03/30/2020
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On the Subject of Metaphor
by Lyndon H. LaRouche, Jr.
Part I of II
Reprinted from FIDELIO Magazine, Vol . 1 No.3 , Fall 1992
Most figures and graphics are not included online....
That outline of our proposition now given, we examine the set of relationships among names, thought-objects, and our universe. Let us speak of three domains. First, the domain of thought-objects, within the sovereign bounds of the individual's mental-creative life. Second, the relevant plane of sense and communication media. Third, within the physical universe, behind the superficiality of sense-experience, an underlying governing agency of principle, which controls the lawful behavior of the universe, and which will "recognize" certain of our changes in forms of actions with a favorable response.
Reference Figure 12. We have person A, a secondary-school teacher, and also an experimenter. We have person B, a student, and an observer of the experiment being performed. There is the experimental subject, X. A acts upon X. Student B observes X, and also observes A's actions upon X throughout the experiment. A communicates, reciprocally, with B, a communication which precedes and accompanies the experiment, and which continues after the experiment's completion.
A, beginning from a thought-object in his own mind, provokes the replication of that thought-object within the mind of student B. This occurs through the method of Socratic negation, as applicable to a case which meets the requirement to be a true paradox. Consider an example, related to the Cusa isoperimetric paradox, which illustrates this phase of the transactions among A, B, and X, in this illustration; consider the proof of the uniqueness of the five Platonic solids.30
Take three great circles which can be moved about on the surface of a sphere and arranged at any inclination one to another, as if they were hoops having the same radius as the sphere. Experimenting with such hoops, it will be discovered that when they are arranged such that their respective circumferences mutually divide one another into four equal arcs, the surface of the sphere is partitioned into eight equal, regular spherical triangles. The six points of pairwise intersection of the hoops will be found to form the vertices of an octahedron (see Figure 13).
Do the same for four and six hoops. For four hoops, the pairwise intersection occurs at twelve points, coinciding with the twelve vertices of a cuboctahedron (the truncation through midpoints of edges of the cube or octahedron). The surface of the sphere is thus partitioned into eight equal and regular spherical triangles and six equal and regular spherical quadrilaterals. Each great circle is divided by the others into six equal arcs.
Using six hoops, thirty points of pairwise intersection result, forming the vertices of an icosidodecahedron (the truncation through midpoints of edges of the icosahedron or dodecahedron). The surface of the sphere is partitioned into twelve equal and regular spherical pentagons and twenty equal and regular spherical triangles. Each great circle is divided by the others into ten equal arcs.
It can then be proven that there are no other partitions of the sphere resulting in the division of the great circles into equal arcs. From the limiting case of six hoops, which permits the construction of twelve pentagonal faces, is demonstrated the primacy of the dodecahedron, and relative uniqueness of the five Platonic solids.31 From the six-hooped figure containing dodecahedron and icosahedron, the cube, octahedron, and tetrahedron may be readily derived.
The Golden Section may then be conveniently demonstrated as the ratio of radius to chord on the dodecagon formed by inscription in each of the six great circles, or, alternatively, as one of the many well-known internal relationships of the pentagon, formed by projection of the spherical pentagon onto a plane. In either case, the derivation of this ratio from the construction upon the sphere is to be stressed, rather than derivation from a pentagon or pentagonal division of the circle, presumed as given or constructed by algebraic artifice.
This approach has shown several points which are of crucial importance:
1. The necessity of deriving these regular polyhedra from regular spherical triangles, quadrilaterals, and pentagons is shown. This correlates with our earlier study of the paradoxical effort to square the circle. The construction of the polyhedra is bounded externally by spherical action.
2. That, only regular division of the sphere's surface by the factors 3, 4, and 5 succeeds. Thus, the dodecahedron corresponds to the upper limit of construction, since it is derived from fivefold division. No regular polyhedron of hexagonal sides, or larger, is constructible.
3. That all five regular solids are derived from the construction of the pentagonal-sided dodecahedron.
A strong indication of this is the following view of harmonic orderings cohering with the Golden Section.
The customary classroom and related practice, is to explain the construction of the Golden Section as necessary for the construction of the regular pentagon. This seemingly innocent practice has contributed to the circulation of much nonsense, nonsense which is avoided if the Golden Section is situated directly within a proper reading of the simple construction-proof of the uniqueness of the five Platonic solids. Turn, for illustration of the point, to reference again Pacioli's De Divina Proportione.
Pacioli, Leonardo da Vinci, et al., showed that, on the scale of direct sensory observation of ordinary processes, all living processes have an harmonic ordering of growth and morphology of function which coheres, as a Type, with the Golden Section; whereas, all non-living processes, on this scale, have a different Type of characteristic harmonic ordering. This point is later re-stated by Johannes Kepler in various locations, including his Snowflake paper. Modern evidence leaves no doubt of the correctness of that so-qualified observation of Pacioli, Leonardo, Kepler, et al.
