OBE

No, it is not the Queen’s Honours list. It stands for Outcomes Based Education, the latest pedagogical fad. The EU has practically adopted it as its official educational project. Which probably means that to bid successfully for EU funded projects you’d stand a better chance if you insert the OBE here and there.

But I am writing this because in the Wikipedia page Outcomes-based education

I was somewhat amused to (re)read a paragraph such as this:

In a traditional education system, students are given grades and rankings compared to each other. Content and
performance expectations are based primarily on what was taught in the past to students of a given age. The goal of traditional education was to present the knowledge and skills of an older generation to the new generation of students, and to provide students with an environment in which to learn. The process paid little attention (beyond the classroom teacher) to whether or not students learn any of the material.

and guess what reference they cite at the end of this paragraph? The
Constance Kamii and Ann Dominick 1998 paper on the “harmful effects” of algorithms in Grades 1-4. They also quote this gem from the paper:

“The teaching of algorithms is based on the erroneous assumption that mathematics is a cultural heritage that must be transmitted to the next generation.”

And whole countries build their educational policies on such “findings”.

Alan Turing: The Imitation Game

Of course, what makes Turing special to so many of us is not the detail of his life so much as the *meaning* that each of us as individuals draw out of who Turing was and what he left behind. And this explains why “The Imitation Game” grips and moves so many of us, with its emphasis on the inner mind and experience of the man. The current IMDb user rating is 8.4 out of 10:

http://www.imdb.com/title/tt2084970/ratings

compared with the 8.1 (for instance) of 4 Oscar winner “The King’s Speech”; or 7.9 for 3 Oscar winner “Avatar”. 8 Oscar winner “Slumdog Millionaire” gets 8.0.

On Rotten Tomatoes “The Imitation Game” gets a creditable 90% critics rating; and 95% Audience Score (the audience score higher than for the three Academy ‘Best Pictures’ mentioned). See:

http://www.rottentomatoes.com/m/the_imitation_game/

But here is Christian Caryl writing on “A Poor Imitation of Alan Turing” in the influential New York Review of Books:

http://www.nybooks.com/blogs/nyrblog/2014/dec/19/poor-imitation-alan-turing/

Let’s take an overview of this carefully argued piece. The ‘traitor’ comments, also appearing in a Guardian item: http://bit.ly/17alkL1 (referring to Turing’s “cryptographic work at Bletchley Park from 1939-45″), show a misunderstanding both of movie and the historical and social context). We will be guided by the sort of computational broad sweep handed down to us by Alan Turing – just skip the next paragraph if you want.

The problems with a movie as complex and dependent on higher order impact and meaning as “TIG” is that the appreciation and evaluation of the movie is not algorithmic. The observer may doggedly limit attention to the detail (as one certainly would in evaluating a mathematical proof, but not if watching Shakespeare’s Henry IV, Part 1), or perform a higher order computation with a quite different result to that of someone else (we don’t all like the same movies). Much deep information theory is based on quite basic intuitions – we observe that someone “could not see the wood for the trees”; or, converely, in the UK people say “look after the pennies and the pounds will look after themselves”. The emergence of Turing, person and work, certainly does depend on detail. The jury is still out on who is right about the balanceand dependencies between the levels of meaning. People we have high regard for will come to quite different conclusions. It was Alan helped us to understand the emergence of ‘incomputable’ higher order information, and what it means for human versus algorithmic thinking.

What about “the portrayal of Alan Turing in the movie” as “an arrogant and obnoxious dick” as one of my distinguished correspondents wrote? Or, conversely, those who described the movie as a piece of hagiography? Back in 2012 I wrote a piece for the Guardian Northerner on “Alan Turing and the bullying of Britain’s geeks”:
http://www.theguardian.com/uk/the-northerner/2012/jun/20/alan-turing-geeks

and got a number of very moving letters of appreciation for bringing this sensitive issue into the open.

I wrote: “In 2002, an American study found that 94% of school students with Asperger’s syndrome faced torment from their peers and commented:

‘Some of their behaviors and characteristics that others see as ‘different’ make these children easy targets for frequent and severe bullying. Having Asperger’s Syndrome means these children are part of a vulnerable population and are easy targets.’ “

It’s conjectural, but probable (see the quote from Alan’s brother John in the Guardian piece), that Alan did fit with this, and was misinterpreted as ‘arrogant’ at times. He was by many accounts a lovely man with a great sense of humour – with children especially, and those like Robin Gandy that he was close to. But scriptwriter Graham Moore is the right person to get the balance right – there are some great interviews on the web, here
is one: http://deadline.com/2014/12/graham-moore-interview-screenwriter-imitation-game-alan-turing-1201337316/

He describes how:

“I had been a lifelong Alan Turing obsessive. Among incredibly nerdy teenagers, without a lot of friends, Alan Turing was always this luminary figure we’d all look up to.”

