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The mathematician is entirely free, within the limits of his imagination, to construct what worlds he pleases. What he is to imagine is a matter for his own caprice; he is not thereby discovering the fundamental principles of the universe nor becoming acquainted with the ideas of God. If he can find, in experience, sets of entities which obey the same logical scheme as his mathematical entities, then he has applied his mathematics to the external world; he has created a branch of science.
Aspects of Science, 1925.
Loci: Convergence
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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.