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Framework of Possibility
The isotropic vector matrix gives us a description of the symmetry of space. We can think of this matrix as a framework of possible directions and configurations of ordered space, or more simply, as a frame of reference. It is a network of vectors specifically situated to model nature's eternal tendency toward equilibrium. Lines are forces, length is magnitude, and all is in balance. The IVM weaves together a number of synergetics ideas: minimum system of Universe, vector equilibrium (both exhibiting four planes of symmetry), twelve degrees of freedom, complementarity of octahedra and tetrahedra, space-filling, and stability (exclusively a product of triangulation). In so doing, it sets the stage for an energetic mathematics, and systematizes further investigation.
The IVM also provides an alternative to the XYZ system's absolute origin. Every vertex in the IVM can be considered a temporary local origin, which, as reinforced by Fuller's use of the concept of "systems," is consistent with the requirements of describing Scenario Universe. ["All points in Universe are inherently centers of a local and unique isotropic-vector-matrix domain..." (537.11).] There can be no "absolute origin" in a scenario.
Finally, by describing such a wide variety of ordered polyhedraand thereby clarifying the relationships between different shapesthe IVM supports Fuller's concept of "intertransformability." Countless potential shapes and transformations can be systematically represented within this omnisymmetrical matrix; it is a framework of possibility.
Invention: Octet Truss
Our familiarity with the IVM enables us to visualize and appreciate Fuller's "Octet Truss." Awarded U.S. Patent 2,986,241 in 1961, this structural framework is so widespread in modern architecture that one might assume buildings have always been constructed that way. Again, as the story goes, the invention can be traced to 1899 when Bucky was given toothpicks and half-dried peas in kindergarten. So extremely farsighted and cross-eyed that he was effectively blind (until he received his first pair of eyeglasses a year later), Bucky Fuller did not share the visual experience of his classmates and therefore lacked the preformed assumption that structures were supposed to be cubical. Thus, as other children quickly constructed little cubes, young Bucky groped with the materials until he was satisfied that his structures were sturdy. The result, much to the surprise of his teachers (one of whom lived a long, long life, and periodically wrote to Fuller recalling the event) was a complex of alternating octahedra and tetrahedra. He had built his first Octet Trussalso the first example of what was to become a lifetime habit of approaching structural tasks in revolutionary ways.
The experience had a great impact on the four-year-old, as he recounted in a 1975 lecture:
All the other kids, the minute they were told to make structures, immediately tried to imitate houses. I couldn't see, so I felt. And a triangle felt great! I kept going 'til it felt right, groping my way...(3)
The truss's omnisymmetrical triangulation distributes applied forces so efficiently that the resulting strength of such an architectural framework is far greater than predicted by conventional formulae:
The unitary, systematic, nonredundant, octet-truss complex provides a total floor system with higher structural performance abilities than engineers could possibly ascribe to it through conventional structural analysis predicated only upon the behavior of its several parts. (650.11)
Struts can be all one length, thus simplifying construction, while the minimal volume-to-material ratio inherent in the geometry of the tetrahedron (4) maximizes resistance to external loads. The intrinsic stability of triangulation together with efficient dispersal makes this system the most advantageous possible use of materials in a spaceframe configuration:
It is axiomatic to conventional engineering that if parts are horizontal," they are beams; and the total floor ability by such conventional engineering could be no stronger than the single strongest beam in the plural group. Thus their prediction falls short of the true behavior of the octet truss by many magnitudes... (650.11)
The octet truss takes the conceptual matrix into physical realization, and thus embodies Fuller's design science concept of using geometric principles to human advantage.
We can now appreciate the difference between diamond and graphite. Both consisting of carbon atoms, the former is exquisitely hard and clear, the latter soft and grey, and their differences are due to geometry. Carbon atoms in the structure of diamond take advantage of the strength of tetrahedra; their organization can be thought of as a double octet truss, two intersecting matrices with the vertices of one overlapping the cells of the other. Stabilized by the high number of bonds between neighboring atoms, which also allow forces to be distributed in many directions at once, the configuration is supremely invulnerable. In contrast, carbon atoms in graphite are organized into planar layers of hexagons-triangulated and stable in themselves, but not rigidly connected to other layers. As a result, separate layers are able to shift slightly with respect to each other, which does not mean that graphite lacks all stability, but rather that it is relatively soft. This softness enables graphite to leave visible residue on the surface of paper, thus performing its useful function in pencils. A more illustrative although less widely recognized application is that these sliding layers make graphite a powerful lubricant. The comparison provides a spectacular example of synergy: rearrangement of identical constituents produces two vastly different systems.
Thus we see that nature also employs design science.
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