986.040 Greek Geometry

986.041 It was a very different matter, however, when in supposed scientific integrity mathematicians undertook to abstract the geometry of structural phenomena. They began their geometrical science by employing only three independent systems: one supposedly “straight”-edged ruler, one scribing tool, and one pair of adjustable-angle dividers.

986.042 Realistically unaware that they were on a spherical planet, the Greek geometers were first preoccupied with only plane geometry. These Greek plane geometers failed to recognize and identify the equally important individual integrity of the system upon whose invisibly structured surface they were scribing. The Euclidean mathematicians had a geocentric fixation and were oblivious to any concept of our planet as an includable item in their tool inventory. They were also either ignorant ofor deliberately overlooked the systematically associative minimal complex of inter-self-stabilizing forces (vectors) operative in structuring any system (let alone our planet) and of the corresponding cosmic forces (vectors) acting locally upon a structural system. These forces must be locally coped with to insure the local system’s structural integrity, which experientially demonstrable force-interaction requirements are accomplishable only by scientific intertriangulations of the force vectors. Their assumption that a square or a cube could hold its own structural shape proves their oblivousness to the force (vector) interpatternings of all structurally stable systems in Universe. To them, structures were made only of stone wallsand stone held its own shape.

986.043 The Ionian Greeks seem to have been self-deceived into accepting as an absolute continuum the surface of what also seemed to them to be absolutely solid items of their experiencewhether as randomly fractured, eroded, or ground-apart solids or as humanly carved or molded symmetrical shapes. The Ionian Greeks did not challenge the self-evident axiomatic solid integrity of their superficial-continuum, surface-face-area assumptions by such thoughts as those of the somewhat later, brilliantly intuitive, scientific speculation of Democritus, which held that matter might consist of a vast number of invisible minimum somethingsto which he gave the name “atoms.” All of the Euclidean geometry was based upon axioms rather than upon experimentally redemonstrable principles of physical behavior.

986.044 Webster’s dictionary defines axiom (etymologically from the Greek “to think worthy”) as (1) a maxim widely accepted on its intrinsic merit, and (2) a proposition regarded as self-evident truth. The dictionary defines maxim as (1) a general truth, fundamental principle, or rule of conduct, and (2) a saying of a proverbial nature. Maxim and maximum possibly integratingly evolved as “the most important axiom.” Max + axiom = maxim. The assumption of commonly honored, customarily accredited axioms as the fundamental “building-blocks” of Greek geometry circumvented the ever-experimentally-redemonstrable qualifying requirement of all serious scientific considerations.

986.045 The Ionian Greeks assumed as fundamental geometric components their line-surrounded areas. These areas’ surfaces could be rough, smooth, or polished just as the smooth surface of the water of the sea could be roughened without losing its identity to them as “the surface.” Looking upon plane geometry as the progenitor of subsequently-to-be-developed solid geometry, it seemed never to have occurred to the Euclideans that the surface on which they scribed had shape integrity only as a consequence of its being a component of a complex polyhedral system, the system itself consisting of myriads of subvisible structural systems, whose a priori structural integrity complex held constant the shape of the geometrical figures they scribed uponthe polyhedral system, for instance, the system planet Earth upon whose ground they scratched their figures, or the stone block, or the piece of bark on which they drew. Even Democritus’s brilliant speculative thought of a minimum thing smaller than our subdimensional but point-to-able speck was speculative exploration a priori to any experimentally induced thinking of complex dynamic interactions of a plurality of forces that constituted structuring in its most primitive sense. Democritus did not think of the atom as a kinetic complex of structural shaping interactions of energy events operating at ultra-high-frequency in pure principle.

986.046 Cubical forms of wood and stone with approximately flat faces and corner angles seemed to the Euclidean-led Ionians to correspond satisfactorily with what was apparently a flat plane world to which trees and humanly erected solid wooden posts and stone columns were obviously perpendicularergo, logically parallel to one another. From these only-axiomatically-based conclusions the Ionians developed their arbitrarily shaped, nonstructural, geometrical abstractions and their therefrom-assumed generalizations.

986.047 The Greeks’ generalized geometry commenced with the planar relationships and developed therefrom a “solid” geometry by in effect standing their planes on edge on each of the four sides of a square base and capping this vertical assembly with a square plane. This structure was then subdivided by three interperpendicularly coordinate linesX, Y, \text{and}Zeach with its corresponding sets of modularly interspaced and interparalleled planes. Each of these three sets of interparallel and interperpendicular planes was further subdivisible into modularly interspaced and interparallel lines. Their sets of interparallel and interperpendicular planar and linear modulations also inherently produced areal squares and volumetric cubes as the fundamental, seemingly simplest possible area-and-volume standards of uniform mensuration whose dimensioning increments were based exclusively on the uniform linear module of the coordinate systemwhose comprehensive interrelationship values remained constantergo, were seemingly generalizable mathematically quite independently of any special case experiential selection of special case lengths to be identified with the linear modules.

986.048 The Euclidean Greeks assumed not only that the millions of points and instant planes existed independently of one another, but that the complex was always the product of endlessly multipliable simplexesto be furnished by an infinite resource of additional components. The persistence of the Greeks’ original misconceptioning of geometry has also so distorted the conditioning of the human brain-reflexing as to render it a complete 20th-century surprise that we have a finite Universe: a finite but nonunitarily-and-non-simultaneously accomplished, eternally regenerative Scenario Universe. In respect to such a scenario Universe multiplication is always accomplished only by progressively complex, but always rational, subdivisioning of the initially simplest structural system of Universe: the sizeless, timeless, generalized tetrahedron. Universe, being finite, with energy being neither created nor lost but only being nonsimultaneously intertransformed, cannot itself be multiplied. Multiplication is cosmically accommodated only by further subdivisioning.

986.049 If the Greeks had tried to do so, they would soon have discovered that they could not join tetrahedra face-to-face to fill allspace; whereas they could join cubes face-to-face to fill allspace. Like all humans they were innately intent upon finding the “Building-Block” of Universe. The cube seemed to the Greeks, the Mesopotamians, and the Egyptians to be just what they needed to account their experiences volumetrically. But if they had tried to do so, they would have found that unit-dimensioned tetrahedra could be joined corner-to-corner only within the most compact omnidirectional confine permitted by the corner-to-corner rule, which would have disclosed the constant interspace form of the octahedron, which complements the tetrahedron to fill allspace; had they done so, the Ionians would have anticipated the physicists’ 1922 discovery of “fundamental complementarity” as well as the 1956 Nobel-winning physics discovery that the complementarity does not consist of the mirror image of that which it complements. But the Greeks did not do so, and they tied up humanity’s accounting with the cube which now, two thousand years later, has humanity in a lethal bind of 99 percent scientific illiteracy.