Another on “Trial and Triumph” about the development of the script and the philosophy underlying the film: http://deadline.com/2014/12/graham-moore-interview-screenwriter-imitation-game-alan-turing-1201337316/

Just recently, towards the end of November, I got a very moving message,
which for me brought out a key significance of the film and the special
understanding of all concerned with it:

“I followed your article in the guardian in 2012 from many previous articles about Alan Turing. He is quickly becoming a special interest , having known nothing about him a few days ago. I am inspired and in absolute awe of this man, someone who sees life differently. A possible member of team aspie too.
I thank you for your positive link to the aspergerians / aspies and how we should be seen as valuable to society, and thank you for recognising that so often we are not. As someone who was bullied relentlessly for being different , I am always grateful that someone has any positive light to
share on our experiences.
I am aspie, as is my daughter and son. I hope to share all the stories and information so Alan’s story will stay well known. Anyway, I have said what I wanted to, have a good weekend. Thank you.”

Apart from the science (a headline part of the Turing legacy for many of us) this is an issue reminds me why an Oscar for Alan does mean a lot. If “The Imitation Game” gets an Oscar, it will be seen by many as being for Alan Turing. Never before has there been a movie hero like him, not mad or disabled, but a man who did great things *because* of who he was – a hero for the information age. And – through Turing – we would have bullied thoughtful amazing kids coming out of schools all around the world … with Oscars! Oscars for being like Alan.

And “TIG” director is very much on-board with this aspect of his movie – Genevieve Hassan quoted him for the BBC under the heading “Alan Turing film celebrates ‘difference'”: http://www.bbc.co.uk/news/entertainment-arts-29137930

The LA Times has Morten Tyldum “on doing justice to Alan Turing”, talking about a number of issues, including the old chestnut about “TIG” not including enough gay sex:
http://www.latimes.com/entertainment/envelope/la-et-mn-en-morten-tyldum-20150101-story.html

It just so happens I was at a showing of the movie with a group including a clever mathematical teenager, and someone asked him if Alan Turing in the movie reminded him of anyone. The implication was accepted with a quiet smile communicated something very like pride …

Just this wekend WIRED magazine had this nice piece from author “Walter Isaacson on The Imitation Game and Making Alan Turing Famous”: http://www.wired.com/2015/01/geeks-guide-walter-isaacson/
where we read:

“One of the reasons I wrote this book [The Innovators] is because I wanted to make people like Alan Turing famous,” Isaacson says in Episode 131 of the Geek’s Guide to the Galaxy podcast. “And now I must admit that Benedict Cumberbatch, by playing him, has done that a thousand times better than I ever could have.”

What a nice man!

Apologies to for an untypical update. To be honest, I’m swamped with media items dealing with the movie, and thoughts arising from it. Those of us who started out 5 or 6 years ago, with even those professionally involved with computers and information not having even heard of Turing … it’s an amazing experience to have friends, relations, students who have suddenly ‘taken ownership’ of the Turing story via this movie.

Some are already going to Alan Turing biographies by Andrew Hodges, Jack Copeland, David Leavitt and others; visiting Bletchley Park, the Science Museum, the HNF museum in Paderborn, MOSI in Manchester, or the Computer History Museum in Mountain View, California; or even braving Turing’s mathematics, computer science, philosophy and logic, and artificial intelligence via “Alan Turing His Work and Impact”, Charles Petzold’s “The Annotated Turing”, or various other excellent books – maybe a list is in order for a future update.

Anyway, it’s already approaching 4am. So just time before breakfast to dig out a few links to interesting “TIG” related links, and next time a return to shorter and more diverse Turing news.

