Those who have embarked on a problem of physical constructive geometry, whether
to derive the five regular solids from the sphere or to construct a cube whose volume is
double another cube, will eventually realize a certain profound relationship between the
physical objects of construction, their mind and the conceptions of these objects, and the
self-developing methods of construction as they present themselves in the material and
in the paradoxes which are to be confronted. Even though this relationship has concerned mankind for thousands of years, never
has it presented itself with so much epistemological importance than through the work of
Johannes Kepler. Kepler's method, which has shaken and still shakes the foundations
of modern science, could be best described, in the words which Beethoven used headlining his "Grosse Fugue": "as rigorous as it is free." Or in other words, as disciplined as it is playful. These few words are able to give a glimpse into the true scientific method of investigating the nature of the creation of our universe and, significantly, mankind's ever-changing relationship to the universe, so defined by our discoveries in the principles of art and science. True scientists commit to demonstrating their knowledge of the universe out of love for their fellow man which, in turn of the other side of the same coin, is expressing piety toward the Creator. Such was the example set by Kepler in his investigation into Harmony.
According to Kepler, like those early Pythagoreans before him, the question of
Harmony should tend to arise alongside the age-old question of the true motions of the heavenly bodies. For if planets are arranged so gracefully in the heavens about a common "fount of motion," then assuredly there must be a common cause of the relationships of
these bodies, which in concert are expressed in Harmonic proportions. The crux of this
investigation builds up to Book V, but first Kepler investigates the boundaries of what he
calls "Congruence" or the relationships of regular figures to themselves and one another.
For, as he mentions, the Latin word "congruentia" has the same meaning for the Greek
The second and third parts of this pedagogical, entitled "The Volumes Experiment" and "The Elusive Snub Cube," respectively, will stretch outside the boundaries of Kepler's writings, but not of his method. It is not intended to steer away from the content of Kepler's work, but to affirm the creative process already explicit within it. Also, the reader is strongly advised to actually construct as many of the diagrams
as possible. Assuredly, without experiencing the act of geometric construction, you will
not gain full understanding of key concepts, and thus not gain the ability to develop the
concepts even further. Your mind will benefit so much more for performing the labor,
inasmuch as your room will improve with interesting decoration.