With school half-term fast approaching my kids are bracing themselves. Unlike most children, they get a little nervous about an impending holiday. With a mathematician for a dad they often find themselves embarking on some nerdy, maths-inspired adventure.

It all started one year when we were on holiday in Andalusia. My family were happy to join the tourists who flock to the Alhambra in Granada, Spain. For me, the Moorish palace is one of the meccas of mathematics. If I had to choose one building to which to be banished, as a man obsessed with symmetry, I would choose the Alhambra. What gets my blood racing are the symmetrical tiles that cover every available surface of the palace.

Muslim texts forbid the depiction of things with souls. So the Moorish artists found more geometric ways to express themselves. Armed with a sophisticated mathematical intuition, they began to cover the ceilings, walls, floors and gardens of their palaces with tiles of different geometric shapes and colours.

Each wall was like a new canvas, challenging the artist to create a different, original way of covering the façade. The tiles repeat themselves left and right, up and down. Sometimes the tiles have left-right reflectional symmetry such as that found in the human face. But there are also more subtle symmetries at work in the walls of the Alhambra.

Symmetry for a mathematician means being able to pick up all the tiles, move them around and then set them back down again so that they fit into the original outline. Symmetry is like a magic trick: close your eyes, I move the tiles and when you open your eyes the tiles look as they did before I moved them.

But can you make a science out of this art? Is there a way to say that two walls have the same underlying group of symmetries although they look physically very different? Are the symmetrical possibilities endless, or are there limits to what symmetries can exist?

It took until the beginning of the 19th century for the creation of the mathematics that would reveal the symmetrical secrets of the Alhambra. A young revolutionary called Evariste Galois developed a language called group theory that allowed mathematicians to articulate the story of symmetry. His life, alas, was cut tragically short before he could fully realise the potential of his discovery. He was shot dead, aged 20, in a duel over love and politics. But by the end of the 19th century mathematicians used the ideas of Galois to prove that, however hard you tried, there are only 17 underlying groups of symmetries that one can depict on the walls of the Alhambra.

The drawings of M.C. Escher , the Dutch artist, were inspired by visits that he made to the palace. Not bound by the Muslim prohibition of drawing things with souls, his designs are full of angels and demons, lizards and fish. But, however hard Escher tried, the mathematics of Galois proves that there cannot be an eighteenth group of symmetries that he could squeeze out of these pictures.

The task for my family’s half-term visit to the Alhambra was to discover whether the Moorish artists had found examples of all 17 different sorts of symmetries. We set out on our mathematical treasure hunt, trying to find as many symmetries as we could. Often we’d think that we’d got a new design only to discover that it was an old one in new clothes.

We spent two days combing the walls as groups of tourists flew past us unaware of the hidden mathematical beauty in front of them. By the end we’d discovered examples of all 17 . . . well, nearly. There was one group of symmetries that proved quite elusive. Eventually we found a wall that almost gave us the missing one. The only trouble was that some of the tiles were the wrong colour. Once we’d painted the blue tiles black we had our seventeenth symmetry. But before I get inundated with letters complaining that I’ve defaced the Alhambra let me reassure you that the repainting was done in our minds. Mathematicians don’t like to get their hands dirty.

Finding Moonshine by Marcus de Sautoy is published by Harper Perennial tomorrow at £8.99. To order it for £8.54 inc p&p call 0845 2712134 or visit timesonline.co.uk/booksfirst

Conundrum

How many different ways can you put a dice down inside a square outline on a table?

Answer

24. There are 6 faces that can be showing on the top of the cube. Each of these faces can be rotated in 4 different orientations. So that makes a total of 24. These correspond to the rotational symmetries of the cube.

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