Introduction Congruence Truncations Volumes Snub

©2006 LaRouche Youth L.L.P.
 Constructibility and Congruence In Books I and II, Kepler draws attention to certain problems of constructive geometry, i.e. the knowability of regular polygons and the relationship of these figures to each other, which, physically, can express harmonic proportions. The question is raised as to why certain polygons are constructible, others to more remote degrees than others, but also some, like that of the heptagon, are unknowable and not constructible. Does this perhaps mean that the heptagon is not harmonic since it is not constructible? For how do you draw relations to something which is not knowable and thus has no geometric relationship to anything that is knowable? This is a very crucial question which comes up when Kepler draws the distinction between constructibility and congruence. Constructibility refers to the regular polygons and their differing degrees of knowability, which can be carried infinitely by the doubling of their sides, etc. Congruence refers to the relationships of these figures in the plane and in space. When investigating the property of congruence in the plane, you'll notice that there is a bounded number of constructible figures that will "fill the space" with no gaps. Some constructible figures can fill the space by themselves, some constructible figures need other figures to be congruent, and then there are some constructible figures which not only cannot fill the space by themselves, but cannot even form congruence with other figures. Among these three examples, there can be a perfect congruence or an imperfect congruence. They are perfect if the pattern can be continued outwards, and imperfect if it cannot. Below are a few examples of these congruences (see figure). When investigating the property of congruence in space, a property which shows itself in bodily form, one's mind will tend to take what is knowable from the constructible figures and use them to fashion solid bodies. For example, when one has discovered how to construct an equilateral triangle, eventually, the question that arises is how many of these triangles can come together to form a solid body. You will probably be able to make several different bodies with just a triangle, but there are only three that can be constructed such that every vertex, or corner, of the solid has the same number of triangles at that meeting point. Kepler would call this congruence in space to be most perfect, since the triangles come together leaving no gap, and are all of the same shape-- a shape which happens to be equilateral and equiangular. However, merely constructing the solid out of paper, wood, or whatever material possible, does not completely make the structure knowable, as will be seen later in the case of the snub cube. If you carry this experiment further with a square and a pentagon, you will find that there are two more of these congruences, one body for the square and one for the pentagon (see figure). These five regular figures have many remarkable properties; to begin with, a few things should be mentioned. All of their vertices lie on a sphere. They all have equal edges, equal faces, and equal angles. I challenge the reader to try to construct a sixth figure, fitting the properties described above. It is also important to note the dual relationship they have to each other. For example, by taking the midpoints of each face of the cube, you will obtain the octahedron inside of it, and vice versa, between the two solids. This is also true for the icosahedron and dodecahedron. But the tetrahedron duals itself (see fig.) All this Kepler describes, most significantly, in his Mysterium Cosmographicum, where he constructed a model of the solar system in which the distances of the planetary orbits were physically expressed by the nesting of these five regular solids. Kepler adds, among one of his original discoveries, that "semiregular" figures like the rhombus, which has equal edges but two different angles, can also create a most perfect solid congruence. The two most significant congruences are shown here (see figure). However, unlike the regular solids, their vertices do not lie on the same sphere, but in fact, lie on two different spheres and also, there is a specific rhombus for each of the two solid bodies. Also, each rhombic solid has two different types of vertices. The one rhombic solid has three rhombi at a vertex and four rhombi at another and the other rhombic solid has three rhombi at a vertex and five rhombi at another. Kepler, in his "On the Six Cornered Snowflake" (1611), mentions how the rhombus provoked a problem of geometry such as constructing solid bodies, similar to the five regular solids, made purely out of rhombi. In demonstrating how playful Kepler is with pursuing principles of the physical universe, he is exploring this in the context of a written gift for his patron, John Matthew Wacker, in which Kepler hypothesizes about why the snowflake always forms in six-fold symmetry. The idea of the rhombi also comes up when Kepler asks the question of why bees, a living process, form the cells of their comb in six-fold symmetry. He notices that at the base, or keel, of the cell is a configuration of three rhombi that looks similar to the three rhombi of one of the vertices of the twelve faced rhombic solid. Another living process, the pomegranate, also has 'loculi' inside the fruit's structure which, when squeezed together as the fruit grows, form the twelve-faced rhombic shape. As Kepler points out, the solid made of twelve rhombi can fill space with no gaps by stacking them, just like the cube. This can be taken as a special type of solid congruence. Kepler makes a connection between the material necessity of these forms' existence and the "soul" of each thing. For example, in the pomegranate, it is not what the pomegranate is made of or any chemical agent that causes the rhombic shapes in the loculus, but there is a soul or life-principle governing the plant which is assisted by material necessity. The pomegranate is a part of a principle of life, a principle which is not seen directly through the senses, but whose effects you can see in the growth and development of the fruit, effects such as the loculi of the fruit tending toward the shape of the rhombic congruence. However, what kind of principle is governing the formation of a snowflake? After all, each snowflake is unique, just like trees and plants never grow exactly like each other. But a snowflake is created in a much different process. There aren't two snowflakes that mate or seed each other to propagate another. Yet snowflakes and plants take the form they do for a reason, which is all happening as a part of a universal principle, whether a principle of living processes or non-living processes. Lyndon LaRouche expounds upon this idea of higher orderings of principles in his "Vernadsky and Dirichlet's Principle." Another significant form of congruence is what Kepler calls solid congruences which are perfect in an inferior degree. There are thirteen in number, which are called the Archimedean solids (see figure). Keep in mind that these congruences can be formed, as Kepler explains it, by exhausting the different types of constructible polygons that can come together to form solid angles, and also form solid bodies whose vertices all lie on the same sphere. Now, a distinction to be made here is the difference between the congruence of a prism and the congruence of an Archimedean. These examples of prisms, which fulfill the properties of plane figures forming a solid angle and whose vertices all lie on a sphere, would be considered by Kepler to form an imperfect congruence because the larger plane figure doesn't occur more than twice. Pyramids, except for the tetrahedron, would also be considered imperfect congruences because the larger figure, or the base, do not occur more than once (see figure). If you notice, among the thirteen Archimedeans; six of them share properties of the cube and octahedron; six of them share properties of the icosahedron and dodecahedron; and one of them shares the properties of the tetrahedron. Then the next question: If the Archimedeans share the properties of the five regular solids, then what exactly is the relationship between them? On to the Truncations!