The De quinque corporibus regularibus is a treatise on Euclidean geometry, on the five regular geometric solids inscribable in a sphere, the regular polyhedra, or solids whose surface is formed by polygonal faces, which compared to all other polyhedra have the characteristic of having all faces, edges, and angles equal, of equal size. The five regular polyhedra are: the triangular-based pyramid, namely the tetrahedron (four triangular faces, equilateral triangles), the cube also called hexahedron (six square faces), the octahedron (eight triangular faces), the dodecahedron (twelve pentagonal faces) and the icosahedron (twenty triangular faces). The painter often used regular polyhedra to construct the space within which he set the paintings.
Madonna del Parto, the drawing is in the space marked by a dodecahedron
De prospectiva pingendi, the most famous treatise, is divided into three parts: the “drawing”, that is, how to paint individual figures, the “commensuratio”, that is how to arrange them in space, and the “coloro”, that is, how to color them. In particular, the essay focuses on the second part, examining the projection of surfaces, geometric bodies, and more complex volumes, such as parts of the human body, investigated looking for a scientific basis for their representation. The work, one of the fundamental treatises on the figurative arts of the Renaissance, also referred to topics of solid geometry discussed in the previous writing of Piero della Francesca, the De quinque corporibus regularibus. From these works Leonardo da Vinci then wrote the Treatise on Painting.
Flagellation of Christ by Piero della Francesca. The perspective lines, the vanishing point, and the cube lines within which the left scene is set are highlighted.
One of Piero della Francesca’s best friends, the mathematician Luca Pacioli (Borgo Sansepolcro, about 1445 – Borgo Sansepolcro, June 19, 1517), developed in his treatise De divina proportione what was written by the painter in De prospectiva pingendi. The printed edition of 1509 consists of three distinct parts: the first deals with the golden ratio and its applications in the various arts; the second is a treatise on architecture that refers to the theory of Vitruvius; the third part is the Italian translation of Piero della Francesca’s De quinque corporibus regularibus on the five regular solids. Therefore, the Tuscan friar was then accused of plagiarism by Vasari. At the end of the three parts there are two sections of illustrations, the first with the capital letters of the alphabet drawn using a line and compass by Luca Pacioli himself and the second with the 60 tables by Leonardo da Vinci.
Some of the 60 tables by Leonardo da Vinci designed for De divina proportione by Luca Pacioli
Portrait of Luca Pacioli (1495), attributed to Jacopo de’ Barbari, Capodimonte Museum