Piero della Francesca's Polyhedra

    Piero della Francesca (1410(?) - 1492) was an outstanding 15th century Renaissance artist, both a mathematician and a painter.  His genius in developing the new methods of perspective and employing them in his paintings has withstood the centuries. However, his well-deserved mathematical reputation was lost and only has been regained in this century.  That loss was largely due to the fact that Pacioli plagarized Piero's major writings shortly after Piero's death, incorporating them into his published books without attribution.  For centuries, the mathematical contributions of Piero were questioned.  However, three of Piero's mathematical manuscripts were discovered in the early 20th century, and his reputation regained.

    Piero 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.

    Above are Piero's drawings of the truncated tetrahedron.  Note the solid form of the drawing --- edges of the rear surface are not drawn --- in contrast with the transparent form of the following images:
    The above illustration is Piero's drawing of an icosahedron inscribed in a cube.  This illustration comes from the manuscript of Piero's Libellus De Quinque Corporibus Regularibus, now in the Vatican library. The fact that each vertex of an icosahedron can be placed thusly in a face plane of a cube is a beautiful Platonic relationship discovered by Piero, the only one not described in Euclid's Elements.  As such it is a genuinely new piece of polyhedral mathematics, and marks the beginning of the Renaissance fascination with polyhedral geometry.

    A clearer image, at left, comes from the printed version of  Pacioli's De Devina Proportione, which plagarized Piero's manuscripts, but did have the positive effect of widely disseminating his discoveries.
     
     

    Exercise: On the facing page of his Libellus, Piero shows another interesting and original construction.  I stared at this for quite a while before figuring out what it is.  I recommend you stare at the illustration at right for a while to see if you can make it out.  Hint: it involves two Platonic solids.

    Answer: Piero is showing a nonstandard method of inscribing a cube in a regular octahedron.  The eight vertices of the cube lie on eight of the twelve edges of the octahedron.  It differs from the standard method of inscribing a cube in an octahedron in that they are rotated 45 degrees relative to one another, about one of the 4-fold axes. This is not a standard Platonic relationship because only one set of their three pairs of 4-fold symmetry axes are aligned.

    I have no idea how Piero thought about this construction, but it is interesting to note some geometric and artistic consequences of it: 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.

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

    Virtual Polyhedra, (c) 1998, George W. Hart