This year marks the quincentenary of the death in 1492 of Piero della Francesca whose paintings have always seemed to me to have some special attraction for anyone who is mathematically inclined. I would like to celebrate this anniversary by describing a brief geometrical exploration – and some consequent speculations – prompted by looking at one of his paintings. It is not known for certain when Piero della Francesca painted his strange, enigmatic picture of the flagellation of Christ , now to be seen in the Palazzo Ducale, Urbino, which is on the cover of this edition. It is usually held to be one of his later works, painted in his late forties before that last period of his life which he seems to have devoted exclusively to mathematics. The experts disagree not only about the date of the painting, but also about its meaning and purpose. In particular, it is not known, though it is much discussed, which contemporary persons might have been portrayed as the three curiously aloof figures in the foreground.