The Harmony of the World
In Greek, the word harmonia means "congruent," or "fitting together." The Greek words for "harmony" (harmonia) and "number" (arithmos) both derive from the Indo-European root (a)ri, recognizable in such English words as rite and rhythm. In Vedic India, rta meant unity or cosmic order. Kepler considered the Harmonices Mundi (The Harmony of the World, 1619) his greatest work. The text relates his findings about the concept of congruence with respect to diverse categories of the physical domain: regularities in three-dimensional geometry, the relationships among different species of magnitude, the principles of consonance in music, and the organization of the Solar System. Here we seek to highlight issues that provide a flavor of his thought.

1.) Figures
Kepler entitled the first book of The Harmony of the World:
"On the regular figures, the harmonic proportions they create, their source, their classes, their order, and their distinction into knowability and representability." (The Harmony of the World, Book I)
Plato is credited with discovering that only five three-dimensional, convex solids can be formed using regular convex polygons (the so-called Platonic Solids). When a sphere is circumscribed around each shape touching all its corners, the vertices mark off spherical polygons that define the only possible equal divisions of the sphere's surface area.

Earlier, in the 1597 Mysterium Cosmographicum Kepler proposed that a nested arrangement of the Platonic Solids determines the spacing between the planetary orbits.

He expanded his investigation of three dimensional geometric shapes, notably of the semi-regular Archimedean Solids, in The Harmony of the World.