De quinque corporibus regularibus
It was the first of what would become many books connecting mathematics to art through the construction and perspective drawing of polyhedra,[1] including Luca Pacioli's 1509 Divina proportione (which incorporated without credit an Italian translation of della Francesca's work).
[3] The five Platonic solids (the regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron) were known to della Francesca through two classical sources: Timaeus, in which Plato theorizes that four of them correspond to the classical elements making up the world (with the fifth, the dodecahedron, corresponding to the heavens), and the Elements of Euclid, in which the Platonic solids are constructed as mathematical objects.
[4][5] The thirteen Archimedean solids, convex polyhedra in which the vertices but not the faces are symmetric to each other, were classified by Archimedes in a book that has long been lost.
The other two, De prospectiva pingendi and Trattato d'abaco, concern perspective drawing and arithmetic in the tradition of Fibonacci's Liber Abaci, respectively.
[10] Although the early parts of De quinque corporibus regularibus also borrow from this line of work, and overlap extensively with Trattato d'abaco, Fibonacci and his followers had previously applied their calculation methods only in two-dimensional geometry.
[16] The third part includes additional exercises on circumscribed spheres, and then considers pairs of Platonic solids inscribed one within another, again focusing on their relative measurements.
[20] The fourth part of De quinque corporibus regularibus also includes domed shapes like the domes of the Pantheon, Rome or the (at the time newly constructed) Santa Maria presso San Satiro in Milan formed from a ring of triangles surrounded by concentric rings of irregular quadrilaterals, and other shapes arising in architectural applications.
[13][26] However, della Francesca likely wrote his book first in Italian, before translating it into Latin either himself or with the assistance of a friend, Matteo dal Borgo,[27] so its original draft may have been from before Guidobaldo's accession.
[31] Although Giorgio Vasari denounced Pacioli for plagiarism in his 1568 book, Lives of the Most Excellent Painters, Sculptors, and Architects, he did not provide sufficient detail to verify these claims.
[2][32] Della Francesca's original work became lost until, in 1851 and again in 1880, it was rediscovered in the Urbino collection of the Vatican Library by Scottish antiquary James Dennistoun and German art historian Max Jordan [de], respectively, allowing the accuracy of Vasari's accusations to be verified.
