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.
