The Man Who Knew Infinity: a book review

The Man Who Knew Infinity by Robert Kanigel (1991)

I recently, and somewhat belatedly, read Robert Kanigel’s The Man Who Knew Infinity, a biography of the brilliant Indian mathematician Srinivasa Ramanujan. A partly fictionalised film based on the book was released in 2015 (see Scott Aaronson’s review of the film here).

Whewell’s Court, Trinity College, Cambridge, where Ramanujan lived when he first arrived in England in 1914 (photo: Cmglee)

Ramanujan had one of the greatest mathematical intuitions of all time (he himself credited his insights to the goddess Namagiri). However, his brilliant guesses were as likely to be wrong as right. Furthermore, Ramanujan often neglected formal mathematical proofs, so that the work of separating the many diamonds from the occasional paste was frequently left to collaborators (like G. H. Hardy, who invited Ramanujan to England, and who wrote several joint papers with him). There are still results in Ramanujan’s journals which have neither been proved nor disproved (see this perspective on Ramanujan by Stephen Wolfram).

One of Ramanujan’s formulae for π

Interest in Ramanujan seems to have peaked at around the year 2000, according to Google Ngrams (although this does not include the influence of the recent film):

Google Ngrams search for Ramanujan’s name in books

I found Kanigel’s book a very enjoyable read. There is extensive biographical detail, albeit with a few misquotes, and with apparent confusion at times about the language of a century ago (e.g. the word “cult,” used in a technical sense to mean “a particular system of religious belief,” referring to the Brahmin version of Hinduism which Ramanujan followed). Kanigel does not quite succeed in fitting Ramanujan into a larger context – I would have liked a bit more discussion of Ramanujan by other mathematicians. And I cannot help but wonder what would have happened had illness (probably chronic hepatic amoebiasis, although Kanigel suggests tuberculosis) not killed Ramanujan at the tragically young age of 32. I guess nobody can imagine what further mathematics we might have seen.

See here and here for other reviews of the book.

The Man Who Knew Infinity by Robert Kanigel: 3.5 stars


Praising Cambridge

Clare College, Cambridge (my photo)

The University of Cambridge has been mentioned repeatedly on this blog. It is one of the oldest universities in the world – so old, in fact, that nobody is quite certain when it began (it seems to have been up and running by 1226, however). Cambridge is home to the Cavendish Laboratory, where many of the greatest scientific discoveries were made (the electron, the neutron, the structure of DNA, and more). Cambridge is ranked 4th in the world on the Times Higher Education list of engineering institutions. In the humanities, C. S. Lewis moved there from Oxford in 1954.

The university buildings are scattered across the city, which tends to confuse people. I can recall a tourist asking me, on a visit I once made to Cambridge, “Where on earth is the university?”

The Cavendish Laboratory, Cambridge (photo: “RichTea”)

The university is also home to the Cambridge University Eco Racing team, which competes in the World Solar Challenge, and which fielded a rather unusual-looking design in 2013 and 2015:

Cambridge University Eco Racing team’s 2015 WSC entry (photo: CUER)

Science in stained glass

These four wonderful stained-glass windows at Gonville and Caius College, Cambridge commemorate Francis Crick (and the structure of DNA), neurophysiologist C.S. Sherrington (and a neuron), statistician Ronald Fisher (and a Latin square), and logician John Venn (and a Venn diagram). All photos are by Wikipedia user “Schutz.”

WSC Guest Post: An English View

Today we have a guest post from Nigel, a UK-based solar car fan. He explains why Team 12 is his favourite in the Challenger Class for this year’s World Solar Challenge. A follow-up post will give his view of the Cruiser class.

Made in England (Part 1)

Over the past few years I have written a few posts about Solar Cars and Solar races and I have always tried to follow the rules on impartiality. In any case, being from the UK which has a pretty poor record of success in the sport, it is relatively easy not to blow our own trumpet too loudly.

Therefore, if you are prepared to read further, you can be sure that as I explain my choice of favourite cars at this year’s WSC you can be sure that no jingoism is involved. Before I go any further let me be clear, these are not the cars that I am suggesting will win the event. Anything can happen in solar racing and maybe both cars are capable of completing the course but I think it would take a LOT of mishaps to other teams for either car to even get close to the top five. No, these are the cars that I like for other reasons.

My first choice, in the Challenger Class, is Evolution from the Cambridge University Eco Racing team.

