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12 ___________ "All-Space" Filling: New Types of Packing Crates Fuller in his characteristic drive for verbal accuracy updates geometry's conventional term "space filling" with his own more descriptive (and predictably longer) "all-space filling." Here's the puzzle: which of the polyhedra introduced so far can pack together in such a way that all available volume is occupied without any gaps? The concept is not new; ever since closepacking equiradius spheres, we have danced around the issue of space filling, and in the process made most of the discoveries that this chapter will expand upon. The IVM disclosed certain space fillers, while making it clear that other polyhedra did not share this ability. However, filling all space now becomes the focus of our investigation, calling for the systematic analysis that enables new insights and a more thorough understanding. Despite its obvious applicability, space filling is not emphasized in Fuller's work. An ability to "fill all space" is generally mentioned as further description of a given polyhedron rather than providing an investigative starting point for synergetics. Fuller's "operational" approach encourages more experimental exercises, such as packing spheres together, which then lead to space fillers after the fact. In view of our overall goal of researching the characteristics of space, what could be more logical than to ask what fits into it? What shapes are accommodated by space? The notion that space is not a passive vacuum gradually becomes second nature; experience has changed our awareness. Now we want to become ever more exact about these active properties. The existence of an extremely limited group of polyhedra that can pack together to fill all space is one of the more direct illustrations of the specificity of spatial characteristics. The puzzle is quite challenging. Without actually making a horde of tiny cardboard models of the polyhedra in question, these spatial configurations are extraordinarily difficult to visualize—with the sole exception of the obvious space filler, an array of cubes. Once again, to gain more experience with the concept, we revert to the plane, or to be more accurate, the page. In deference to Bucky's strict precision, we must acknowledge that the theoretical "plane" is a nondemonstrable concept, but we can certainly (and quite appropriately) discuss filling up a page. Plane Tessellations
The mathematical title may be somewhat intimidating, but plane tessellations are actually quite familiar. Derived from the Latin word for "tiling," tessellation, in mathematics, refers to planar patterns of polygons. Because of the ease of working with flat patterns, we begin our study of space filling with the analogous situation in the plane. Which regular polygons fit together edge to edge to fill a page? We can fill up a page with squares, as anyone who has ever seen graph paper knows, and the pattern created by equilateral triangles is almost as familiar. Another successful tessellation is found on many bathroom floors covered by tiny hexagonal tiles (Fig. 12-la). With these immediately apparent examples, we might begin to suspect that any regular polygon can fill a page. However, a little experimentation quickly reveals that we have already exhausted the possibilities.
Three regular pentagons (equilateral and equiangular) placed side by side leave a 36-degree angular gap—not nearly wide enough to accommodate a fourth pentagon. Five-sided tiles are thus disqualified. Three heptagons simply cannot fit around a single point, as we saw in Chapter 4; octagons meet a similar fate, as of course do any polygons with more sides. We are suddenly confronted with a very limited group of plane fillers: triangles, squares, and hexagons. Opening up the field to allow combinations of regular polygons while maintaining equivalent vertices expands the inventory only slightly. Eight "semiregular" tessellations join the three patterns above (Fig. 12-lb). Still an impressively small group. These eleven tilings can be categorized in terms of the different rotational symmetries exhibited by each, and within this group every category of repeating planar pattern is represented. It is fascinating to reflect on the implications of these results: for example, a wallpaper designer can only create what the limitations inherent in the plane will allow. These cumulative experiences—especially those as straightforward as plane tessellations—nurture our growing awareness of spatial constraints. And with each step, our knowledge of the elegant precision of this order expands synergetically. Filling Space
Cubes stack neatly together to fill space. What other polyhedra exhibit this property? Attempts to fit regular tetrahedra together are quickly frustrated; likewise for octahedra. However, working together, the two shapes can fill space indefinitely. None of this is new information: we discovered the complementarity of octahedra and tetrahedra while exploring isometric arrays of both spheres and vectors. Subsequently, multiplication by division uncovered the octahedron hiding inside every tetrahedron. The octahedron—tetrahedron marriage is clearly an eternal bond. Neither icosahedra nor pentagonal dodecahedra can fill all space. ["Icosahedra, though symmetrical in themselves, will not close-pack with one another or with any other symmetrical polyhedra" (910.01).] The cube thus stands alone among regular polyhedra. Complementarity But let's reevaluate our apparently simple array of cubes. As the obvious solution to filling all space with a single polyhedron, this packing seems to provide the most straightforward information about the shape of space. But look further. What if you could see the cubes' face diagonals? An implied tetrahedron awaits visibility. Now imagine filling in the necessary diagonals, so that the inscribed tetrahedron—surrounded by four eighth-octahedra—appears in each cube. At every junction of eight cubes, the octahedral parts come together and form one complete octahedron around each cubical corner (Fig. 12-2). As rectilinear boxes are unstable without diagonal bracing, a stabilized packing of cubes turns into an Octet Truss. Whether visible or not, the octet symmetry is implicit in the configuration.
The Greeks failed to get at the triangulated heart of their stack of cubes, philosophizes Bucky, for "like all humans they were innately intent upon finding the 'Building Block' of Universe." Had they experimented with arranging tetrahedra vertex to vertex and been confronted with the inescapable octahedral cavities, he continues, they would have anticipated the physicists' 1922 discovery of "fundamental complementarity."... 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. (986.049b) Fueled by the developments of twentieth-century physics, Fuller spoke frequently and emphatically of the "inherent complementarity" of Universe. He cites two examples in particular out of the many provided by quantum physics. First, the "complementarity principle" announced by Niels Bohr (1885-1962) in 1928, which goes hand in hand with Heisenberg's indeterminism. Bohr summarizes the basic feature of quantum physics by stating that experimental evidence cannot be comprehended within a single frame, but rather must be understood as "complementary," or partial, information. The totality of a phenomenon must therefore be represented through more than one complementary part, for all aspects cannot be accurately measured simultaneously. Fuller also calls our attention to a later Nobel-winning development made by two Chinese physicists working in the U.S. In 1957, Tsung Dao Lee and Chen Ning Yang were honored for their discovery that "parity" is not conserved In weak interactions, for the subatomic particles involved show "handedness." |

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