Monday 18, December 2017

Artiscience Library and Reading Room at Summerhall

Artiscience Library

ASCUS is delighted to offer access to the Artiscience Library at Summerhall. This library contains around 25,000 books on art and science. The library, collated by Colin Sanderson, is available as a reading room and for research and residency work. If you are interested in accessing the library please see details below. If you have ideas for working in or with ASCUS and the library around artiscient projects please contact both Colin (artiscience@aol.com) and James (howie@ascus.org.uk).

Artiscience, n., 1. the theory and practice of integrating art(s) and science(s), 2. knowledge of relations between the arts and sciences

Hence, artiscient, adj., exhibiting or practising artiscience

The collection of books and other items in the Artiscience Library at Summerhall are devoted to historical and contemporary relations between the arts and sciences, in all senses of those terms, and from pre-history to the present day. Some foreign language materials are also available, something we should like to build upon.

The Library shall be available to Readers on a daily basis by appointment. to use the Reading Room simply call or text in advance to arrange a time to, Colin Sanderson on 0789 996 4250, who shall be on hand to provide information about the content and purposes of the Library and to offer, if required, advice on any artiscient project you may have in mind.

Ramanujan’s Congruences

Ian Stewart

Colin Sanderson: “A couple of years ago, asked for a single paragraph on what he thought was one of the most beautiful equations, Ian Stewart, Emeritus Professor of Mathematics, Warwick University, kindly provided us with the following text on Srinivasa Ramanujan and his equation.  

      The project for which we asked it could not be taken forward, but with the opening in cinemas tomorrow (8 April) of the film “The Man Who Knew Infinity,” starring Dev Patel as Ramanujan and Jeremy Irons as G. H. Hardy, Ian has kindly permitted us to share his text now, for which many thanks.   

  1. H. Hardy was the author of “The Mathematician’s Apology” (1940), which is surely the most quoted modern book on the art and beauty of mathematics.   C. P. Snow, whose “two cultures” lecture was the subject of the Science Book Reading Group meeting we held in the Artiscience Library last month, knew Hardy at Cambridge in the 1930s and wrote a preface for a later edition (1967).

      To learn more about mathematics and art, Ramanujan, Hardy and Snow, do visit the library at Summerhall.”

 

Ramanujan’s Congruences

Ian Stewart

Some of the most beautiful and surprising formulas in mathematics were discovered by the self-taught Indian mathematician Srinivasa Ramanujan, and published in 1919. This Indian mathematician was also known as an amateur playing betsson: online casino . They concern the partition function p(n), which is the number of different ways to write an integer n as a sum of integers. For example 4 can be written as 4, 3+1, 2+2, 2+1+1, or 1+1+1+1. Since there are five distinct partitions here, p(4) = 5.

Ramanujan observed empirically that if n is of the form 5k+4, then p(n) is divisible by 5. If it is of the form 7k+5, then p(n) is divisible by 7. If it is of the form 11k+6, then p(n) is divisible by 11. He proved the first two of these observations using the astonishing formulas

Eqn1

Eqn2

 

where (q)¥ is the Pochhammer symbol

(q)¥ = (1-q)(1-q2)(1-q3)…

The details don’t matter—the point is that the right-hand side of Ramanujan’s first equation is clearly a multiple of 5, and the right-hand side of the second is a multiple of 7. Later G. H. Hardy proved the third observation using a method found in Ramanujan’s unpublished writings.

Ramanujan asserted that no similar result could be valid for numbers other than 5, 7, and 11. But in 2000 Ken Ono proved that similar results occur for any divisor that is not a multiple of 6. For example, p(4063467631k+30064597) is always a multiple of 31. It is hardly surprising that Ramanujan, in the days before computers, could not spot this.

The beauty of these theorems rests on two features. The formulas involved are quite unlike anything found in mathematics before (the element of surprise) and the final result is simple and elegant.

One comment

  1. This sounds like an amazing project! Definitely going to have to check it out 🙂