Intuitive Geometry
Fuller took an intuitive approach to his studies, often going into exhaustive empirical detail while at the same time seeking to cast his findings in their most general philosophical context.
For example, his sphere packing studies led him to generalize a formula for polyhedral numbers: 2 P F2 + 2, where F stands for "frequency" (the number of intervals between balls along an edge) and P for a product of low order primes (some integer). He then related the "multiplicative 2" and "additive 2" in this formula to the convex versus concave aspects of shapes, and to their polar spinnability respectively.
These same polyhedra, developed through sphere packing and related by tetrahedral mensuration, he then spun around their various poles to form great circle networks and corresponding triangular tiles on the surface of a sphere. He exhaustively cataloged the central and surface angles of these spherical triangles and their related chord factors.
Fuller was continually on the lookout for ways to connect the dots, often purely speculatively. As an example of "dot connecting" he sought to relate the 120 basic disequilibrium LCD triangles of the spherical icosahedron to the plane net of his A module.(915.11Fig. 913.01, Table 905.65)
The Jitterbug Transformation provided an unifying dynamic in this work, with much significance attached to the doubling and quadrupling of edges that occurred, when a cuboctahedron is collapsed through icosahedral, octahedral and tetrahedral stages, then inside-outed and re-expanded in a complementary fashion. The JT formed a bridge between 3,4-fold rotationally symmetric shapes, and the 5-fold family, such as a rhombic triacontahedron, which latter he analyzed in terms of the T module, another tetrahedral wedge with the same volume as his A and B modules.
He modeled energy transfer between systems by means of the double-edged octahedron and its ability to turn into a spiral (tetrahelix). Energy lost to one system always reappeared somewhere else in his Universe. He modeled a threshold between associative and disassociative energy patterns with his T-to-E module transformation ("E" for "Einstein").(Fig 986.411A)
Synergetics is in some ways a library of potential "science cartoons" (scenarios) described in prose and not heavily dependent upon mathematical notations. His demystification of a gyroscope's behavior in terms of a hammer thrower, pea shooter, and garden hose, is a good example of his commitment to using accessible metaphors. (Fig. 826.02A)
His modular dissection of a space-filling tetrahedron or MITE (minimum tetrahedron) into 2 A and 1 B module served as a basis for more speculations about energy, the former being more energy conservative, the latter more dissipative in his analysis.(986.422921.20, 921.30). His focus was reminiscent of later cellular automaton studies in that tessellating modules would affect their neighbors over successive time intervals.
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Famous quotes containing the words intuitive and/or geometry:
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