900.01 Definition: Modelability
900.10 Modelability 
900.11 Modelability is topologically conceptual in generalized principle independent of size and time: ergo, conceptual modelability is metaphysical. 
900.12 Conceptual formulation is inherently empirical and as such is always special case sizing and always discloses all the physical characteristics of existence in time. 
900.20 Synergetics 
900.30 Model vs Form 
900.31 Model is generalization; form is special case. 
900.33 Forms have size. Models are sizeless, representing conceptuality independent of size. 
901.00 Basic Disequilibrium LCD Triangle
901.01 Definition 
901.02 The Basic Disequilibrium 120 LCD Spherical Triangle of synergetics is derived from the 15greatcircle, symmetric, threeway grid of the spherical icosahedron. It is the lowest common denominator of a sphere's surface, being precisely 1/120th of that surface as described by the icosahedron's 15 great circles. The trigonometric data for the Basic Disequilibrium LCD Triangle includes the data for the entire sphere and is the basis of all geodesic dome calculations. (See Sec.612.00.) 
Fig. 901.03 Fig. 901.03 
901.03 As seen in Sec. 610.20 there are only three basic structural systems in Universe: the tetrahedron, octahedron, and icosahedron. The largest number of equilateral triangles in a sphere is 20: the spherical icosahedron. Each of those 20 equiangular spherical triangles may be subdivided equally into six right triangles by the perpendicular bisectors of those equiangular triangles. The utmost number of geometrically similar subdivisions is 120 triangles, because further sphericaltriangular subdivisions are no longer similar. The largest number of similar triangles in a sphere that spheric unity will accommodate is 120: 60 positive and 60 negative. Being spherical, they are positive and negative, having only common arc edges which, being curved, cannot hinge with one another; when their corresponding angleandedge patterns are vertexmated, one bellies away from the other: concave or convex. When one is concave, the other is convex. (See Illus. 901.03 and drawings section.) 
901.10 Geodesic Dome Calculations 
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. 
901.15 For this reason the greatcircle interior mapping of the symmetrically superimposed other sets of 10 and 6 great circles, each of which^{__}together with the 15 original great circles of the icosahedron^{__}produces the 31 great circles of the spherical icosahedron's total number of symmetrical spinnabilities in respect to its 30 midedge, 20 midface, and 12 vertexial poles of halfasmanyeach axes of spin. (See Sec. 457 .) These symmetrically superimposed, 10 and 6greatcircles 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 120LCDtriangled 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 threeandonly 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 highfrequency, triangulated, spherical structures and structural subportions of spheres; for within only one disequilibrious LCD triangle were to be found all the spherical chordfactor constants for any desired radius of omnisubtriangulated spherical structure. 
902.00 Properties of Basic Triangle 
Fig. 902.01 
902.01 Subdivision of Equilateral Triangle: Both the spherical and planar equilateral triangles may be subdivided into six equal and congruent parts by describing perpendiculars from each vertex of the opposite face. This is demonstrated in Fig. 902.01, where one of the six equal triangles is labeled to correspond with the Basic Triangle in the planar condition. 
Fig. 902.10 
902.10 Positive and Negative Alternation: The six equal subdivision triangles of the planar equilateral triangle are hingeable on all of their adjacent lines and foldable into congruent overlays. Although they are all the same, their dispositions alternate in a positive and negative manner, either clockwise or counterclockwise. 
Fig. 902.20 
902.20
Spherical Right Triangles: The edges of all spherical
triangles are arcs of
great circles of a sphere, and those arc edges are measured
in terms of their central angles
(i.e., from the center of the sphere). But plane surface
triangles have no inherent central
angles, and their edges are measured in relative lengths
of one of themselves or in special
case linear increments. Spherical triangles have three
surface (corner) angles and three
central (edge) angles. The basic data for the central
angles provided below are accurate to
1/1,000 of a second of arc. On Earth

902.22 The spherical surface angle BCE is exactly equal to two of the arc edges of the Basic Disequilibrium 120 LCD Triangle measured by their central angle. BCE = arc AC = arc CF = 20° 54' 18.57". 
Fig. 902.30 
902.30 Surface Angles and Central Angles: The Basic Triangle ACB can be folded on the lines CD and CE and EF. We may then bring AC to coincide with CF and fold BEF down to close the tetrahedron, with B congruent with D because the arc DE = arc EB and arc BF = arc AD. Then the tetrahedron's corner C will fit exactly down into the central angles AOC, COB, and AOB. (See Illus. 901.03 and 902.30.) 
902.31 As you go from one spherefoldable greatcircle set to another in the hierarchy of spinnable symmetries (the 3, 4, 6, 12sets of the vector equilibrium's 25 greatcircle group and the 6, 10, 15sets of the icosahedron's 31greatcircle group), the central angles of one often become the surface angles of the nexthighernumbered, more complex, greatcircle set while simultaneously some (but not all) of the surface angles become the respective next sphere's central angles. A triangle on the surface of the icosahedron folds itself up, becomes a tetrahedron, and plunges deeply down into the congruent central angles' void of the icosahedron (see Sec. 905.47 ). 
902.32 There is only one noncongruence the last wouldbe hinge, EF is an external arc and cannot fold as a straight line; and the spherical surface angle EBF is 36 degrees whereas a planar 30 degrees is called for if the surface is cast off or the arc subsides chordally to fit the 906030 right plane triangle. 
902.33 The 6 degrees of spherical excess is a beautiful whole, rational number excess. The 90degree and 60degree corners seem to force all the excess into one corner, which is not the way spherical triangles subside. All the angles lose excess in proportion to their interfunctional values. This particular condition means that the 90 degrees would shrink and the 60 degrees would shrink. I converted all the three corners into seconds and began a proportional decrease study, and it was there that I began to encounter a ratio that seemed rational and had the number 31 in one corner. This seemed valid as all the conditions were adding up to 180 degrees or 90 degrees as rational wholes even in both spherical and planar conditions despite certain complementary transformations. This led to the intuitive identification of the Basic Disequilibrium 120 LCD Triangle's foldability (and its fallinability into its own tetravoid) with the A Quanta Module, as discussed in Sec. 910 which follows. 
Next Section: 905.00 