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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 sixty-degree angles meet, they create a flat surface. Therefore, if six faces are to surround a convex vertex of a polyhedron, their angular total must be less than 360 degreeswhich produces the "angular takeout". Those interested in exploring how to calculate individual edge lengths and surface angles can refer to Appendix A, "Chord Factors," and to Appendix C for a list of additional sources; other readers should simply be aware that the values will be highly irregular numbers. Having noted that, to assure convexity, the surface angles must add up to less than 360 degrees at each vertex, we can reflect on how extremely small the gaps at each vertex in this structure will be.
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 secondsor 355.5556 degrees. Thus, the "takeout" angle-instead of being concentrated at a few points, with 60 degrees removed from each of twelve verticesis 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 mathematicsan 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 degreesor one tetrahedronless that the sum of the vertexes times 360 degreesergo 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.)
Finally, knowing the different types of angles in geometry will be helpful. The nomenclature is straightforward. Surface angle is by now familiar, referring to a comer angle of a polyhedral face (Fig. 6-6a).
Dihedral angles are the angles between adjacent faces, on the inside of a system. (Fig. 6-6b).
A central angle corresponds to a polyhedral edge and is measured from the exact volumetric center of a system to each end of the edge (Fig. 6-6c). Central angles provide a way to find exact edge lengths of a given system for any desired radius. (4) Central angles provide an effective way to record relative lengths, for remember, an angle is independent of the lengths of its sides. This means that we can list the complete set of edge lengths for a complex polyhedral or geodesic system in terms of the central angles corresponding to each edge, and then directly calculate exact lengths for any desired radius, with the help of a simple trigonometric equation. This process might sound complicated at first, but is more expedient than the alternative, which is to specify one set of edge lengths, applying to only one "special-case" system. Data for any different size would then have to be completely recalculated, step by step, from scratch. Central angles give us data for the general case, applicable to any particular realization of the same shape. In architecture, this allows us to build a geodesic dome of any size from one set of central-angle calculations. A pocket calculator is all that is required to simply multiply the desired radius by twice the sine of half the central angle. (4)
An axial angle is the angle between the edge of a polyhedron and an adjacent radius (Fig. 6-6d).
That finishes this chapter, but the subject of angle is never far removed from any discussion in synergetics.
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