Kepler divides
The Harmony of the World into five long chapters: the first is on regular polygons; the second is on the congruence of figures; the third is on the origin of harmonic proportions in music; the fourth is on
harmonic configurations in astrology; the fifth is on the harmony of the motions of the planets. , the stellated dodecahedra (
small and
great) and the
Platonic solids assigned to
elements.
Chapter 1 and 2 Chapters 1 and 2 of
The Harmony of the World contain most of Kepler's contributions concerning
polyhedra. He is primarily interested with how polygons, which he defines as either regular or semiregular, can come to be fixed together around a central point on a plane to form congruence. His primary objective was to be able to rank polygons based on a measure of sociability, or rather, their ability to form partial congruence when combined with other polyhedra. He returns to this concept later in
Harmonice Mundi with relation to astronomical explanations. In the second chapter is the earliest mathematical understanding of two types of
regular star polyhedra, the
small and
great stellated dodecahedron; they would later be called Kepler's solids or Kepler Polyhedra and, together with two regular polyhedra discovered by
Louis Poinsot, as the
Kepler–Poinsot polyhedra. He describes polyhedra in terms of their faces, which is similar to the model used in
Plato's
Timaeus to describe the formation of
Platonic solids in terms of basic triangles. While medieval philosophers spoke metaphorically of the "music of the spheres", Kepler discovered physical harmonies in planetary motion. He found that the difference between the maximum and minimum angular speeds of a
planet in its orbit approximates a harmonic proportion. For instance, the maximum angular speed of the Earth as measured from the Sun varies by a
semitone (a ratio of 16:15), from
mi to
fa, between
aphelion and
perihelion.
Venus only varies by a tiny 25:24 interval (called a
diesis in musical terms). At very rare intervals all of the planets would sing together in "perfect concord": Kepler proposed that this may have happened only once in history, perhaps at the time of creation. Kepler reminds us that harmonic order is only mimicked by man, but has origin in the alignment of the heavenly bodies: Kepler discovers that all but one of the ratios of the maximum and minimum speeds of planets on neighboring
orbits approximate musical harmonies within a margin of error of less than a diesis (a 25:24 interval). The orbits of Mars and Jupiter produce the one exception to this rule, creating the inharmonic ratio of 18:19.
Chapter 5 Chapter 5 includes a long digression on astrology. This is immediately followed by Kepler's
third law of planetary motion, which shows a constant proportionality between the cube of the semi-major axis of a planet's orbit and the square of the time of its orbital period. Kepler's previous book,
Astronomia nova, related the discovery of the first two principles now known as Kepler's laws. ==Recent history==