Partition Numbers Behave Like Fractals, Says Mathematician
    By News Staff | January 21st 2011 03:29 PM | 4 comments | Print | E-mail | Track Comments
    For hundreds of years, mathematicians even as great as Leonhard Euler(1) have tried to make sense of partition numbers, the basis for adding and counting.  Progress has been made but there has never been a full theory to explain partitions.   Answers have always led to more questions.

    Basically, partitions are not considered by some to be part of number theory at all but to keep it simple for now, in number theory, a partition of a positive integer (n) is a way of writing n as a sum of positive integers.    

    Here is an example from the Classic Encyclopedia:
    To form the partitions of 6 we take first 6; secondly, 5 prefixed to each of the partitions of 1 (that is, 51); thirdly, 4 prefixed to each of the partitions of 2 (that is, 42, 411); fourthly, 3 prefixed to each of the partitions of 3 (that is, 321, 3111); fifthly, 2 prefixed, not to each of the partitions of 4, but only to those partitions which begin with a number not exceeding 2 (that is, 222, 2211, 21111); and lastly, 1 prefixed to all the partitions of 5 which begin with a number not exceeding 1 (that is, 11111 I); and so in other cases.
    Mathematicians love to count and counting the number of ways that a number can be partitioned has fascinated them since Euler made a dramatic breakthrough in understanding the partition function by writing down the generating series for it.

    Ferrers diagrams showing the partitions of positive integers 1 through 8. They are so arranged that images under the reflection about the main diagonal of the square are conjugate partitions.  Credit: Wikipedia

    Want to try it yourself without the pen and paper?  Wolfram Alpha has a web version.

    Emory mathematician Ken Ono says he will unveil new theories that answer the famous questions about partition numbers and prove that partition numbers behave like fractals, which you may know as those gimmicky random things so popular in the 1990s. They say they have unlocked the divisibility properties of partitions, and developed a mathematical theory for "seeing" their infinitely repeating superstructure - and that they have devised the first finite formula to calculate the partitions of any number. 

    "Our work brings completely new ideas to the problems," says Ono, who will explain the findings in a public lecture at 8 p.m. Friday on the Emory campus. "We prove that partition numbers are 'fractal' for every prime. These numbers, in a way we make precise, are self-similar in a shocking way. Our 'zooming' procedure resolves several open conjectures, and it will change how mathematicians study partitions."

    "Ken Ono has achieved absolutely breathtaking breakthroughs in the theory of partitions," says George Andrews, professor at Pennsylvania State University and president of the American Mathematical Society. "He proved divisibility properties of the basic partition function that are astounding. He went on to provide a superstructure that no one anticipated just a few years ago. He is a phenomenon."

    The work was funded by the American Institute of Mathematics (AIM) and the National Science Foundation. Last year, AIM assembled experts on partitions, including Ono, to attack some of the remaining big questions in the field. Ono, who is a chaired professor at both Emory and the University of Wisconsin at Madison, led a team consisting of Jan Bruinier, from the Technical University of Darmstadt in Germany; Amanda Folsom, from Yale; and Zach Kent, a post-doctoral fellow at Emory.


    (1) Euler gets mentioned a lot around here but he was not a one-dimensional math guy.  He could repeat Virgil's Aeneid from memory and even tell people the first and last line of any page in the edition which he used.


    One thing about this is misleading.  It is very easy to assume, for example, that “321” refers to the number 321, rather than as being shorthand for 3+2+1.  Also the somewhat quaint way that the 1911 Encyclopaedia describes things is not always helpful.
    However, partitions are interesting and powerful things.  Following this article I looked up the following book:


    Euler: The Master of Us All by William Dunham and found how Euler proved that, for any number, the number of partitions where all the summands are distinct (none is repeated) is equal to the number of partitions where all the summands are odd.
    This is only one of the many mathematical treasures in this wonderful book!
    Robert H. Olley / Quondam Physics Department / University of Reading / England
    I think to non-mathematicians there can be some confusion but who else is reading about partitions?   Well, you and me, but only 600 other people.  

    Greatest mathematician ever?  I'd make that argument.
    I think that any choice about the greatest mathematician ever is a very subjective speculation. But I doubt that any would argue that a list of the greatest mathematicians would contain:
    • Newton
    • Leibniz
    • Cantor
    • Euler
    • Gauss
    • Hilbert
    • Fermat
    • Pascal 
    • Riemann
    • Archimedes 
    • Godel
    • Riemann
    Of course any such list would be incomplete in my opinion, as it does exclude many non European figures from that made influential contributions to maths, but since they are not widely known in the west, they fade into obscurity. These would include Arabic, Indian and Chinese mathematicians. 


    Some significant discoveries by Middle Eastern and Indian mathematicians appear to have surfaced in Europe a couple of hundred years later, but evidence of transmission has been lost.

    However, there is plenty of evidence for certain cases.  By far the most important of these is the work of Al-Khwarizmi, whose work forms the basis for algebra as we know it.  His algebra treatise Hisab al-jabr w'al-muqabala is the most famous and important of all his works, and its title gives us the word “algebra”.

    Here is an interesting history of how the Hindu-Arabic number system was transmitted to Europe.

    Robert H. Olley / Quondam Physics Department / University of Reading / England