The film made a great start at Telluride and Toronto film festivals, and here’s a typical review “Benedict Cumberbatch gives Oscar worthy performance” from The Independent: http://ind.pn/1owfFRt

The big event in the UK was the October 8 BFI premiere in London. Some of us northerners (well enabler Daniela and I) caught it in Manchester that night, a grand experience. And I managed a hasty Guardian Northerner blog “The Imitation Game: how Benedict Cumberbatch brought Turing to life” for the day before the premiere: bit.ly/1EqLWnr

A further piece for “The Conversation”:
http://theconversation.com/imitation-game-brings-to-life-the-real-alan-turing-pioneer-of-the-computer-age-32517
prompted Graham Moore to message us on Twitter “Thanks! I adored your
tremendous piece yesterday—Brilliant links between AT’s sexuality and his
imitation game.”
There was another in November on “Imitation Game will finally bring Alan
Turing the fame he so rightly deserves”:
http://theconversation.com/imitation-game-will-finally-bring-alan-turing-the-fame-he-so-rightly-deserves-34324

About this time, everything went crazy, and did umpteen interviews – along with anyone else knew something about Turing and was willing to give their time, eg this on The Colin McEnroe Show for Connecticut Public Radio:
http://wnpr.org/post/cracking-code-alan-turing

Also this weekend, Radio Times announced “Benedict Cumberbatch’s The Imitation Game is already a Hollywood award winner” – going on to explain “The Alan Turing biopic wins Best Picture at the Capri, Hollywood International Film Festival in Italy, the first of a busy awards season for the British film and its Sherlock star”: http://bit.ly/1vHPzxp

Cumberbatch himself is getting huge amounts of praise – this from the Baltimore Magazine is typical “The Imitation Game Benedict Cumberbatch shines as the strange genius who broke the Nazi code”:
http://www.baltimoremagazine.net/2014/12/29/the-imitation-game

An important element of the Oscar campaign is the intervention of the Harvey Weinstein company earlier last year. These are fearsomely determined and able people, but with the sort of concern for the worthwhile that has guided them to a string of successes, including “The King’s Speech”. And an interesting piece from The Wrap suggests that they have realised that the strongest card in their hand is Alan Turing himself. It’s what drew so many creative people to the movie in the midst of the Turing centenary celebrations, including Black Bear Pictures founder Teddy Schwarzman when Warner dropped the movie in 2012. According to The Wrap “Weinstein Revs Up ‘Imitation Game’ Awards Campaign by Promoting Alan Turing, not Benedict Cumberbatch”:
http://bit.ly/1wNObcO Of course, it’s the Turing-Cumberbatch package is a specially potent one.

A recent development has been the muttered suggestions of conspiracies and hatchet jobs in relation to such pieces as that of Christian Caryl in the New York Review of Books. But despite the sharpness of tone, and timing to coincide with Oscar voting, hidden agendas and conspiracies are yet to be uncovered. The Independent treads carefully with “Oscars 2015: The Imitation Game and Selma criticised for being loose with facts as voting season begins”: http://ind.pn/1wNPvfU
While Deadline is more colouful with “‘Tis the Season: ‘Foxcatcher’, ‘Big
Eyes’ Latest Oscar Contenders Under Attack”: http://deadline.com/2015/01/oscars-foxcatcher-selma-big-eyes-controversy-1201339223/
And Jen Yamato starting her article with:

“Oscar voting opened Monday, and like clockwork, the haters have come calling. As Deadline’s Pete Hammond wrote on Monday, ’tis the season for controversy over fact-based awards contenders.”

I hope we have given a reasonable idea of the state of things at the start of 2015. My Google alert is sending me literally dozens of Turing related items a day, many duplicating and cannibalising each other, and a full report would take a week or more. It’s now gone 6am, enough for us all.

There is still lots to tell concerning Turing developments not directly directly to “The Imitation Game”. Please send any items – not to do with the film – by next weekend.

All best

Barry

__________________________________________________________________________
ALAN TURING YEAR http://www.turingcentenary.eu
ASSOCIATION COMPUTABILITY IN EUROPE http://www.computability.org.uk
Email: pmt6sbc@leeds.ac.uk
Facebook: www.facebook.com/pages/The-Alan-Turing-Year/199853901070
http://en-gb.facebook.com/people/Alan-Turing-Year/100000473465821
https://www.facebook.com/TuringImpact
https://www.facebook.com/Alan.Turing.Institute
Twitter: http://twitter.com/AlanTuringYear
__________________________________________________________________________

Yagmur Denizhan: Performance-based control of learning agents and self-fulfilling reductionism.

Yagmur Denizhan: Performance-based control of learning agents and self-fulfilling reductionism. Systema 2 no. 2 (2014) 61-70. ISSN 2305-6991. The article licensed under the Attribution-NonCommercial-NoDerivatives 4.0 International License. A PDF file is here.

