Plus Blog

May 29, 2013

Is there a perfect voting system? In the 1950s the economist Kenneth Arrow asked himself this question and found that the answer is no, at least in the setting he imagined.

Polling station

Kenneth defined a voting system as follows. There is a population of voters each of whom comes up with a preference ranking of the candidates. A voting system takes these millions of preference rankings as input and by some method returns a single ranking of candidates as output. The government can then be formed on the basis of this single ranking.

For a voting system to make any democratic sense, Kenneth required it to satisfy each of the following, fairly basic constraints:

  1. The system should reflect the wishes of more than just one individual (so there's no dictator).
  2. If all voters prefer candidate x to candidate y, then x should come above y in the final result (this condition is sometimes called unanimity).
  3. The voting system should always return exactly one clear final ranking (this condition is known as universality).

He also added a fourth, slightly more subtle condition:

  1. In the final result, whether one candidate is ranked above another, say x above y, should only depend on how individual voters ranked x compared to y. It shouldn't depend on how they ranked either of the two compared to a third candidate, z. Arrow called this condition independence of irrelevant alternatives.

Arrow proved mathematically that if there are three or more candidates and two or more voters, no voting system that works by taking voters' preference rankings as input and returns a single ranking as output can satisfy all the four conditions. His theorem, called Arrow's Impossibility Theorem helped to earn him the 1972 Nobel Prize in Economics.

You can find out more in the Plus articles Which voting system is best? and Electoral impossibilities.

April 22, 2013
Chaos movie still

"Everything flows. Everything is movement."

From the makers of Dimensions comes a great free online movie exploring dynamical systems, the butterfly effect and chaos theory by means of stunning visuals accompanied by a beautiful musical score. The dynamics of the weather, the three-body problem, Smale's horseshoe — it's all there, including cute lego athletes whizzing around a Lorenz attractor.

"The film is for everybody," says the mathematician Étienne Ghys, one of its creators. "It is split into nine chapters. The level [of the mathematics] continuously increases from chapter one to chapter nine and, in principle, one should not be frustrated if one does not watch the film until the very end." So whatever your level of maths there is something in it for you.

The field of dynamical systems was created by Henri Poincaré some 125 years ago. "Its purpose is to understand the motion of mechanical systems, like celestial bodies for instance, without solving any equations," says Ghys. "The trick is to use pictures to prove theorems: this is just what is needed for a movie."

Poincaré was indeed something of a pioneer in the use of pictures. "During the nineteenth century, especially in France, there was a period in maths during which pictures were considered a taboo," says Ghys. "They were supposed to be evil: the origin of mistakes. Under the influence of mathematicians like Poincaré mathematics, or at least some parts of maths, has become more visual and movies are indeed a wonderful tool to explain things. One picture is worth a thousand words!"

So just imagine how many words a whole movie can replace! Watch Chaos: a mathematical adventure.

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April 12, 2013
The scales of justice

The scales of justice

Every year the London Mathematical Society (which despite its name represents academic mathematicians throughout the UK) puts on a pair of popular lectures that are well worth seeing. This year's speakers will be Ray Hill, showing how the misuse of mathematics can lead to miscarriages of justice and how to prevent them, and a good friend of Plus, Vicky Neale, talking about intriguing problems to do with adding numbers and their elegant solutions (if they exist!).

The lectures will take place in London on 25th June 2013 at 7pm and in Birmingham on 20th September 2013 at 6:30pm. Admission is free but you need to register for a ticket, either online or by emailing popularlectures@lms.ac.uk.

April 8, 2013

Find out why physicists believe they might be in a lecture by John D. Barrow at the Bath Royal Literary and Scientific Institution on April 13th.

John D. Barrow

John D. Barrow FRS has been a Professor of Mathematical Sciences at the University of Cambridge since 1999, carrying out research in mathematical physics, with special interest in cosmology, gravitation, particle physics and associated applied mathematics. He is the author of over 420 articles and 19 books, translated in 28 languages, exploring the wider historical, philosophical and cultural ramifications of developments in mathematics, physics and astronomy. Most importantly (to us anyway), he is the director of the Millennium Mathematics Project of which Plus is a part.

The lecture starts at 7:30pm and you can book tickets here (£8/£6 concession).

