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Standard mathematics has recently been rendered obsolete by the discovery that for years we have been writing the numeral five backward. This has led to reevaluation of counting as a method of getting from one to ten. Students are taught advanced concepts of Boolean algebra, and formerly unsolvable equations are dealt with by threats of reprisals.

In Howard Eves' Return to Mathematical Circles, Boston: Prindle, Weber, and Schmidt, 1988.

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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.