Cambridge’s Evolution at home

There are several reasons why I like this car and the first is simple – it is not a Catamaran. In the past all sorts of weird and wonderful shapes of car have turned up at WSC and events around the world with varying levels of success. This year, however, 29 of the 30 teams in the Challenger class have brought what we now call catamarans with the only differences being where the driver sits and how many fairings they have. I do not blame the teams, clearly it is the case that cars have evolved like this because it is a design which best answers the criteria given in the rules of the event.

So, some cars will better designed, with more aerodynamic features, and some will have a better finish than others but we can be almost certain that a catamaran will win the race. That’s fine, there are lots of areas that teams can focus on to move the sport, and engineering, forward such as weight reduction, battery technology, energy efficiency and so on – but all of that is hidden from view. What we see is one catamaran after another, different colours, different teams but still catamarans.

Another view of Cambridge’s Evolution

Thank heavens for Evolution. When CUER made their car called Resolution for the 2013 running of the event it was quite a departure from the norm. The car was tiny, made possible because it used Gallium solar cells, it was bullet shaped and the array was inside the car. Ok, it was not built as well as it could have been and events proved that the design went too close to the edge of what was safe. But thankfully we still have young people who are prepared to test where the edge is, not by gradually moving closer to it but by standing on the edge and then moving backwards. This is what CUER did and Evolution is a very aptly named car.

Resolution in Australia in 2013

This car is wider and lower than its predecessor and the team claims that it has the same, or better, aerodynamic performance. The centre of gravity has been lowered and moved in relation to the centre of roll. It is also much better finished than Resolution, is easier to drive and, the team claims, uses less energy. Some might claim that it is stubbornness that made this team persist with this design, I prefer to call it determination and here’s why I’m glad that they did.

Of all the cars at WSC, in any class, I believe that Evolution is closest to what is required for our future transportation needs. I actually think that this car is racing in the wrong class. There has been much talk about the rules of the Cruiser Class, particularly the scoring system, but for me the fundamental flaw in the rules is the one that defines the class – “They must be designed to carry two or more occupants.”

WHY, WHY, WHY, have this rule? The most practical car carries one person!

Let me explain, I have a wife and two adult children. We all work in different places at different times and we go out to different places so we have a car each. Our cars are made to carry a total of 19 people so, for most journeys, that’s 4 people, 4 cars and 15 spare seats. Add in my neighbours and we have 10 cars, 12 people and 39 spare seats. So, all of these extra empty seats and all of the extra metal needed to enclose then is being carted around the country day after day. What we need are small single occupant cars for the vast majority of journeys and, I believe, Evolution is closer to answering that need than Nuna8, Tokai Challenger or any other catamaran.

The organizers of WSC may not thank me for saying this but the solar part of this challenge is a gimmick. It’s a necessary gimmick – if this was the World Electric Car Challenge it would have died a death years ago – and it provides a great place for engineers to advance their knowledge but, in reality, the best place for solar cells is on the roof of your house.

Yes, the catamarans are needed to keep this fantastic event alive but they are not the future of transport. For that we need to look to EVOLUTION.

World Solar Challenge: Team 12

12  Cambridge University (Evolution)

One of the more interesting entrants in the World Solar Challenge is Cambridge University’s teardrop-shaped Evolution, which looks very different from most of the others in the Challenger class. Cambridge’s car uses gallium arsenide solar cells (which the race rules limit to 3 m2) on a tilting plate. This plate can track the sun, giving potentially very high energy output. A transparent canopy over the plate maintains low aerodynamic drag. It will be very interesting to see how this beautiful car performs!

In 2013, Cambridge University’s earlier Resolution vehicle withdrew after a crash. However, the new vehicle is more stable on the road than Resolution, and so this year we are likely to see a fair test of Cambridge’s innovative design. Good luck, team 12!

Update: see also this interesting interview.

For up-to-date lists of all World Solar Challenge 2015 teams, see:

Thermodynamic visualisation

This plaster model was made by the great James Clerk Maxwell in 1874 (the photograph was by taken by James Pickands II, 1942). This historic artefact is one of three copies, held in museums around the world, including the Cavendish and the Sloane Physics Laboratory at Yale.

