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A human being is a part of the whole, called by us "Universe," a part limited in time and space. He experiences himself, his thoughts and feelings as something separated from the rest, a kind of optical delusion of his consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us. Our task must be to free ourselves from this prison by widening our circle of compassion to embrace all living creatures and the whole of nature in its beauty. Nobody is able to achieve this completely, but the striving for such achievement is in itself a part of the liberation and a foundation for inner security.

In H. Eves, Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.

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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 feature discussing double-entry bookkeeping.  But Pacioli compiled and wrote other texts.  In De divina proportione of 1509, he discusses 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.

Below 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 removes the sides to better reveal the compete structure of the polyhedron.  These later “nets” of vertices and edges illustrate the artist’s graphic genius.