Tea and Symmetry

For the past week I have been sitting down at about 3pm with a pot of tea and a copy of Finding Moonshine by Marcus du Sautoy. The book takes a journey through 12 months in his life . It sets in historical and personal context his work in mathematics and its connection to symmetry. The chapter for October: the Palace of Symmetry is delightful. It details his search, accompanied by his son Tomar, for the 17 underlying symmetries in two dimensions. This is carried out within the Palace of the Alhambra, looking at the Moorish tiling patterns. Interesting to find on p63 a confirmation of my party piece on symmetry:-

There are only 17 different types of wallpaper!

If anyone challenges this then I explain that, as identified by their symmetry groups, there are only 17 possibilities. You can have birds, you can have cartoon characters, you can have elephants even, but there are only 17 basic ways in which you can have them! I did learn something though on p83. If you allow permutations of two colours you can add a further 46 symmetry groups. No. I think I'll stick to 17. It's demonstrably a prime and it should get the NANs going(New Age Numerologists).

The book concludes as I read yesterday, downing my pot of Darjeeling, with the search for the Monster. This beast has a large number of possible symmetries, How large? Ridiculously large 808.... - it contains 54 digits before the decimal point. If you had a mind to get your head round it you would have to consider that it exists, if that is the right word, in 196,883 dimensions. Feeling a little dizzy? Don't, just trust him he is a mathematician.

What, I hear you ask in the sternest of prudential tones, is the use of all this business?

Ah well now that is a story for another time!