The Page 69 Test: "Symmetry"

Wednesday, March 12, 2008

"Symmetry"

Marcus du Sautoy is Professor of Mathematics at the University of Oxford and a Fellow of Wadham College. He is the author of Symmetry: A Journey Into the Patterns of Nature (Finding Moonshine, UK title) and The Music of the Primes.

He applied the Page 69 Test to Symmetry and reported the following:

Quite extraordinary how page 69 captures so many themes in the book. In fact symmetry, the subject of the book, is at the heart of the very number 69. I think 69 is the first number which only has rotational symmetry and no reflectional symmetry. This rather spookily is also the theme of the passage on page 69. The setting for the page is the Alhambra in Granada, a palace dedicated to symmetry. The book attempts to weave a historical narrative of mankind's journey to unlock the secrets of symmetry together with a very personal narrative of what it is like to be a practicing mathematician. In this chapter, I take a trip to Granada to explore the symmetries painted on the walls by the Moorish artists in the Alhambra. It is a chapter where the two themes, the personal and historical, synthesize in a particularly organic way. The aim of this chapter is to introduce some of the big themes that will dominate the rest of the narrative. What is symmetry? How can we articulate the fact that two walls have the same symmetry although they may look very different? What are the limitations of symmetry? How can we prove that there are in fact only 17 different symmetries that can be represented on the two dimensional wall? And did the Moors find all 17? Joining me on my journey through that Alhambra is my son, Tomer. He is my Passepartout on my journey through symmetry and acts as a set of innocent eyes, almost like the reader's eyes, to counter my more mathematical perspective on the world. On page 69 we encounter a wall full of rotational symmetry but without any reflectional symmetry, just like the number 69.

Page 69:

[Fig. 25]

Caption: The entrance to the Alhambra

What makes the images at the entrance to the Alhambra a regular tiling and not a Roman mosaic or Escher cheese sandwich is that each piece can be lifted and shifted (either up or down, left or right) and eventually it will sit perfectly on a copy of itself. But there is more regularity here than in just the individual movement of each piece. I can take a copy of the whole picture, shift it horizontally or vertically, and lay it down again so that it exactly matches the original picture. This is what imbues it with a sense of the infinite. The symmetry in the wall contains a message – a programme, if you like – which stipulates exactly how the tiles will be laid out as the wall is expanded, even to the infinite reaches of the universe.

But there is more to the symmetry of this wall than simple repetition. How can we articulate what that symmetry is, though? How can we express the fact that one wall has more symmetry than another? Is it possible even to pin down precisely what we mean when we say that two walls have the same symmetry?

The reason there is more symmetry in this wall than simple repetition is that there are other ways I can pick the picture up and place it down on a shadow of itself. Instead of simply shifting it left or right, up or down, I can turn it before I lay it down. For example, if I keep the centre of one of the eight-pointed stars fixed and rotate a copy of the picture by 90° around this point, the shapes line up perfectly on top of the original picture.

I am intrigued to see what Tomer makes of the design. His initial reaction is that it hasn’t got any symmetry. He is looking for lines that he can fold the image along so that the two sides of the picture match up, as one of the psychologist Rorschach’s inkblot images. Immediately he can see that this isn’t possible here. Intriguingly, the design that...