Abstract: This paper presents a systemic analysis made in an attempt to explain why half a century after the prime years of cybernetics students started behaving as the reductionist cybernetic model of the mind would predict. It reveals that self-adaptation of human agents can constitute a longer-term feedback effect that vitiates the efficiency and operability of the performance-based control approach.

From the Introduction:

What led me to the line of thought underlying this article  was a strange situation I encountered sometime in 2007 or 2008. It was a new attitude in my sophomore class that I never observed before during my (by then) 18 years’ career. During the lectures whenever I asked some conceptual question in order to check the state of comprehension of the class, many students were returning rather incomprehensible bulks of concepts, not even in the form of a proper sentence; a behaviour one could expect from an inattentive school child who is all of a sudden asked to summarise what the teacher was talking about, but with the important difference that –as I could clearly see– my students were listening to me and I was not even forcing them to answer. After observing several examples of such responses I deciphered the underlying algorithm. Instead of trying to understand the meaning of my question, searching for a proper answer within their newly acquired body of knowledge and then expressing the outcome in a grammatically correct sentence, they were identifying some concepts in my question as keywords, scanning my sentences within the last few minutes for other concepts with high statistical correlation with these keywords, and then throwing the outcome back at me in a rather unordered form: a rather poorly packaged piece of Artificial Intelligence.
It was a strange experience to witness my students as the embodied proof of the hypothesis of cognitive reductionism that “thinking is a form of computation”. Stranger, though, was the question why all of a sudden half a century after the prime years of cybernetic reductionism we were seemingly having its central thesis1 actualised.

Ivor Grattan-Guinness obituary

From The Guardian, by Tony Crilly

Energetic historian of mathematics and logic

When Ivor Grattan-Guinness, who has died aged 73 of heart failure, became interested in the history of mathematics in the 1960s, it was an area of study widely considered to be irrelevant to mathematics proper, or something that older mathematicians did on retirement. As an undergraduate at Oxford, he found that mathematics was presented drily, with no inkling of the original motivations behind its development. So Ivor set himself the task of asking “What happened in the past?” – as opposed, he said, to taking the heritage viewpoint of asking “How did we get here?”

Read in full.

Ivor Owen Grattan-Guinness, historian of mathematics and logic, born 23 June 1941; died 12 December 2014

MBE to a maths clubs volunteer

From BBC:

A man who runs free maths classes for primary age children has been recognised in the New Year Honours list with an MBE.

Gbolahan Bright has been running the Bright Academy maths clubs for primary age children in London and Essex for the past 20 years.

“I have gained a lot from this society. I have been blessed and it would have been ungrateful of me if I did not give back,” he said.

Of the 500 or so children who have taken the classes, about 50 gained their GCSE while still at primary school.

Read more.

Alexander Grothendieck: some recollections

I first saw Alexander Grothendieck (AG) when as a raw research student I attended his lecture at the 1958 Edinburgh ICM.

I knew of his Tohoku paper as this was a source for Dick Swan’s 1958 lectures at Oxford on “The Theory of Sheaves”, where I was charged with working with Dick on writing up the notes. Now I saw this amazing figure telling Serre and others what was what! I also heard a comment of Raoul Bott: “Grothendieck was extraordinary as he could play with concepts, and also was prepared to work very hard to make arguments almost tautological.”; which sounded good to me.

Of course over the years I heard much more, and indeed organised at Bangor a Meeting on Toposes in 1972. I knew of his surprising departure from IHES in 1970.

In early 1982 I read a paper by Jack Duskin which referred to Grothendieck’s interest in \(n\)-categories. I was at that time concerned about very negative views expressed in the UK on my work with Philip Higgins on higher groupoids in algebraic topology, and on plans to develop this work with Jean-Louis Loday. Indeed such views did not much change till in 1984 I was able to report on the work with Loday on the nonabelian tensor product.

Since I had planned to go to a conference in Marseilles-Luminy in June, 1982, I wrote to AG with a load of offprints and preprints explaining that I had heard of this interest and wondered if I could visit him to discuss it.

He wrote back to say that he was very out of things, so a meeting may not be useful, but that he did write three letters to Larry Breen in 1975 and would I be interested in seeing them. Of course I replied in essence “Yes please”. The letters were in French, two of them typed, and the third 40 pages of handwritten A5 paper not so easy to decipher. So I translated and typed it as best as I could, and checked it out with others at the conference and with Larry Breen.

