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VSOCOLSKY RAISES THE BAR OF THE ANALYSIS OF PIERO DELLA FRANCESCA AT THE METROPOLITAN. SHE SAYS:

‘“Piero della Francesca is the first of a series of artists whose geometric explorations involved the gradual rediscovery of the Archimedean solids. He rediscovered the five Archimedean solids which are truncated Platonic solids: the truncated tetrahedron, truncated octahedron, truncated cube, truncated dodecahedron, and truncated icosahedron. These were presumably known to Archimedes, but that was not known in Piero’s time. In addition, he rediscovered the cuboctahedron (mentioned by Plato). These six solids are employed in new and substantial geometric problems — calculating the edge length of the solid given the radius of their circumsphere.

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Piero’s drawing of an icosahedron inscribed in a cube comes from the manuscript of Piero’s Libellus De Quinque Corporibus Regularibus, now in the Vatican library. There are three different ways to so inscribe a cube within the same octahedron. Superimposing the three (and then erasing the octahedron) gives a well-known compound, the compound of three cubes. Dualy, there are three different ways of so placing an octahedron around the same cube. Superimposing the three gives the compound of three octahedra. Interestingly, Escher employed both of those compounds.”

To see the drawings, go to
http://www.georgehart.com/virtual-polyhedra/piero.html

‘For a recent study of Piero’s life and times, see Marilyn A. Lavin (ed.), Piero della Francesca and his Legacy, 1995.’