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Concept
De quinque corporibus regularibus
Summary
De quinque corporibus regularibus (sometimes called Libellus de quinque corporibus regularibus) is a book on the geometry of polyhedra written in the 1480s or early 1490s by Italian painter and mathematician Piero della Francesca. It is a manuscript, in the Latin language; its title means [the little book] on the five regular solids. It is one of three books known to have been written by della Francesca.
Along with the Platonic solids, De quinque corporibus regularibus includes descriptions of five of the thirteen Archimedean solids, and of several other irregular polyhedra coming from architectural applications. It was the first of what would become many books connecting mathematics to art through the construction and perspective drawing of polyhedra, including Luca Pacioli's 1509 Divina proportione (which incorporated without credit an Italian translation of della Francesca's work).
Lost for many years, De quinque corporibus regularibus was rediscovered in the 19th century in the Vatican Library and the Vatican copy has since been republished in facsimile.
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. Two apocryphal books of the Elements concerning the metric properties of the Platonic solids, sometimes called pseudo-Euclid, were also commonly considered to be part of the Elements in the time of della Francesca. It is the material from the Elements and pseudo-Euclid, rather than from Timaeus, that forms della Francesca's main inspiration.
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.
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