|[ To Contents ]|
A Final Philosophical Note
Fuller pointed out that sphere-packing models encourage us to conceive of area and volume in terms of discrete quanta instead of as the physically impossible continuums promoted by traditional geometry:
Because there are no experimentally-known "continuums," we cannot concede validity to the concept of continuous "surfaces" or of continuous "solids." The dimensional characteristics we used to refer to as "areas" and "volumes," which are always the second- and third-power values of linear increments, we can now identify experimentally, arithmetically, and geometrically only as quantum units that aggregate as points, both in system-embracing areal aggregates and... as volume-occupant aggregates. The areal and volumetric quanta of separately islanded "points" are always accountable numerically as the second and third powers of the frequency of modular subdivision of the system's radial or circumferential vectors. (515.011)
He argued that sphere polyhedra, having the advantage of visibly separate subunits, illustrate the otherwise invisible truth about physical reality. An awareness of the particles inherent in all physical systems (on some level of resolution) is nurtured, because in the sphere packings it is so logical to express volume and area in terms of number of units. The examples of the VE and icosahedron models demonstrate how the terms for expressing length (frequency) and area (number of particles) are actually related by a formula (10f ² + 2)as would be necessary in a new geometry.
Fuller attributed the precedent for thinking about volume in terms of quanta to Amadeo Avogadro (1776-1856) and his discovery that equal volumes of all gases, under the same conditions of pressure and temperature, contain the same number of molecules. Avogadro thereby identified volume with number of molecules a long time ago. We take this a step further, by remembering that, although we tend to conceive of volume as a spatial continuum, our convention for quantifying an amount of space uses number of imaginary cubeseven if that quantity usually involves an extraneous partial cube (or fraction) tacked on to a whole number. We shall discuss the subject of volume more fully in Chapter 10. For now, we simply lay the groundwork with the evidence that polyhedra can be constructed out of a multitude of spheres, at different frequencies, and that the resulting models play an important role in satisfying Fuller's criteria for a geometry consistent with Universe.
|[ To Contents ]|