Unfortunately, too frequently, those who point to this distinctive Platonic coherence of living processes with the Golden Section, either degrade this connection to a kind of cabalistic speculation, or simply present the Golden Section itself as a section in a circle, without showing necessity, in such popularized terms as to leave the matter of harmonic ordering vulnerable to a false charge of numerological mystification. This latter negligence appears whenever we might misdefine the Golden Section in terms of either, simply, "the Golden Mean," or as simply the derivation of the pentagon, by construction from a given circle.
If the following, restated, preconditions of rigorous treatment are satisfied, in defining the Golden Section, the risk of misleading mystification is avoided.
First, the Golden Section is located as a necessary, (intrinsic) metrical characteristic of negative spherical curvature, as nothing other than the characteristic distinction of the spherical generation of a subsumed, constructed dodecahedron.
Second, the five Platonic solids are recognized as each and all subsumed by the construction of a single one among them, the dodecahedron.
Third, this topic, of spherical determination of the Platonic solids' uniqueness, is referenced from the standpoint of the method we indicated above, for recognizing and solving the deep paradox inhering in Archimedean squaring of the circle. In short, that the spherical action, of a different, higher species than any polyhedron, bounds externally, and thus determines the constructible existence and metrical characteristics of the species of polyhedra in general.
These points are underscored by comparing the paradoxical process of squaring the circle to the way in which harmonic orderings coherent with the Golden Section bound externally the linear Fibonacci series (see Figure 14)32 This may then be compared with Johannes Kepler's distinction between packings contrary to, respectively, negative and positive spherical curvatures (see Figure 15). In short, the Golden Section is a determined, necessary limit of packing of the type illustrated by the Fibonacci "growth" series under the constraint of negative curvature. With that observation, the premises for mystification evaporate.
That material covered by teacher A, the teacher brings the student's attention to the work of Huygens and his successors on the subjects of tautochrone and brachistochrone.33 This leads the student through (a) the elements of the cycloid, (b) the proof, by Huygens, that the cycloid is a tautochrone, and (c) the proof, by Johann Bernoulli et al.,that the tautochrone is also the brachistochrone (see Figure 16).
The teacher, A, then reviews the work which was referenced by Johann Bernoulli, Huygens' Treatise on Light,34 as the next unit of study in B's classroom. In this setting, A includes relevant references to the subject of light and hydrodynamics in the Leonardo da Vinci Codices, in the work of Fermat, and the treatments of a universal principle of least action by Fermat, Leibniz, and the Bernoullis. The geometrical construction employed as proofs, together with the Bernoulli experiment itself, are, combined into one, the experiment X; the Bernoulli experiment itself, is the relevant physical experiment.
This experiment shows implicitly that the universe portrayed by René Descartes and Isaac Newton does not exist. First, the tautochrone/brachistochrone equivalence, for the case of a constant relative speed of light, shows that the notion of physical function in our universe requires that family of non-linear, non-algebraic functions which is derived from the isoperimetric principle. This notion of non-algebraic function supersedes all those notions of arithmetic-algebraic function derived from a notion of pairwise, linear causal interrelationship as primary. Thus the refutation of Descartes and Newton. Whereas, the non-algebraic and algebraic conceptions conflict respecting a notion of causal principle, the algebraic view is shown to be axiomatically false.
This signifies that the Cartesian domain is axiomatically false in conception from the outset. Isaac Newton's case is ultimately the same, but historically of greater ironical interest.
Newton refers to what he admits to be an absurdity of his mathematical-physics scheme, that it represents the universe as "running down," in the sense of a mechanical time-piece.35 This "clock-winder" topic is a featured element within the Leibniz-Clark-Newton correspondence later.36 Later, during the 1850's, Rudolf Clausius, at the prompting of Lord Kelvin, employed the assistance of the mathematician Herman Grassman to codify the so-called "universal entropy" dogma,37 or "Second Law of Thermodynamics," which is nothing but a nineteenth-century version of Newton's seventeenth-century "clock-winder" fallacy. The key reference-point for discussion here, is that the seventeenth-century Newton, unlike the nineteenth-century Clausius, Kelvin, Helmholz, Rayleigh, and Boltzmann, states clearly that the fallacy of "universal entropy" erupts within his physics as a consequence of a defect embedded within his choice of mathematics.
This represents an important challenge for teacher A. B asks, "Does entropy exist?" "Yes," replies A, "but not as a governing principle of the physical universe." B is perplexed by this. A explains, by reference to Kepler, "Remember our studies of Kepler's work?"
"Remember our review of this matter in our study of Kepler's Snowflake paper?" Positive curvature is associated with non-living functions, such as the snowflake, which do exhibit entropy as an included characteristic. However, negative curvature requires a non-entropic ordering cohering with the limiting implications of the Golden Section.
The point here is, that, in a universe super-densely packed with spherical bubbles,38 the envelope of all positive curvatures is a negative curvature. Thus, although some phase-states of our universe are entropic, other phase-states are not. Up to recent decades, we have known that the astrophysical realm, like living processes, is negentropic; we have found, as, for example, so-called "cold fusion" illustrates this, that the extremes of microspace are also characteristically negentropic.
