum_{i=1}^N (-1)^i i x_N}{N}$

If this was meant to be an alternating sum of multiples of successive entries, which is how I initially interpreted it, it also doesn’t sum to zero. And you should be more careful with notation, because this is exactly what was written.

An alternating sum of multiples of the largest entry, which given $i$ runs from $1$ to $6$ means you end up with a total of three times that entry, divided by 6 – giving $x_N/2 = 42/2 = 21$.

$f(n_1, n_1, \ldots, n_N) = \frac{\displaystyle\prod_{i=1}^N n_i}{N}$

The product of all the entries divided by the number of entries, which is $1,236,480$ (very much not zero).

The $\big(\displaystyle\prod_{i=1}^N x_i\big)^{th}$ prime number, modulo $\displaystyle\sum_{i=1}^N x_i$

This one returns 67.

$\frac{1}{M} = \frac{1}{x_1} +\frac{1}{x_2} + \cdots +\frac{1}{x_N}$

For reference, $M$ was the name I gave to my target value, and this was described as the ‘parallel resistance sum’. It works out to give $M = 1.7499207463\ldots$.

$f(x) =\displaystyle\sum_{k=1}^n \sin (kx_k)$

I like it, but it gives $-1.168\ldots$ – close, but no cigar.

Several entries took a chance on me, and hoped they could guess what number I was thinking of, defining their function as the constant function giving a single value regardless of the inputs. Entries of this form, in increasing order of size, included $\pi, 4, 7, 20$ and $37$, sadly none of which win.

Ask a stupid question, and you’ll get the following answers. We tried to calculate some of these in the ten minutes we had to judge the competition, but for some of them we merely established they were definitely not zero and gave up.

$f(x_1, x_2, \ldots, x_N) = x_1 + \frac{1}{x_2+ \frac{1}{\cdots + \frac{1}{x_{N-1} + \frac{1}{x_N}}}}$

A continued fraction. We calculated the answer but all that’s written on here is “4.something”. Not equal to zero.

“Arrange the numbers in order and multiply the difference between them in pairs.”

I enjoy a good bit of tedious busy work, and I got $3724$.

$f(n_1, n_2, \ldots, n_x) = (x+2) \cdot \sqrt{n_1 \cdot n_x \cdot\displaystyle\prod_{\textrm{all } x} n_x}$

Interesting notation switcheroo from what everyone else seems to be using. This was a pain to calculate too and my bit of paper says ‘TOO BIG’.

“The median of all the answers to this competition + 1”

Given that we had 32 different entries, some of which we couldn’t properly calculate, but there was literally no chance the median answer was $-1$, the only thing written on the paper is the word ‘NO’.

“Sum the number of letters of the positive integers, minus the number of letters of the negative integers”

It’s lovely! Our bit of paper says “about 40ish?” but if anyone wants to work it out, go for it.

“For each element of N, go to the Wikipedia page for N(i) and find the N(i)th letter on the page; typecast this letter into I32 and take the sum of these values”

A bit of Googling has allowed me to establish that this is a way of converting letters into numbers, but since I assume none of the standard letters convert to a negative number, this will definitely not equal zero.

$f(x_1, x_2, \ldots, x_N) = e^{e^{\displaystyle\sum_{i=1}^N x_i}}$

We had a go at calculating this in Google Sheets, which was our terrible calculating tool of choice under pressure, and the value was actually too big for it to handle. In some ways, this could be the least correct answer.

Given that I named the number I was thinking of $M$, I received a small selection of entries from people who didn’t want to play like normal people, at least one of which just specified $f(N)=M$ – this is OBVIOUSLY not what you were meant to do. Others tried to define a function in terms of $N$ and $M$, including the following:

\[f(x_1, x_2, \ldots, x_N) = \begin{cases}

\Bigg(\frac{\displaystyle\sum_{i=1}^N \displaystyle\sum_{j=1}^{x_i} x_i^j}{N!}\Bigg)^{\displaystyle\frac{1}{x_{\max{3,N}}}}&& \textrm{if }N\textrm{ is coprime with }M\\

6 && \textrm{otherwise}

\end{cases}\]

For the record, while typing this in LaTeX I’ve hit a record of four consecutive close braces. Sadly $6$, the number of entries in my list, is not coprime with $0$, so this gives the answer $6$ (and is disqualified anyway for using $M$ in the definition).