If you can't get to Bath you can also read Barrow's Plus article Are the constants of nature really constant? or listen to Barrow in the accompanying podcast.

April 3, 2013

The London Knowledge Lab have a fascinating Maths-Art workshop planned for this month, a "mostly" hands-on experience entitled "Art and the Möbius strip".

Animated Klein bottle

The Möbius band is one-sided (Image Konrad Polthier)

Many artists have explored the curious properties of the Möbius strip, with Max Bill and M. C. Escher among the most famous exponents. After a brief survey of this art and a basic mathematical overview, Simon Morgan and John Sharp will explore new aspects of this fascinating object as a starting point for potential new art. The session will be mainly practical because the properties of the Möbius strip can only be explored through hands-on experience. Please bring scissors, tape, and large paper sheets (e.g. old newspapers)!

The workshop is on Thursday 11th April at 6.00pm at the London Knowledge Lab, 23-29 Emerald St, London, WC1N 3QS. It is free and you don't need a ticket but an email to lkl.maths.art@gmail.com is appreciated to assist with planning. You can view past Maths-Art seminars on their YouTube channel and you can read more about the Möbius strip and its cousin the Klein bottle on Plus.

March 27, 2013

The legendary mathematician Paul Erdős would have turned 100 today. Aziz S. Inan, Professor of Electrical Engineering at University of Portland and a friend of Plus, has sent us this fitting tribute.

Erdos

Paul Erdős would have turned 100 today! Image: Kmhkmh.

Paul Erdős (26 March 1913-20 September 1996, died at 83) was an influential Hungarian mathematician who spent a significant portion of his later life living out of a suitcase and writing papers with those of his colleagues willing to provide him room and board. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory. He published more papers than any other mathematician in history, working with hundreds of collaborators. He wrote over 1,500 mathematical articles in his lifetime, mostly with co-authors. Erdős is also known for his "legendarily eccentric" personality.

Erdős strongly believed in and practiced mathematics as a social activity, having over 500 collaborators during his life. Due to his prolific output, his friends created the Erdős number as a humorous tribute to his outstanding work and productivity. The Erdős number describes the "collaborative distance" between a person and mathematician Paul Erdős, as a measure by authorship of mathematical papers.

Today, Tuesday, 26 March 2013, marks Erdős' centennial birthday. As I was looking at numbers related to Erdős' birthday, I noticed some interesting numerical coincidences and connections. I decided to report my findings in this article, as a centennial brainteaser birthday gift for Erdős.

  • Erdős' birth date, 26 March, can be expressed in day-month date format as 26-3, or simply 263. Coincidentally, the 263rd day of 2013 and any other non-leap year is 20th of September, the day Erdős died in 1996.
  • Erdős' 100th full birthday in day-month-year date format can be written as 26-3-2013 or simply, 2632013. This number can be expressed in terms of its prime factors as follows:
      \[ 2632013=19 \times 83 \times 1669 . \]    
    So Erdős' 100th birthday expressed as 2632013 is divisible by the 263rd prime number 1669, where 263 represent Erdős' birth date, 26 March! Also, if its prime factor 1669 is split in the middle as 16 and 69, these two numbers add up to 85, where the 85th day of 2013 (and any other non-leap year) is amazingly 26 March! In addition, the prime 83 represents Erdős' death age and 19 is equal to half of the reverse of 83.
  • Erdős' birth year 1913 is the 293rd prime number, 293 is the 62nd prime number, the reverse of 62 is 26, again the day number of Erdős' birthday.
  • Erdős' 100th full birthday in month-day-year date format is 3-26-2013, or simply, 3262013. Interestingly enough, the prime factors of the reverse of 3262013 are given as follows:
      \[ 3102623 =29 \times 83 \times 1289. \]    
    Here, 1289 is the 209th prime number where 209 represents Erdős' death date, 20 September. In addition, again, prime 83 corresponds to his death age.
  • 1289 is also one of the prime factors of the reverse of Erdős' full birthday expressed in day-month-year date format as 26031913 since
      \[ 31913062 = 2 \times 1289 \times 12379. \]    

Thanks for transforming mathematics into a universal social activity through your modesty and humbleness Paul Erdős, and have a happy 100th birthday!

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