901.10 Geodesic Dome Calculations

901.11 When two great-circle geodesic lines cross, they form two sets of similar angles, any one of which, paired with the other, will always add to 180. (This we also learned in plane geometry.) When any one great circle enters into or exits froma spherical triangle, it will form the two sets of similar angles as it crosses the enclosing great-circle-edge-lines of that triangle.

901.12 As in billiards or in electromagnetics, when a ball or a photon caroms off a wall it bounces off at an angle similar to that at which it impinged.

901.13 If a great-circle-describing, inexhaustibly re-energized, satellite ball that was sufficiently resilient to remain corporeally integral, were suddenly to encounter a vertical, great-circle wall just newly mounted from its parent planet’s sphere, it would bounce inwardly off that wall at the same angle that it would have traversed the same great-circle line had the wall not been there. And had two other great-circle walls forming a right spherical triangle with the first wall been erected just as the resilient ball satellite was hitting the first great-circle wall, then the satellite ball would be trapped inside the spherical-triangle-walled enclosure, and it would bounce angularly off the successively encountered walls in the similar-triangle manner unless it became aimed either at a corner vertex of the triangular wall trap, or exactly perpendicularly to the wall, in either of which cases it would be able to escape into the next spherical area lying 180 ahead outside the first triangle’s walls.

901.14 If, before the satellite bouncingly earned either a vertexial or perpendicular exit from the first-described spherical triangle (which happened to be dimensioned as one of the 120 LCD right triangles of the spherical icosahedron) great-circle walls representing the icosahedron’s 15 complete great circles, were erectthus constructing a uniform, spherical, wall patterning of 120 (60 positive, 60 negative) similar spherical, right triangleswe would find the satellite sphere bouncing around within one such spherical triangle at exactly the same interior or exiting angles as those at which it would have crossed, entered into, and exited, each of those great-circle boundaries of those 120 triangles had the wall not been so suddenly erected.

901.15 For this reason the great-circle interior mapping of the symmetrically superimposed other sets of 10 and 6 great circles, each of whichtogether with the 15 original great circles of the icosahedronproduces the 31 great circles of the spherical icosahedron’s total number of symmetrical spinnabilities in respect to its 30 mid-edge, 20 mid-face, and 12 vertexial poles of half-as-many-each axes of spin. (See Sec. 457.00.) These symmetrically superimposed, 10- and 6-great-circles subdivide each of the disequilibrious 120 LCD triangles into four lesser right spherical triangles. The exact trigonometric patterning of any other great circles orbiting the 120-LCD-triangled sphere may thus be exactly plotted within any one of these triangles.

901.16 It was for this reason, plus the discovery of the fact that the icosahedron among all the three-and-only prime structural systems of Universe (see Sec. 610.20)required the least energetic, vectorial, structural investment per volume of enclosed local Universe, that led to the development of the Basic Disequilibrium 120 LCD Spherical Triangle and its multifrequenced triangular subdivisioning as the basis for calculating all high-frequency, triangulated, spherical structures and structural subportions of spheres; for within only one disequilibrious LCD triangle were to be found all the spherical chord-factor constants for any desired radius of omnisubtriangulated spherical structure.

901.17 In the same way it was discovered that local, chord-compression struts could be islanded from one another, and could be only tensionally and non-inter-shearingly connected to produce stable and predictably efficient enclosures for any local energetic environment valving uses whatsoever by virtue of the approximately unlimited range of frequency-and-angle, subtrianglestructuring modulatability.

901.18 Because the 120 basic disequilibrious LCD triangles of the icosahedron have 2\sfrac{1}{2} times less spherical excess than do the 48 basic equilibrious LCD triangles of the vector equilibrium, and because all physical realizations are always disequilibrious, the Basic Disequilibrium 120 LCD Spherical Triangles become most realizably basic of all general systems’ mathematical control matrixes.

901.19 Omnirational Control Matrix: Commensurability of Vector Equilibrium and Icosahedron The great-circle subdivisioning of the 48 basic equilibrious LCD triangles of the vector equilibrium may be representationally drawn within the 120 basic disequilibrious LCD triangles of the icosahedron, thus defining all the aberrationsand their magnitudesexisting between the equilibrious and disequilibrious states, and providing an omnirational control matrix for all topological, trigonometric, physical, and chemical accounting.