Image 2
The Platonic Solids as a Model for the Solar System
from: Mysterium Cosmographicum by Johannes Kepler (1571-1630)
Format 5 : 9 (Minor Seventh)
Johannes
Kepler was convinced from his youth onwards that the harmonic order of
the universe could be proved empirically. His investigation of this was
groundbreaking for the modern approach to natural science: according to
his Laws of Celestial Mechanics, still applied today, a planet's
distance from the sun can be calculated from its observed orbital
period-but this does not explain why it has exactly this distance for
this orbital period. This was precisely the question that interested
Kepler, and this type of observation is typical for the harmonic way of
thinking: it is a matter of the identification of morphological
correspondences, which comprise a “physiognomic” statement for the
observer. His first approach to this subject was geometric:
When one attempts to assemble regular three-dimensional solids from
equilateral polygons, it becomes apparent that only five such solids
are possible. From equilateral triangles, three solids can be built:
the tetrahedron from four, the octahedron from eight, and the
icosahedron from twenty triangles. From squares, only the cube can be
built, and finally the pentagon-dodecahedron can be built from twelve
pentagons. These five solids were first described by Plato, and are
therefore called the “Platonic solids”-they can be seen in Image 3. Now
each solid can be placed inside a sphere, so that all the corners of
the solid touch the surface of the sphere on the inside; and another
sphere can be placed inside each solid, so that the sphere's surface is
tangent to the center of each surface of the solid from the inside.
Kepler then discovered that when these solids are placed inside each
other so that the outer sphere around one solid is also the inner
sphere inside the next, the radii of the spheres accurately correspond
to the distances between the planets. This is illustrated as a model in
Image 2, the lower figure being a magnification of the central part of
the upper figure.
Admittedly, the outer planets (the first of which, Uranus, was
discovered two hundred years after Kepler's time) have no place in this
model, since only five such solids are geometrically possible; and
Kepler himself was not completely satisfied with the precision of the
ratios. But this discovery encouraged him to look further wherever he
could for clues to the harmonic structure of the universe.
Peter Neubäcker

Image 3
The Platonic Solids and their Relationships
from: Harmonices Mundi by Johannes Kepler (1571-1630)
Format 8 : 9 (Whole-Tone)
In 1619, Johannes Kepler published Harmonices Mundi - the Five Books of Universal Harmonics,
which he considered his own most significant work, since it described
the musical harmony of the solar system which he had spent his life
seeking and had finally found.
In the first two books of this work, he derives qualities of numbers
from geometric investigations of the ways in which the regular solids
can be arranged. Image 3 comes from the second book; here he puts the
regular figures together into three-dimensional bodies. The Platonic
solids appear again, here connected to the elements, fire, water, air,
and earth, and to the cosmos. A few figures derived from these solids
also appear, such as the star-solids first described by Kepler.
In the third book, he transfers the number qualities found in the first
two books into the audible domain, using the monochord, and develops an
all-embracing musical theory. In the fourth book he applies the
insights gained in geometry and music to astrological observations.
In the fifth book, the astronomical part, he presents his great
harmonic discovery: the realization of musical harmonies through the
movements of the planets. After first musically examining the distances
and orbital periods of the planets from various viewpoints, and
obtaining no satisfying results, he then examines the angular
velocities of the planets, as an observer on the sun would perceive
them, and establishes that these velocities, at the perihelion and
aphelion of each planet, when positioned on the monochord, yield the
most beautiful musical harmonies-for each individual planet as well as
for the various planets compared with one another. This result also
applies to the outer planets discovered later, as calculated in modern
times; it appears to be a universally valid principle of the structure
of our solar system.
Astonished at his discovery, Kepler writes: “...Whether people read
this book now or later does not matter. It can wait a hundred years for
the readers, if God Himself waited six thousand years for His work to
be seen...”