AG became enthusiastic about the idea of \(\infty\)-groupoids, and also about potential applications of a 2-dimensional van Kampen Theorem to compactifications of modular toposes of complex curves. I have not been able to pursue the latter ideas.

In October I sent him a comment explaining that our strict \(\infty\)-groupoids modelled only very restricted homotopy types, which I now call “linear homotopy types”, since there is no quadratic or higher information, such as Whitehead products. This led him to start thinking hard about various weak versions of \(\infty\)-groupoids, and their value in modelling homotopy types.

Gradually, our correspondence developed and became very friendly, and he eventually signed himself as “Yours affectionately, Alexander”. Tim Porter also joined in the correspondence, which continued in a friendly even chatty tone.

I early asked him if I could circulate any of the correspondence, and he was happy with that. This circulation led to others getting in contact with him.

In the end, my correspondence with him amounted to about 78 letters.

In early 1983, I and Larry Breen received the first part of the manuscript called “Pursuing Stacks”, “written in English in response to a correspondence in English”, and prefaced by a “letter to Quillen”, which was in fact never answered. I duly circulated this work to a few people. By the end of June, the manuscript had amounted to over 600 pages! It was written in the form of a diary, typed but with handwritten corrections. AG insisted that if it was to be published, it should be “as is”, since he felt that young people should be aware that even well known people make mistakes. I am glad to say that Pursuing Stacks is being edited by Georges Maltsiniotis and is to be published by the Société Mathématique, together with his correspondence with various people in these years.

Alexander had clearly hoped I would work on some of this. But in April 1982 Jean-Louis Loday and I came up with the notion of nonabelian tensor product of groups as arising from some pushouts of crossed squares, and realised that this put the importance of a complete proof of our conjectured van Kampen type theorem on a higher urgency.

In 1985 I has arranged a visit to Toulouse, and asked Alexander if I could visit him on the way. He picked me up at Orange rail station and took me back to his home, Les Aumettes, out in the country, and a long way from Montpellier where he worked. He showed me to the room where I would sleep, and hoped I would not be disturbed by a Buddhist shrine in the room. I said that was no problem. He had kindly bought some sausages for my meals, as he was a strict vegetarian. Meals were preceded by a Buddhist chant, which sounded very good to me. While it was light, he went to the garden and chopped up some vine roots for the stove.

The following day he drove me out into the mountains for a picnic lunch, and then back to his house for some supper. I regret that my photographs of this were jinxed by being overlapped with those of my following visit!

We discussed many things over the two evenings. He showed me a beginning draft of “Recollte et Semaille”. I sometimes wonder if I could have used more nous to dissuade him from some of what seemed to be some attacks on people. But I had little knowledge of the people and of the background, and did not want to argue with my kind host.

He was not so keen to discuss mathematics, but I did try to steer the conversation round to that. One comment stuck in my mind when he claimed he could compute the “Teichmüller Groupoid” seemingly by induction on the handles, using clutching functions. I have never understood this, but it led to thinking about what might be higher groupoid structures for symmetries of graphs, and so for the notion of symmetry for the topos of graphs and morphisms. This idea was helpful to a joint supervision with Chris Wensley of the thesis of John Shrimpton on symmetries of directed graphs.

The drive

The next day we drove to Monpellier for my lecture. Alexander had a somewhat battered estate car, avoided motorways and drove fast. We picked up two hitchhikers who must have been amazed at the conversation in the front, as I was going on about double groupoids. Finally we arrived outside the Department, a little late, and had the following conversation: “Do you mean to say that \(n\)-fold groupoids model homotopy \(n\)-types?” “Yes.” “I suppose you have said that at some time in your letters?” “Yes!” “But that is absolutely beautiful!”
Alexander dressed very informally, with an open crocheted sweater, and liked to talk informally to the students at Montpellier. At some time he remarked: “There is something enervating about the atmosphere of a great research institute.” Perhaps this is some background to his departure from IHES in 1970, and his then apparent rejection of mathematics. This rejection was taken quite seriously by the mathematical community, and yet it is clear that in his time at Montpellier he was writing many thousands of pages of mathematics. He even sent 2000 pages of notes on his approach to Galois theory to two USA mathematicians, who in the end left them in storage!

For a few years 1982–1991 he was in contact with many mathematicians.