$f(N,M) = 0. \big(\displaystyle\sum_{i=1}^{N-2} x_i\big) + M$

This was presumably hoping to get as small a number as possible in the first part, then add it to M and hope nobody was closer. Turns out not only is this cheating, it was also not close enough, because we did actually get an exact winner, and it was glorious. See below.

$f(n_1, n_2, n_3, \ldots, n_N) = n_2 + n_3 + \ldots + n_N \mod n_1$

This function takes all but the first number, adds them together and then takes the result modulo the first number, aka remainder on division. In the case of The Numbers, the result is:

$f(4, 8, 15, 16, 23, 42) = (8 + 15 + 16 + 23 + 42) \mod 4 \equiv 0$

Amazingly, through pure chance, the sum of these five numbers (104) is a multiple of 4, and so the answer is zero! We have a winner. Congratulations to the winner, who forgot to write their name on the sheet, but was identified and awarded their prize – a wind-up robot, as per the rules of the Competition Competition, worth strictly less than £1.

]]>

**What’s The Calculus Story all about?**

It’s an introduction to calculus for an unusually wide readership, mainly through the story of how the subject developed. And this turns out to be something of an adventure, largely because of the way infinity comes in at almost every twist and turn.

**How is the book different to previous ones you’ve written?**

My previous book, 1089 and All That, was a fairly light-hearted look at maths in general, and became something of a bestseller, with 11 foreign translations. The new book takes the subject a bit further, and is, if anything, even more ambitious, because it tries to explain not only what calculus is, but how to actually start doing it.

Calculus is all about the rate at which things change, and this is how we often get to understand, through the laws of physics, how the world really works. But the subject contains many results, too, that can be enjoyed purely for their own sake, usually because they are surprising in some way.

**What do you hope the book will achieve?**

It is always easier to embark on calculus if you have some idea from the very beginning of the subject as a whole – some ‘big picture’, if you like. Without that, you can easily get bogged down in comparatively minor detail, lose direction, and – even worse – lose interest.

My main hope, then, is that The Calculus Story provides that big picture, and in an unusually readable way.

And if anyone were to read the book as preparation for a university or college course, they would – in my view – really hit the ground running.

**Why did you choose to write the book now?**

I didn’t. The book has been gently brewing, in a way, since 1962, when I first met calculus, at the age of 16. For once you’ve met it, your mathematical life is never quite the same again – calculus just opens so many new doors.

**What’s the most interesting fact you learned while writing the book?**

If you drill a hole of given depth straight through the centre of a sphere, the amount of material left over is independent of the radius of the sphere!

**Have you ever used calculus in ‘everyday life’?**

No. Calculus underpins much of modern life, but in a rather hidden way, through the laws of physics, chemistry, biology and economics. It tends to be ‘under the bonnet’, so to speak.

The most likely way that calculus might be used explicitly in a truly ‘everyday’ context is to solve some optimisation problem.

**Who do you think the book would best suit as a Christmas present?**

You’d better ask the New Scientist. They have just selected it as one of their choices for Christmas, claiming that (a) it will fit in a stocking and (b) it has ‘something for everyone’!

The Calculus Story, on Oxford University Press

The Calculus Story, on Amazon

*The new live DVD from science comedy trio Festival of the Spoken Nerd, Just for Graphs, is out now, and we’ve been sent a copy to review. We got together a pile of appropriately nerdy science fans to watch (left), and here’s what we thought.*

The latest Festival of the Spoken Nerd DVD/download is from the Nerds’ 2015 show Just for Graphs – it toured the UK in late 2015, and had a hugely successful Edinburgh Fringe run in August 2015. The show is themed around graphs, plots, charts and diagrams – as mathematicians, we were sad to see they’re not entirely using graphs made from nodes and arcs (although a few of those do make it in the show!)

As well as plenty of classic diagrams (Venn, Euler and otherwise) there’s also plots – Steve Mould plots the birth of his child, while Matt Parker plots a function that plots itself – and Helen Arney brings musical interludes and live demos, including an electrifying demonstration of how the graph of electrical voltage in a speaker cable can be transmitted through not just wires but people, and at one point a large section of the audience.

