Calculating Pi Using Darts

As you may have noticed, it is summer. However, just because the academic season has ended there is still lots of maths that can be done. Yesterday, having been inspired by a quick video I found online, I decided to undertake an interesting experiment. I tried to calculate 𝛑 using darts. In doing this, I found that although this is something I have never done before, it is remarkably easy. All you need to try this at home is a dart, a dart board or alternative to stop the darts, a piece of square paper, a pencil and a compass.

The first thing I did was use the pencil and compass to draw the shape below with the circle completely inside the square just touching the centre of each of the square’s edges.

If we call the length of the square x, the area of the square is x2 . The radius of the circle is x/2. Therefore the area of the circle is 𝛑x2/4.

Thus the ratio of the area of the circle to the area of the square is 𝛑/4.

Thus the ratio of the area of the circle to the area of the square times by 4 is 𝛑.

This also proves that the length of the square is unimportant, which is apparent anyway.

How would one approximate the area of the circle and square? This is where the darts come in because the number of darts landing in a particular area over a large enough number of throws will be proportional to the area as long as the throws are random.

Therefore if I throw lots of darts at the shape above,  4 x the number of darts that land in the circle/ the number that land on the whole shape should equal 𝛑.

 

I tried this and I threw lots of darts at the pattern above and then I decided to see how close I got. My approximation for 𝛑 was 3.96 (3 s.f.). That is not a typo, 3.96, not 3.16. Oh dear! I did some thinking about why I was so far from the true value. This method is never going to work too well for a number of reasons e.g. The circle I had drawn does not fit the square correctly. However, it was clear the biggest issue was arising from my throwing. Each time I threw I aimed for the centre of the target and although I only occasionally play darts I was getting the dart usually within about 5 inches of the centre of the target. This resulted in me getting way too many darts in the square than would be random. Therefore, I made a few changes.

I firstly changed the target to the shape below, (dimensions irrelevant). The ratio of the area of the circles to the area of everthing is still 𝛑/4.

I felt this design meant even if I threw consistently my results would be good. The net thing to do was make my throwing more random. For this I had some help. I ‘borrowed’ my brother to throw darts as well to try and reduce systematic error because the darts are only thrown with my style. Also, to make our throws more random we threw from different angles, threw on the move, we threw with our opposite hands, threw underarm and even threw blindfolded, which is not recommended. The places the darts landed looked very random, but did we get an accurate result? We showed 𝛑 is approximately 3.16. This value is quite close.

To improve the accuracy further we probably needed to throw more darts and use a better template. However, if you follow these steps correctly and do enough dart throwing you can get a pretty good value of 𝛑.

I hope you enjoyed this post and please try this at home and write in the comments what your approximation of 𝛑 is and if you made any changes to the experiment tell me them too.

Thank you for reading.

Mathematicians Interviewed - 4. Professor Damian Rössler

The latest eminent mathematician to kindly answer my questions is Professor Damian Rössler. He is currently a Professor of pure mathematics at the University of Oxford Mathematical Institute and a tutor and fellow of Pembroke College. His research interests primarily focus on number theory and algebraic geometry. This interview is one of my favourites not only because I had the pleasure of meeting Professor Rössler in June, but I found his thoughts on the mathematical way of thinking inspiring.

1. What inspired you to become a mathematician?

I have found mathematics fascinating from an early age. There seemed to be a promise of

an intellectual garden of Eden, where intellectual investigation can uncover incontrovertible but also widely applicable truths. I still see mathematics in that way and I have now even come to believe that the mathematical way of thinking is a model for any intellectual activity - but this might just be the natural bias of someone who has dedicated a large part of his life to one subject.

2. What is your favourite area of mathematics and why?

I am potentially interested in any area of mathematics but since the end of my studies, I have mainly been interested in algebraic geometry and its applications to number theoretic problems. I find that I am very often more interested in geometric problems than combinatorial of analytic ones.

3. If you could discover any conjecture or problem what would it be and why?

There are two conjectures that I find absolutely fascinating (but I have no hope whatsoever that I shall ever prove them). The Hodge conjecture:

https://en.wikipedia.org/wiki/Hodge_conjecture

and the conjecture of Birch and Swinnerton-Dyer:

https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture

4. What is your favourite maths book and why?

I suppose that my favourite graduate textbook is "Eléments de Géométrie Algébrique”

by A. Grothendieck and J. Dieudonné (see https://en.wikipedia.org/wiki/%C3%89l%C3%A9ments_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique) but my favourite undergraduate textbook is the book “Calculus" by M. Spivak.

5. What do you think could be done to encourage more interest in maths in children and young people?

I think that a good way to do this is to teach them Euclidean Geometry, which is the very origin of mathematics. The core of the mathematical activity is the proof of theorems and this is what historically distinguished Euclidean Geometry from other contributions to mathematical techniques.

6. What advice would you give to a 16 or 17 year old who is thinking of studying maths at university?

I think that mathematics can be a gate to many other subjects, like physics, computer science, biology, finance and in its applied form it also has many industrial applications, eg Big Data. So studying mathematics is not a bad idea for the undecided. On the other hand, it is a course that requires a lot of dedication and work.

7. What breakthroughs do you think are imminent in maths?

It is impossible to answer that question… mathematics is like an unstable chemical compound and you never know what will come next.

8. Who is your favourite mathematician past or present and why?

One of my favourite mathematicians is Daniel Quillen (see https://en.wikipedia.org/wiki/Daniel_Quillen ).His writings are an absolute model of clarity and lucidity.

9. What do you feel is your greatest contribution to mathematics?

I am quite proud of my elementary proof, with the German mathematician Richard Pink, of the Manin-Mumford conjecture. See

http://people.maths.ox.ac.uk/rossler/mypage/pdf-files/mma.pdf

Review: 1089 and All That by David Acheson

I recently had the pleasure of reading 1089 and all that by David Acheson and I wanted to share my thoughts on it in this review.

1089 and All That by David Acheson is a wonderful little mathematics book I recently read. One thing I really enjoyed about 1089 is that it is written in such a way that it is very easy to pick up and only read a couple of pages, yet even in that short time learn some maths. This book is very accessible to anyone and you almost don’t need to know any maths before you read the book.  I feel 1089 would appeal to a wide range of people, because of this. For example, this book would be very interesting for GCSE to A level students who want to learn something beyond the syllabus without reading a textbook. I feel the real strength of this book is that it covers a wide variety of mathematics in a concise book. This does mean that the detail of each topic is not extensive. However, I felt there was sufficient detail to allow anyone to understand the problems and conjectures discussed. The other reason I like this book so much is that the problems are presented in a very  clear and lucid way so that when the mathematics appears it is very easy to understand. Acheson also uses diagrams and real world problems very cleverly to help the reader understand the wider importance of maths in the real world. It is hard to pick a favourite chapter in this book because the topics range from 3,000 year old geometry to recent areas such as chaos theory. A favourite of mine is the chapter on pi. I enjoyed this so much because Acheson discussed that pi’s importance is not only because it is equal to the area to circumference ratio of circles. Acheson elevates the importance of pi by highlighting that pi is actually important to almost all areas of pure and applied mathematics. I also enjoyed the chapter on great mistakes in mathematics. Here Acheson stresses how important it is for mathematicians to assume nothing when conducting a proof. He explores a famous case where

Mathematicians were changing the order of an infinite series when counting it’s limit. However, they assumed that the order of the sequence was unimportant as is the case for a finite series. However, mathematicians found a paradox where the limit of the infinite sum appeared to take multiple values when the order of the sequence was changed. This is impossible. The only conclusion was that the order one counts an infinite series does change the limit.