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Cosmic Railroad Tracks: Great Circles
Any planar closed loop drawn on the surface of a sphere is necessarily a perfect circle, as a result of the sphere's steady curvature. Such loops qualify as either "great" or "lesser" circles, and the distinction is defined in mathematics as follows. A great circle is formed by the intersection of a plane passing through the center of a sphere with the surface of that sphere. Any other circle, no matter what size, is lesser. The center of a great circle coincides with the sphere's center. In short, a great circle is an equatorfound in any angular orientation, but always around the fattest part of its sphere.
What's so great about a great circle? Above all, it provides the shortest route between any two points on a sphere. This geometric fact is not obvious in many cases; for example, looking at a globe it appears that the logical route between two points situated some distance apart on the Tropic of Cancer involves traveling along their shared "lesser circle" band (Fig. 14-1a). However, the principle is made more obvious by Fuller's juxtaposition of two extreme cases. He describes a small lesser circle near the North Pole of an imaginary globe, and labels two points A and B (Fig. 14-1b). As with the larger Tropic of Cancer, the eye naturally travels from A to B along their mutual lesser-circle path, without suspecting that this represents the "long way around." Fuller then redraws the same lesser circle in a new locationsuperimposed over the globe's equator so that A and B both fall on the horizontal great circle. The shortest route between A and B is suddenly obvious. Likewise, between any two points on a sphere the most expedient route will be a great-circle segment; Fuller's example makes it easy to see that a lesser-circle path will always present a detour.
Other special characteristics: Any two great circles on a sphere must intersect twicespecifically, at two points 180 degrees apart. There is no other possibility: unless they are actually the same circle, two great circles can neither avoid each other altogether, nor intersect only once, nor intersect more than twice. Finally, the junction of two great circles inscribes two pairs of equal and opposite angles on the sphere's surface, whose two angular values add up to 180 degrees; the statement is equivalent for the intersection of two Euclidean "straight lines." Lesser circles do not share this property; in fact, their intersection produces opposite angles which are necessarily unequal. In conclusion, on the surface of a sphere, only great circles have the geometric characteristics of "straight lines"
As we examine different aspects of great circles, you will notice that much of this material is quite complicated; the patterns are too intricate to be readily visualized in the mind's eye. Furthermore, connections drawn between the geometric models and physical phenomena are unusually speculative. However, as Fuller's Synergetics devotes considerable attention to great circles, anyone who has tried to decipher these sections will welcome full coverage. Referring back and forth from the text to the drawings will be essential.
Why Are We Talking About Spheres?
The vertices of regular and semiregular polyhedra lie on the surface of an imaginary sphere, which is to say that all vertices are equidistant from a polyhedron's center. Given this fact, we can picture spherical versions of each polyhedron, in which the polyhedral edges have stretched outward to become great-circle arcs and the faces have expanded into curved surfaces, as if each shape had been blown up like a balloon. Figure 14-2 shows a spherical tetrahedron, octahedron, and icosahedron as examples. Comparing these systems with their planar counterparts, it is clear that polyhedral edges are actually chords of great-circle arcs. We can conclude that the shortest distance between two events of a system always involves a great circle.
The concept (rather than the reality) of a spherei.e., an omnisymmetrical container-acts as a frame of reference for polyhedral systems. Spherical polyhedra thus introduce new versions of familiar characters. The topological information (that is, the numbers and valencies of vertices, edges, and faces) of any polyhedron are displayed on a spherical canvas. An obvious consequence of this type of display is that shape is no longer a variable. Shape similarities, which are so rigorously accounted for by A and B modules, are thus ignored; our investigation now focuses in on topological, or surface, characteristics. Transforming polyhedra into balloons temporarily equalizes shape and size, providing a "common denominator" for other comparisons. The process develops a somewhat unorthodox chart.
New Classification System
However, the chart is not yet complete. Simply projecting edges and faces onto a spherical surface does not teach us anything new. We have yet to exploit the nature of the sphere.
Spheres suggest spin. That's how synergetics initially arrives at the omnidirectional form. Spin any system in all directions, and ultimately the action will have defined a circumscribing spherical envelope. Fuller places considerable emphasis on the "spinnability" of systems, arguing that as everything in Universe is in motion, the different axes of spin inherent in systems are worthy of investigation.
All polyhedra have three sets of topological aspects: vertices, edges, and faces. These sets correspond to three types of axes of rotational symmetry (or "spin") which connect pairs of either polar-opposite vertices, mid-edge points, or face centers. As a polyhedron spins about any one of these axes, an implied great circle is generated midway between the two poles, in other words, at the equator. Equators corresponding to all existing axes of symmetry can be simultaneously represented on a spherical surface, creating an exhaustive chart of the topological symmetries of a given system. Each symmetrical polyhedron has its own great-circle diagram, which Iincorporates all its axes of rotational symmetryor axes of spin, in Fuller's terminology. The patterns generated by related polyhedra may include some of the same circles, as determined by common symmetries; however, the complete chart of a polyhedron is exactly shared only by its dual, as will be shown below.
Great circles reveal a new aspect of polyhedral "intertransformability," a novel (if obscure) means of detecting symmetrical relationships among systems. We can anticipate the emergence of decidedly unfamiliar patterns in this game for their resemblance to the source polyhedron is sometimes subtle. Finally, in pursuing this study we discover new variations on upper limits and minimum cases. Great circles thus provide another tool with which to detect inherent spatial constraints.
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