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Once in a long while, a "generalized principle" takes recognizable shape and emerges out of the vast sea of man's cumulative findings. For Fuller, these principlescharacterized as true in every caseare the real wealth of society. Applications may not always be immediately clear, but if an inventory of "generalized principles" is made accessible, he reasoned, humanity will put them to use sooner or later.
The "principle of angular topology" was recognized by the mathematician and philosopher René Descartes (1596-1650), but the title of course is Fuller's. Perhaps by giving Descartes's remarkable discovery a new title, Bucky hoped to excite the kind of attention he felt it deserved.
In every polyhedral system, the sum of the angles around all the vertices is exactly 720 degrees less than the number of vertices times 360 degrees, or (360° × V) - 720°. True for the tetrahedron, true for the crocodile. In Fuller's words, every system has exactly 720 degrees of "takeout."
If this principle seems complicated, it is only because the words are hard to follow, but the following image should make it easier. Picture a paper conethe shape of an ice-cream cone without the ice cream. Notice that a cone, having an opening at the base, is not a closed system. Now, split the paper cone open by slicing a straight line from its pointed tip to the circular hole, and then spread the piece of paper out fiat on the floor like a rug (Fig. 6-3). There is now an angular gap left by the paper, where the floor shows through. That gap is the "takeout angle," the angular difference between a flat map and a cone.
In the same manner, you can slice open some number of edges of a polyhedron until its surface can be spread out like a rug. The resulting map, similar to a dressmaker's pattern, is called in geometry a polyhedron's net. A net contains all faces of a polyhedron, some of them separated by angular gaps; it is a flat pattern which can be folded along the edges, and taped together to generate its polyhedron. Figure 6-4 shows nets of an icosahedron and an octahedron. The principle of angular topology states that the sum of the angular gaps, no matter how simple or complex the system, will be exactly 720 degrees.
Or go through these procedures in reverse: start with flat paper and cut out one pie-shaped segment to make a cone. Then keep going, cutting out more slices, just until the point at which the paper can be closed off into a system. This point is reached when you have taken out exactly 720 degrees. It's a prerequisite to closure; there is no leeway. To check, you can measure all the scraps that have been taken out; the results are always the same: however irregular your cuts or strange your resulting closed system, the total takeout must be 720 degrees. This consistent total presents a generalized principle for closed systems.
The surface angles of any tetrahedron (regular or not) also happen to add up to 720 degrees. (Four triangles: 4 × 180° = 720°.) Fuller certainly isn't going to let that one slip by! The "difference between the visibly definite system and the invisibly finite Universe [i.e., plane] is always exactly one finite invisible tetrahedron..." (224.10).
Consider once again the variety of systems. This principlea first cousin of Euler's lawdescribes an extraordinary consistency. The "720-degree excess" is an appropriate parallel to Euler's "constant 2" in that there are 720 degrees in two complete revolutions. Both are counterintuitive: Euler's law reveals that the number of edges is always exactly two less than the vertices plus faces, no matter how complex the system, just as the angular "takeout" is 720 degrees whether the surface angles themselves add up to a total of 720 degrees, as in the tetrahedron, or to 57,600 degrees, as in the "four-frequency icosahedron." (Don't worry, that structure will be explained below.) Table III shows the results for a few different polyhedra, verifying the constant "excess" of 720 degrees.
Table III reveals another notable consistency: the sum of the surface angles in every polyhedron is a multiple of the tetrahedron's 720 degrees (column 5). This calls to mind our earlier observation that the number of edges in many ordered polyhedra is a multiple of the tetrahedron's six. (Refer to Chapter 4.)
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