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Angular Takeout: An Example A complicated system such as the four-frequency icosahedron provides an especially good illustration of this remarkable consistency. The structure is an irregular polyhedron with 320 triangular faces, and is based on the symmetry of the icosahedron. Each icosahedral triangle is replaced by sixteen new smaller' triangles, producing the total of 320 faces of this more or less spherical structure. Chapter 15 will describe the origin of high-frequency icosahedral enclosures in detail, but for now, we can understand that the faces are irregular triangles. As shown in Figure 6-5, most vertices join six triangles, and we recall from Chapter 4 that if six
If all 162 vertices are equally distant from the structure's center, the average total of the surface angles at each will be 355 degrees, 33 minutes, and 20 seconds—or 355.5556 degrees. Thus, the "takeout" angle-instead of being concentrated at a few points, with 60 degrees removed from each of twelve vertices—is distributed among many component triangles. Imagine cutting open this system and spreading out its multifaceted net. Each angular gap would be very narrow, averaging 4 degrees, 26 rninutes, and 40 seconds (4.4444 degrees.) If we were to draw this net, the pencil thickness itself would be a nuisance. Supposing each edge is approximately one inch long, the outermost or widest part of each gap will be less than one-twelfth of an inch. In addition to being miniscule, these numbers are typically quite irregular, not at all simple fractions of degrees. Nevertheless, these gaps, calculated (say) to six decimal places, add up to exactly 720.000000 degrees, no matter how many vertices in the system, or how irregular the distribution. One interesting implication of the principle of angular topology is further demonstration of the impossibility of the traditional sphere as defined by mathematics—an unreachable planar 360 degrees around every one of an infinite number of vertices. "The calculus and spherical trigonometry alike assume that the sum of the angles around any point on any sphere's surface is always 360 degrees" (224.11). Fuller goes on to point out that in order to achieve a closed system, there must be 720 degrees taken out, distributed throughout the vertices, thereby invalidating this assumption. "The demonstration thus far discloses that the sum of the angles around all the vertexes [sic] of a sphere will always be 720 degrees—or one tetrahedron—less that the sum of the vertexes times 360 degrees—ergo one basic assumption of the calculus and spherical trigonometry is invalid" (224.11). In other words, since the 720-degree takeout is a prerequisite to closure, even a sphere has to have infinitesimally less than 360 degrees around any given point on its surface. (We are forced to conclude that "infinitesimal" times infinite" here equals 720 degrees.) Angle Types Finally, knowing the different types of angles in geometry will be helpful. The nomenclature is straightforward. A An
That finishes this chapter, but the subject of angle is never far removed from any discussion in synergetics. |

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