Leonardo Da
Vinci and Platonic Solids


The constructions of the Platonic solids are included
in Book XIII of Euclid's Elements. Propositions 13 through 17
describe the construction of the tetrahedron, octahedron,
cube, icosahedron, and dodecahedron in that order. Plato was
mightily impressed by these five definite shapes that
constitute the only perfectly symmetrical arrangements of a
set of (nonplanar) points in space, and late in life he
expounded a complete "theory of everything" (in the treatise
called Timaeus) based explicitly on these five solids. It's
uncertain who first described all five of these shapes  it
may have been the early Pythagoreans  but some sources
(including Euclid) indicate that Theaetetus (another friend of
Plato's) wrote the first complete account of the five regular
solids. Plato conceived the four classical elements as atoms
with the geometrical shapes of four of the five platonic
solids . These are, of course, not the true shapes of atoms;
but it turns out that they are some of the true shapes of
packed atoms and molecules, namely crystals: The mineral salt
sodium chloride occurs in cubic crystals, fluorite (calcium
fluoride) in octahedra, and pyrite in dodecahedra. This
concept linked fire with the tetrahedron, earth with the cube,
air with the octahedron and water with the icosahedron. There
was intuitive justification for these associations: the heat
of fire feels sharp and stabbing (like little tetrahedra).

Air is
made of the octahedron; its minuscule components are so smooth that
one can barely feel it. Water, the icosahedron, flows out of one's
hand when picked up, as if it is made of tiny little balls. By
contrast, a highly unspherical solid, the hexahedron (cube)
represents earth. These clumsy little solids cause dirt to crumble
and breaks when picked up, in stark difference to the smooth flow of
water. Plato made the following identifications:
tetrahedron = plasma ("fire")
icosahedron = liquid ("water")
hexahedron = solid ("earth")




The fifth Platonic Solid, the dodecahedron, Plato
obscurely remarks, ...the god used for arranging the
constellations on the whole heaven (Timaeus 55).
Interestingly, almost 2000 years later, Johannes Kepler
was similarly fascinated by these five Platonic shapes, and
developed his own cosmology from them.. Newton's comments
about the sides of light particles are very reminiscent
of Plato's language in Timaeus. It's also interesting to
compare some passages in Timaeus, such as And so all
these things were taken in hand, their natures being
determined by necessity in the way we've described, by the
craftsman of the most perfect and excellent among things that
come to be... with phrases in Newton's Principia, such
as ...All the diversity of created things, each in its
place and time, could only have arisen from the ideas and the
will of a necessarily existing being... ...all phenomena may
depend on certain forces by which the particles of
bodies...either are impelled toward one another and cohere in
regular figures, or are repelled from one another and
recede... ...if anyone could work with perfect exactness, he
would be the most perfect mechanic of
all... 
Leonardo da Vinci (14511519) was one of the greatest
inventorscientist of recorded history. His genius was unbounded by
time and technology, and was driven by his insatiable curiosity, and
his intuitive sense of the laws of nature. Da Vinci was dedicated to
discovery of truth and the mysteries of nature, and his insightful
contributions to science and technology were legendary. As the
archetypical Renaissance man, Leonardo helped set an ignorant and
superstitous world on a course of reason, science, learning, and
tolerance. He was an internationally renowned inventor, scientists,
engineer, architect, painter, sculptor, musician, mathematician,
anatomist, astronomer, geologists, biologist, and philosopher in his
time. He researched the natural code ( Da Vinci code ). He connected
it with his knowledge about the Golden Section code and Platonic
solids.


Leonardo was a great lover of geometry, and devoted
much time to it starting in his early forties. In 1496 the
Franciscan monk, theorist, mathematician, and writer Fra Luca
Pacioli (ca. 1445ca. 1514) arrives in Milan, where he stays
until 1499 and collaborates with Leonardo on studies of
proportion, geometry, and mathematics. Leonardo most
outstanding polyhedral accomplishment is the illustrations for
Luca Pacioli's book The Divine Proportion. 1498, February
9, Fra Luca Pacioli dedicates the treatise De divina
proportione to Ludovico "Il Moro" (published in Venice,
1509.). (He coined the term "the divine proportion" for what
is otherwise known as "the golden ratio".) The book was very
influential in circulating information about geometry, and
polyhedra in particular.

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.
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 this version of the dodecahedron:
^{}
