October 2007 Archives

This book has been an enthralling and fascinating read for me. Mark Ronan outlines the mathematics of symmetry and details its historical development from Evariste Galois's letter of May 1832 to about 2005.

Group theory is the mathematics of symmetry and the book has greatly helped me to learn and understand the fundamentals of group theory and especially simple finite groups.

Perhaps the most important revelation in the book is that the sum of the squares of the first twenty four integers is a perfect square (= 70^2). This means that every permutation of the 24-tuple list (1, 2, 3, ..., 24) defines a line of length 70. There are 620 448 401 733 239 439 360 000 permutations. This number is very close to Avogadro's number, an interesting coincidence of magnitudes. The size of the Monster, the largest simple finite group, is much larger than this.

Over the past few years I have read so many books and articles about symmetry and spacetime that it is impossible to keep track of all the threads of thought that have passed through my mind. In writing the ECOS 2007 paper about spacetime, shape, symmetry and thermodynamics I wanted to include a reference to a particular idea I had read about, but was unable to find it when I went looking for it. The general notion was that if we look out into the sky, i.e. look out into spacetime, we can see the image of the structure of spacetime at the smallest scales. I found that thought very appealing and sensible.

Today I came across the source that I had not been able to find: the book The Fabric of the Cosmos by Brian Greene. On p. 308 he states:

According to inflation, the more than 100 billion galaxies, sparkling throughout space like heavenly diamonds, are nothing but quantum mechanics writ large across the sky.

I'm glad to have found that again.

My main hobby project right now is Symmetry and its implications for Physics, Mathematics and Engineering. As of today the paper that I presented at the ECOS 2007 conference, entitled 'What Have Spacetime, Shape and Symmetry to Do with Thermodynamics?' is also published at arxiv.org. I'm delighted about that. It is available at http://www.arxiv.org/abs/0710.1242 .

Another project of mine is 'AblBeebl: a Font Family Scheme for Mathematical Symbols, Based on Symmetrically Transformed or Doubled Roman Characters'. I have put a huge amount of time into this and there is still a lot to do to bring it to a reasonable conclusion. I will stick at it doggedly until it is done. I hope to get it published in a journal - preferably with AMS, where Donald Knuth published a description of the Computer Modern font. However, I need to think that aspect out carefully - I would like to provide the symbols in an open-source manner so that scientists and engineers can make use of them freely and develop them further.

Another hobby project I am working on is the inaugural lecture that I am due to give at some stage in connection with the Professorship with which I was conferred earlier this year by DIT. The tentative title is 'Some Rather Mechanical Reflections on Symmetry in Art, Science, Engineering and Mathematics. I want this to be a really great presentation that is interesting and stimulating for all members of the audience, whether academic or not. It's quite a challenge. The most difficult part of this is the mathematics part. In that connection I am attempting to develop my understanding of groups as the language of symmetry.

'Unstrike a Match' is yet another hobby project. This is an exploration of time through a video that shows a match being unstruck. I have made the video and have written text to go with it. It will need to be published on-line, but I haven't decided how this will be.

'Tena, the Digit Ten' is a hobby project that sprung up from AblBeebl, but immediately developed a life of its own. It is very much in the mathematical field: an exploration of the basis of counting. This comes back to symmetry too. Zero is not a counting number. Tenamal counting is zero-less counting in which all digits are treated equally (are symmetric).

There are two projects I became interested in through helping my daughter Aoife with 'Young Scientist' projects. The first relates to a spheroidal image capture device - I would like to develop that concept further as I believe it has great potential as a new type of still or video camera. The second relates to discrete Fourier series. I started to develop some insights into this in helping Aoife to analyse the sound characteristics of flutes. This comes back to symmetry too; in particular, I'm thinking of the symmetrical way in which light waves propagate and the physical significance of imaginary numbers.