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TODAY, with graphics software to do the hard work for us, we take for granted the ability to generate and manipulate complex images. Leonardo da Vinci (1452-1519) had to rely on pen and paper to represent and understand the world, which makes the achievements of this artistic and engineering genius all the more remarkable.

Manipulating geometrical solids in three dimensions was for him an engrossing form of mental sculpture. Since antiquity, mathematicians had been trying to work out how to determine the volume of such solids. In the notebook from which the spread above is taken, he shows how the…