Leonardo da Vinci's Polyhedra

    Leonardo da Vinci (1452-1519) was the quintessential renaissance man: artist, mathematician, scientist, and engineer.  He was a great lover of geometry, and devoted much time to it starting in his early forties.  His most outstanding polyhedral accomplishment is the illustrations for Luca Pacioli's 1509 book The Divine Proportion.  At right is one of the illustrations from that book. The term Ycocedron Abscisus in the title plaque means truncated icosahedron, and the term Vacuus refers to the fact that the faces are hollow. (The drawings are beautifully hand colored like this in the Ambrosiana manuscript, reprinted by Officina Bodoni, 1956, and also by Silvana Editoriale, 1982.)

    These are the first illustrations of polyhedra ever in the form of "solid edges." The solidity of the edges lets one easily see which edges belong to the front and which to the back, unlike simple line drawings where the front and back surfaces may be visually confused.  Yet the hollow faces allow one to see through to the structure of the rear surface.  This is a brilliant new form of geometric illustration, one worthy of Leonardo's genius for insightful graphic display of information.  However, it is not clear whether Leonardo invented this new form or whether he was simply drawing from "life" a series of wooden models with solid edges which Pacioli designed.  If Pacioli designed these models, then he deserves the credit for this new "solid edge" idea, but it is likely that Leonardo designed them.

    There are roughly sixty similar illustrations in the book, mostly in pairs contrasting models with solid faces and models with this solid edge technique, such as these two versions of the dodecahedron:

    The Platonic solids and six of the Archimedeans are shown, including the first presentation of the icosidodecahedron and the first printed image of the rhombicuboctahedron, which had appeared earlier in a painting, the portrait of Pacioli.  Below is the truncated octahedron.  Click on it for a high-resolution image which shows the details more clearly.


    Another popular polyhedron of Renaissance times was the 72-sided Sphere, drawn with six rows of twelve faces.  It illustrates a theorem from Euclid, and as a possible structure for a dome, it symbolized the role of geometry in architecture:

        

    In the printed version of the book are woodcuts based on Leonardo's drawings.  For example, here is the first printed icosidodecahedron and an "elevated" form of it.  For the elevated forms, each face is augmented with a pyramid composed of equilateral triangles.  Many of the solids are treated in this way.  The "elevated icosidodecahedron" below was used as the template for two of Fra Giovanni's spectacular intarsia.

    In addition to the illustrations for Pacioli's book, we find other Leonardo polyhedra of interest.  Here is a mazzocchio drawn by Leonardo in his solid-edge form.


    Interesting doodles of polyhedra can be found scattered throughout Leonardo's drawings.


     Above is one which shows a dodecahedron in the solid edge form, amidst a series of plans for fortifications.
     

    Above is a drawing showing studies for a fountain, in the middle of which we find a doodle of a tetrahedron inscribed in a cube.

    And here, Leonardo draws an interesting packing of cubes, again in the "solid edge" manner.

    500th Anniversary Exhibition

    I feel there should be a commemorative exhibition celebrating the 500th anniversary of the Pacioli/Leonardo book, coming up in 2009.  The exhibit should include reconstructions of some or all of the wooden models and illustrations of their influence on other artists.  I am working towards organizing such an exhibition.
    Virtual Polyhedra, (c) 1999, George W. Hart