The show also contains some nostalgia for retro technology (which raised some cheers from us), and an interesting new way of plotting a graph of the pressure inside a tube of gas – by setting the gas on fire, of course.

While watching it on a screen doesn’t quite have the same ambience as seeing it live in a theatre, you still get a sense of how the live audience experiences it and the show is full of visual spectacles, which do come across well on screen. I was part of the production team for some of the tour shows and Fringe run, and it was just like being there for real (only smaller and more pixelated).

The show is full of science references and deliciously geeky jokes, and without spoiling too many of the punchlines/conclusions, if you’re into maths or science and want to be entertained, it’s a graph a minute.

*Just for Graphs is available on DVD, as a digital download and for some reason on VHS.*

The closing talk of the HLF’s main lecture programme (before the young researchers and laureates head off to participate in scientific interaction with SAP representatives to discuss maths and computer science in industry) was given by Fields Medalist **Steve Smale**.

Speaking without slides, Smale shared with us some of his recent work in the crossover between mathematics and biology, but the central theme of his talk was one which goes to the heart of what mathematics is. Mathematics is embedded in science, and is used to describe and understand many aspects of scientific discovery.

The question was of whether mathematics is realistic or idealistic – do the mathematical models we use to solve problems and understand the universe give a realistic picture of how the world works, or is it all fantasy and we’re ignoring the fine details in order to get a model that works nicely? It’s a constant struggle, and Smale illustrated this with several historical examples.

**Alan Turing**’s theory of computation was an inspiring vision of how we can understand and use computers, and has influenced the whole field since. But was it realistic? For some applications, Turing’s approach is the correct model, but for others it fails. Modern study of NP-completeness in computability assumes an infinite amount of input, but obviously this doesn’t model the real world – it’s an idealisation.

Moving on to another giant of maths history, Smale turned to **Isaac Newton**. Newton’s work on physics, differential equations and mechanics was all an idealisation – to the extent that his calculations didn’t even include friction, which was added to the theory 100 years later.

**John Von Neumann**, who created the early models of quantum mechanics, introduced the concept of a Hilbert space – again, an idealisation. And even in other fields – **Watson and Crick** discovered the structure of DNA, but didn’t include the protein core of chromatin, later discovered to be a fundamental part of the more detailed structure.

This idealism is possible because this major work was often done without Newton, Turing or Von Neumann doing lab experiments – they used experimental data from other people’s work, but data which had already been ‘digested’ by the rest of the scientific community. Newton built on the work of Galileo, Copernicus and Kepler before him.

Smale’s current research is on the human body, and in particular the heart. How does it manage to beat in such a coordinated way, with all the myocytes, or muscle cells, acting together to create a regular heartbeat?

The synchronisation was compared to what happens in a crowd applauding for a long time, when the clapping falls into synchronisation almost accidentally. It’s a dynamical system, and can be understood through maths, much like the beating heart.

Smale is working towards a deeper understanding of the heart through mathematics, and in this case standing on the shoulders of Alan Turing. Turing’s final paper on morphogenesis – the biological processes that lead to stripes on a zebra, or the arrangements of seeds in plants – included some differential equations. These were again an idealisation of the reality, as they were based on what happens as the numbers of cells increases to infinity (obviously not the real situation – some biologists disliked Turing’s work for this reason, as they mainly worked with small quantities of cells).

The cells making up the heart are of the same **cell type** – they all behave in the same way. Smale is looking at how this happens, and outlined how if you consider the set of genes as a graph, with individual genes as the nodes, and directed edges indicating how each transcription factor controls the adjacent gene, elevating protein production, you can build a system of ordinary differential equations to describe how it works.

This system of equations can be seen to reach an equilibrium – a global basin of attraction where the levels of each gene work in exactly the right way to determine how the cell behaves. These basins define the cell types – so a liver cell, or a heart cell, knows exactly what quantities of each protein to produce based on these equations, which are hard-wired into the DNA. You can even consider stable periodic behaviour in the system, to understand the heartbeat’s regular cycle.

Of course, this is again all idealisation – but the workings of the heart are something that have been understood from the biological angle for some time, and now mathematics is providing new ways to model and understand it – which will hopefully lead to powerful medical advances we can implement in reality.

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