The model shows the relationship between volume, energy, and entropy for a fictitious water-like substance, based on theoretical work by Josiah Willard Gibbs. The lines connect points of equal pressure and of equal temperature. Maxwell found the model a useful aid in his research. The model prefigured modern visualisation techniques – today we would use computer software to visualise such surfaces, like this:


H is for Hawk: a book review

H is for Hawk by Helen Macdonald

For quite some time, I have been waiting eagerly for my copy of H is for Hawk to arrive. I became aware of the author, Helen Macdonald, through her blog, so I knew in advance that the writing would be superb. Early reviews of the book reinforced this belief – “lyrical,” said The Guardian; “a soaring triumph,” said The Telegraph; “a dazzling piece of work,” said The Financial Times.

H is for Hawk is a nature book, but an intensely personal one – like Pilgrim at Tinker Creek, perhaps, but much more so. Grief-stricken by her father’s death, Helen Macdonald decides to follow in the footsteps of the (rather disturbed) author T. H. White by training a goshawk. Macdonald describes the goshawk species this way:

In real life, goshawks resemble sparrowhawks the way leopards resemble housecats. Bigger, yes. But bulkier, bloodier, deadlier, scarier, and much, much harder to see. Birds of deep woodland, not gardens, they’re the birdwatchers’ dark grail. You might spend a week in a forest full of gosses and never see one, just traces of their presence. A sudden hush, followed by the calls of terrified woodland birds, and a sense of something moving just beyond vision… Looking for goshawks is like looking for grace: it comes, but not often, and you don’t get to say when or how.

A female goshawk, Accipiter gentilis

Raising large predatory birds is an unusual way of dealing with grief, but we all cope with loss in different ways. The author was indeed “looking for grace,” and it is at least possible that grace takes wingèd form. Macdonald first meets her goshawk, later named Mabel, emerging from a cardboard box:

The hawk’s wings, barred and beating, the sharp fingers of her dark-tipped primaries cutting the air, her feathers raised like the scattered quills of a fretful porpentine. Two enormous eyes. My heart jumps sideways. She is a conjuring trick. A reptile. A fallen angel. A griffon from the pages of an illuminated bestiary. Something bright and distant, like gold falling through water.

Juvenile (left) and adult (right) goshawks

Training a goshawk requires intense observation of the bird’s appearance and behaviour, and Macdonald’s wonderful prose beautifully captures the results of her observation. To take just one example:

The feathers down her front are the colour of sunned newsprint, of tea-stained paper, and each is marked darkly towards its tip with a leaf-bladed spearhead, so from her throat to her feet she is patterned with a shower of falling raindrops. Her wings are the colour of stained oak, their covert feathers edged in palest teak, barred flight-feathers folded quietly beneath. And there’s a strange grey tint to her that is felt, rather than seen, a kind of silvery light like a rainy sky reflected from the surface of a river.

The gaze of this goshawk “filled me with dread,” according to photographer Steve Garvie

Goshawks are not, in any sense, domesticated animals. They are very, very wild, and Macdonald finds that “Hunting with the hawk took me to the very edge of being a human.” She loves Mabel, but without sentimental misunderstandings (and without the unconscious cruelty of T. H. White). She understands that a predator’s role in the natural world is to kill and eat prey. At one point she describes her goshawk as “the bastard offspring of a flaming torch and an assault rifle.” Such a clear understanding of the fact that predators predate is essential to good wildlife management. Nature is what it is, not what we would have it be: “there is a world of things out there – rocks and trees and stones and grass and all the things that crawl and run and fly. They are all things in themselves.”

A goshawk nest with four young birds (photo: Rainer Altenkamp)

I had high expectations of this book, and I was not disappointed – it is one of the best books I have read in a long time. But what of grace? For goshawks in general, it came in the restoration of the British goshawk population. “The wild can be human work,” as Macdonald writes. T. H. White, on the other hand, never recovered from the pain of his dysfunctional childhood. Mabel the goshawk grew to adulthood and flew for many years. And Macdonald herself? I’ll leave the reader to decide whether wild predatory birds can indeed bring healing from pain and grief. However, I was reminded of a certain young wizard who, “in fierce distress,” transforms into a great hawk in order to return to the land of his birth. “Only in silence the word, only in dark the light, only in dying life: bright the hawk’s flight on the empty sky.

Naturally, this brief review can scarcely do justice to a book as complex as H is for Hawk. But anyone interested in birds, in people, in grief, in England, in T. H. White, or in brilliant, lyrical writing that swoops and soars like the goshawk herself – anyone interested in such things should read this book. I’m giving it a very rare 5 stars.

For lengthy excerpts from the book, along with photographs of Mabel, see here and here. Macdonald has also written Falcon, although that is a very different kind of book.

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H is for Hawk by Helen Macdonald: 5 stars