In letters I had explained that Tim Porter and I had grant applications turned down. In 1984 Alexander submitted a grant application under the name “Esquisse d’un programme”, and circulated it to a number of people.

In 2006 I asked Mikhail Kapranov about the influence of these writings in the Soviet Union. He responded: “From what I remember, Gelfand advocated reading both Esquisse and Pursuing Stacks. Voevodsky was very interested in both anabelian geometry and higher stacks. Drinfeld was influenced by Esquisse in his paper on “Drinfeld associator” and a version of the Grothendieck-Teichmueller group appearing in the theory of quasi-Hopf algebras. This is probably the most serious influence on Soviet mathematics of the period.”

In 1986 AG wrote a reference for me which included the following:

“This programme (which I have started pushing through in the volume 1 of
“Pursuing Stacks”) has some substantial overlap with R. Brown’s. Getting
aware of this was the starting point, in 1982, of a very stimulating correspondence between R. Brown and myself, which has been continuing till now. It is
this correspondence mainly, and the friendly and competent interest of Ronnie
Brown in mathematical ramblings, which was the decisive impetus to take up
again and push ahead some of the old ponderings of mine, materializing in the
writing up of “The Modelizing Story” (the volume alluded to above).”

The later story

Gradually, Alexander’s interests turned to dreams and religion, and away from mathematics, so that much of the programme described in “Esquisse” was not pursued by him, though it has been a great stimulus to many, as has Pursuing Stacks. In April 1991 I sent him a postcard from Iona, and he responded in a friendly way, even saying he had taken up mathematics for five months after a gap of four full years; there was also a hint that he expected some end of all things. To my surprise, in mid 1991 he took himself away from the world.

In 2004, reading of comments of many on AG, and on Pursuing Stacks, I realised that many did not know its origins, that I had a kind of treasure in this correspondence, and so sent some of it to Georges Maltsiniotis. He found it very interesting, so in the end I sent him the complete set of originals.

Conclusion

Others have written comprehensively on Alexander’s mathematical achievements. From his letters he also comes across as great writer, with a feel for the rhythm of the English language.
He thus seems to come under Shakespeare’s words: “…as imagination bodies forth the forms of things unknown/ the poet’s pen turns them to shapes, and gives to airy nothing/ a local habitation and a name.” He also made comments on methodology, for example on snobism, speculation, and how specific computations came out of  “understanding”.

It was a thrill to correspond in this way, and I hope that the eventual availability of the correspondence will enhance the picture of Alexander Grothendieck.

Retraining 15,000 teachers?

Philip Nye writes in a paper  Cameron needs to rethink maths and science plan (12 Dec 2014) that

Under No 10’s plan, 15,000 teachers of other subjects will also retrain as maths or physics teachers, as part of a “major push” to boost maths, science and technology skills.

However, Professor Alan Smithers, director of the Centre for Education and Employment Research at the University of Buckingham says: “It’s really easy to say ‘well, physics is science, so therefore there’ll be people teaching biology, or who have done medicine or engineering [degrees] that we can retrain as physics teachers’. But biology is really as different from physics as, say, history is.”

Perhaps the same skepticism can be applied to mathematics.

Mathematics Resilience – making it happen

The Shard Symposium

16th January 2015 10am – 4pm

Evidence is accruing that Mathematical Resilience is fundamental to developing a numerate, empower society. You are cordially invited to attend a symposium designed to explore the next steps to be taken in enabling learners to become Mathematically Resilient.

The symposium is convened to bring together practitioners, funders and researchers to discuss what is happening in enabling learners to develop Mathematical Resilience. It is a precursor to an international conference that will be held jointly by University of Warwick and Open University in November 2015.

The symposium will be held at the Warwick University Business School Offices in The Shard, 32 London Bridge Street, London, SE1 9SG, nearest underground station London Bridge.

A small charge of £20 is payable for registration, this will be made to cover refreshments throughout the day. You can register for the event here.

Hamid Naderi Yeganeh: Mathematical drawings made from segments

A cardioid

This figure is closely related to a cardioid.

This image shows 1,000 line segments. For each \(i=1,2,3,\cdots,1000\) the endpoints of the \(i\)-th line segment are:

\[\left(\cos\left(\frac{2\pi i}{1000}\right), \sin\left(\frac{2\pi i}{1000}\right)\right)\]

and
 
\[\left(\cos\left(\frac{4\pi i}{1000}\right), \sin\left(\frac{4\pi i}{1000}\right)\right).\]