The Franciscan friar, Luca Pacioli (ca. 1445-1509) is best known for his compendium of fifteenth century mathematics, Summa de arithmetica, geometrica, proportioni et proportionalita (1494). This book was intended to be a summary of the known mathematics of the time and included a special section on double-entry bookkeeping. But Pacioli compiled and wrote other texts. In De divina proportione of 1509, he discussed the “golden proportion” and the properties of various polyhedra. Pacioli was fascinated by polyhedra, studied their properties, and constructed wooden models for many of the solids. The friar also befriended many of the artists of the time, including Leonardo Da Vinci. Da Vinci briefly studied geometry with Pacioli but focused on considerations of shape, size, and perspective, descriptive features of objects rather than their theoretical foundations. Da Vinci illustrated Divina proportione, supplying sixty plates for the work.
On the following pages are facsimiles of several of these plates; specifically, those illustrating the sphere, cone, cylinder, pyramid, and the five Platonic solids. For the Platonic solids, Da Vinci supplied two views: a plane view and a “vacua” or empty view where he removed the sides to better reveal the complete structure of the polyhedron. These “nets” of vertices and edges illustrate the artist’s graphic genius. For more information, see "Leonardo da Vinci's Polyhedra," by George Hart.
We thank the Pennsylvania State University Libraries for allowing Convergence to publish the following images of Da Vinci’s illustrations from their copy of the beautiful facsimile of Pacioli’s De divina proportione published by Silvana Editoriale, Milan, Italy, 1982. This facsimile is of the manuscript copy of Pacioli’s Divina held by the Biblioteca Ambrosiana in Milan, one of only three Pacioli made.
For images from the 1509 printed edition of De divina proportione, see:
- A "mathematical treasure" in Convergence providing additional information about Pacioli’s Divina proportione, along with three images from the copy housed at the University of Oklahoma Library, and
- Another "mathematical treasure" in Convergence providing two images from the copy of Pacioli’s Divina proportione housed at Columbia University’s Butler Library.
Index to more Mathematical Treasures