Johannes Kepler's Polyhedra
Johannes Kepler (1571-1630), best known for his three laws of planetary
motion, was one of the most outstanding mathematicians of his day.
In addition to his astronomical accomplishments, he systematized and extended
all that was known about polyhedra in his time. While previous artist/geometers
discovered particular polyhedra, he took a more mathematical approach:
he defined classes of polyhedra, discovered the members of the class, and
proved that his set was complete. For example, Kepler discovered the infinite
class of antiprisms.
Kepler's logical approach to polyhedra does not mean that he was free
of the mysticism of the day. The following illustration, from his
1619 book, Harmonice Mundi, graphically shows the Platonic associations
of the regular solids with the classical elements: The tetrahedron
corresponds to fire, the octahedron to
air, the cube to earth, the icosahedron
to water, and the dodecahedron
to the cosmos or ether:
The lower-left portion of the above figure illustrates Kepler's two star
polyhedra, the small
stellated dodecahedron and the great
stellated dodecahedron. Although earlier illustrations of these
solids exist, Kepler was the first to recognize them as meeting the definition
of regular polyhedra, but with nonconvex pentagram
faces. (An earlier mosaic of the small stellated dodecahedron is attributed
to Uccello and an earlier drawing of the great
stellated dodecahedron appears in the work of Jamnitzer.)
Kepler saw them from a deeper perspective, and in recognition, we now refer
to these as Kepler solids.
The lower-right portion of the above figure illustrates the disassembly
of two rhombic solids which Kepler discovered: the rhombic
dodecahedron and the rhombic
triacontahedron. (An illustration
of an approximation to the triacontahedron is clear in one of Jamnitzer's
monuments drawn fifty years earlier, but Kepler certainly had a much deeper
understanding of the triacontahedron's structure and its relationships
to other polyhedra.) Below, left, Kepler shows how a dodecahedron
can be constructed by adding "roofs" to the six sides of a cube (the construction
which Euclid uses). Below, right, is how Kepler constructs a rhombic
dodecahedron analogously. These figures are from his book Epitome
of Copernican Astronomy.
In, Harmonice Mundi, Kepler also defined the class of convex
uniform polyhedra which we call the Archimedean
solids, unfamiliar with the fact that Archimedes had defined the class
already. All of these polyhedra except one (the snub
dodecahedron, number 13 at the bottom right below) had already appeared
in the work of various Renaissance artists. But those artists were
haphazard, and wrote as if there could be an infinite number of related
polyhedra which they were selecting from. Kepler's important contribution
was to define this class of polyhedra and systematically explore it, to
find all its members and prove his set was complete. In doing so, he realized
that the prisms and antiprisms belong to
the same class, since at each vertex the same combination of regular polygons
In my view, a most artistic contribution of Kepler is to be found in his
model of the solar system. Kepler proposed that the distance relationships
between the six planets known at that time could be understood in terms
of the five Platonic solids. His 1596 book, Mysterium Cosmographicum, proposed the model illustrated below,
in which one Platonic solid fits between each pair of planetary spheres.
(Note the use of Leonardo's style of open faces.)
The outer sphere is that of Saturn; inside it is the sphere of Jupiter.
Kepler made two prototypes in colored paper hoping to have it fabricated
in silver. His original plan was that it would also function as a punchbowl
dispensing assorted beverages.
The image at right is a closeup of the spheres of inner planets, Mercury,
Venus, Earth, and Mars. This is a beautiful astronomical model.
For example, it explains why there are only six planets: How could
there be a seventh planet, when Euclid proved that there are only five
Platonic solids! Of course, the model is completely false, the interplanetary
distances it predicts are not sufficiently accurate, and Kepler was scientist
enough to accept this eventually. But it an excellent example of how truth
and beauty are not always equivalent.