Introduction
Congruence
Truncations
Volumes
Snub

©2006 LaRouche Youth L.L.P.
Untitled Document

Introduction

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 word "harmonia".

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.



Congruence