# Leonardo da Vinci's Geometric Sketches

## Introduction

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.