1.---The world of symmetry has obsessed people for generations. For many, like Escher, this world is an aesthetic one.

But for the scientist, it not only holds aesthetic appeal but underlies the way much of nature behaves. The behaviour of subatomic particles, the way a CD stores information, even the stacking of oranges at your grocers depend on important properties of symmetries. But the scientist needs to be able to understand not only the symmetries of, for example, Escher's 3-dimensional chocolate box (pictured here) but also about the possible symmetries that exist in 4 and higher dimensional spaces. They have had to wait though for the mathematicians of the twentieth century to navigate them through this rich and complex world of symmetry.

2.---This last month has seen the visit to England of one of the master navigators of this world - Professor John Conway. For many years mathematics at Cambridge had been inspired by Conway's colourful presence, but in the early eighties he was lured (like so many other British scientists) to the attractions of academic life in the States. During his time in Cambridge his mathematical and personal charisma guaranteed him almost cult status. The common room would always be buzzing with Conway's ideas or, more likely, with one of the mathematical games he loves to invent.

3.---He was tempted back to England (if only for a month) by the attractions of a birthday party which was organised at the University of Birmingham. The party was to celebrate the tenth anniversary of a publication that he produced, in collaboration with four other authors, shortly before his departure to Princeton.

The Atlas of Finite Simple Groups, as this publication is called, charts the world of symmetry. It is impressive if only for its size - its pages are over half the size of this page. But its physical dimensions pale in comparison with the sheer size of its content.

4.---As with any atlas, this Atlas is the culmination of decades of exploring by over 100 mathematicians (including Conway himself). Their quest, which covers over 15,000 journal pages, tries in some meaningful way to list (or what mathematicians call classify) all the possible finite symmetries that can exist. What helps in making this list is that there are certain basic symmetries from which one can generate all the more intricate symmetries. The way a molecule in chemistry is built of basic atoms provides some sort of analogy. Conway's Atlas which lists all these basic buildings blocks is then the mathematician's version of the chemist's Periodic Table. The group of symmetries of Escher's chocolate box pictured here is the first building block to appear in the Atlas.

5.---Although mathematicians have called these buildings blocks simple groups of symmetries they can be far from simple in their nature. Most of these symmetries fit into well behaved families.

professor John Conway

But there are 26 basic building blocks which fit into no pattern at all. They are highly irregular symmetrical settings which mathematicians have called sporadic. It is the exceptions which often excite mathematicians more than anything. The story of the discovery of these sporadic symmetries is very similar to that of the discovery of subatomic particles. Their existence was often predicted well before the symmetries were discovered.

6.---Perhaps the most fascinating and mysterious sporadic group of symmetries was discovered by Robert Griess at Princeton. He ''constructed'' a sort of snowflake like figure in 196883 dimensional space which had over 1053 symmetries (that's one with 53 zeroes - more than there are atoms in the sun). But this group of symmetries could not be decomposed into a combination of smaller symmetries.

7.---This object Conway christened the Monster. As people studied the Monster, it became clear that it was more than just a mathematician's curiosity but related to fundamental questions of mathematics and physics. People talked mystically about it being the symmetry group of the universe and it began to take on an almost religious status. Although mathematicians have failed so far to find God in the Monster, they have discovered an intimate relation between the Monster and what physicists call string theory. As those who have read their Hawking will know, physicists hope that string theory will provide a true description of the physical world.

8.---The key to this connection between physics and the Monster is a remarkable way to pack 24-dimensional oranges. Your grocer probably arranges his oranges in the shape of a hexagon when he packs them as this is considered to be the packing which wastes least space. (It is striking that mathematicians still can't prove that this packing is the best.) This hexagonal packing also seems most efficient for 4 and 5 dimensional oranges. But, as we increase the dimensions, the 24-dimensional grocer has a special packing which allows him to stack his oranges in a much more efficient manner. The symmetry of this packing connects string theory and the Monster. Although this may seem interesting only to mathematicians and 24-dimensional grocers, this packing has been a breakthrough in the design of very efficient codes which are used today by the 3-dimensional military. As these beautiful properties became apparent and people became better aquatinted with the Monster, Greiss decided to rechristen it the Friendly Giant.

9.--In his talk in Birmingham, Conway, wearing a tee-shirt with 200 places of pi on its back, described his fascination with the Monster. Most of us would be daunted by its immensity but Conway feels at home with such a creature and deserves more than anyone to call it the Friendly Giant. He has the sort of mind which means he wouldn't have to remove his tee-shirt to tell you what was on its back.

10.--Conway has always been obsessed with symmetry. In his lecture he described his fascination with the designs of Escher. ''I have a book of Escher's pictures on my piano. I try to ration myself to an Escher picture a day. Often I can't resist cheating and turning the page early but I always insist on at least going out of the room first before I can turn the next page.'' So obsessed is he with patterns and symmetry, there was rumour that a sixth person was denied a place on the list of authors of the Atlas because his name didn't have six letters with vowels in the right place like the other five authors - Conway, Curtis, Norton, Parker and Wilson.

11.--Conway's other passion for games also infects much of his serious mathematics. Many people are drawn initially to the subject by its playful nature but this can often be forgotten amidst the serious day to day rigours of being a professional mathematician. It is always refreshing therefore to listen to someone like Conway who has retained his playful approach to the subject.

12.--Conway's performances are almost magical in quality. He weaves together what at first sight look like mathematical curios or tricks but by the end of the lecture has arrived via these games at answers to very deep questions of mathematics. These deep insights are preceded by his characteristic laugh as if he too is surprised at where he has arrived. At the same time he has reduced a room of serious academics to playful children. They rush up at the end of the lecture to play with the mathematical toys he produces from a suitcase of tricks that he often carries with him.

13.--But as this birthday party illustrated, there is still much uncharted water in the world of symmetry. Many properties of the Monster still remain deeply mysterious. One of the most exciting mysteries is a phenomenon Conway and his co-author Simon Norton named Monstrous Moonshine. There were certain strange numbers that appeared as Conway and Norton studied the Monster. At some point whilst Conway was browsing through the library he suddenly noticed these numbers appearing in a seemingly unrelated area of mathematics called modular functions. These functions are essential to the proof of Fermat's Last Theorem. This incredible numerological coincidence seems to hint at a deep connection between these two areas but as yet the connection remains somewhat illusive.

14.--Conway describes this discovery as one of the most exciting moments of his life. It is rather like an archaeologist uncovering designs in tombs in Egypt which have only ever been seen before in Mayan tombs in South America. Mathematics is full of such connections and many mathematicians will admit that they are often the greatest incentive in mathematical discovery.

15.--Monstrous Moonshine which seems to be reflecting beams from the unrelated world of modular functions is still one of the most stimulating questions in the area. Like Bottom in A Midsummer Night's Dream, who can resist the mathematical weaver John Conway's call to "Find out Moonshine"?