1053.00 Superficial and Volumetric Hierarchies 
1053.15 Because each of the octahedron's eight faces is subdivided by its respective six sets of spherical "right" triangles (three positive^{__}three negative), whose total of 6 × 8 = 48 triangles are the 48 LCD's vectorequilibrium, symmetricphase triangles, and because 120/48 = 2 1/2, it means that each of the vector equilibrium's 48 triangles has superimposed upon it 2 1/2 positively askew and 2 1/2 negatively askew triangles from out of the total inventory of 120 LCD asymmetric triangles of each of the two sets, respectively, of the two alternate phases of the icosahedron's limit of rotational aberrating of the vector equilibrium. This 2 1/2 positive superimposed upon the 2 1/2 negative, 120 LCD picture is somewhat like a Picasso duoface painting with half a front view superimposed upon half a side view. It is then in transforming from a positive twoand onehalfness to a negative twoandonehalfness that the intertransformable vector equilibriumtoicosahedron, icosahedrontovectorequilibrium, equilibriousto disequilibriousness attains sumtotally and only dynamically a spherical fiveness (see Illus. 982.61 in color section). 
1053.36 Sphere: Volumesurface Ratios: The largest number of similar triangles into which the whole surface of a sphere may be divided is 120. (See Secs. 905 and 986.) The surface triangles of each of these 120 triangles consist of one angle of 90 degrees, one of 60 degrees, and one of 36 degrees. Each of these 120 surface triangles is the fourth face of a similar tetrahedron whose three other faces are internal to the sphere. Each of these tetra has the same volume as have the A or B Quanta Modules. Where the tetra is 1, the volume of the rhombic triacontahedron is approximately 5. Dividing 120 by 5 = 24 = quanta modules per tetra. The division of the rhombic triacontahedron of approximately tetravolume5 by its 120 quanta modules discloses another unit system behavior of the number 24 as well as its appearance in the 24 external vector edges of the VE. (See Sec. 1224.21) 
Fig. 1053.37 
1053.37
Since the surface of a sphere exactly equals the internal
area of the four great
circles of the sphere, and since the surface areas of
each of the four triangles of the
spherical tetrahedron also equal exactly onequarter
of the sphere's surface, we find that
the surface area of one surface triangle of the spherical
tetrahedron exactly equals the
internal area of one great circle of the sphere; wherefore

1053.50 Volumetric Hierarchy: With a nuclear sphere of radius1, the volumetric hierarchy relationship is in reverse magnitude of the superficial hierarchy. In the surface hierarchy, the order of size reverses the volumetric hierarchy, with the tetrahedron being the largest and the rhombic dodecahedron the smallest. 
1053.51
Table: Volumetric Hierarchy: The space quantum equals
the space domain
of each closestpacked nuclear sphere:
(Footnote 10: The octahedron is always double, ergo, its fourness of volume is its prime number manifest of two, which synergetics finds to be unique to the octahedron.) 
1053.51A
Table: Volumetric Hierarchy (revised): The space quantum
equals the
space domain of each closestpacked nuclear sphere:

1053.60 Reverse Magnitude of Surface vs. Volume: Returning to our consideration of the reverse magnitude hierarchy of the surface vs. volume, we find that both embrace the same hierarchical sequence and have the same membership list, with the icosahedron and vector equilibrium on one end of the scale and the tetrahedron on the other. The tetrahedron is the smallest omnisymmetrical structural system in Universe. It is structured with three triangles around each vertex; the octahedron has four, and the icosahedron has five triangles around each vertex. We find the octahedron in between, doubling its prime number twoness into volumetric fourness, as is manifest in the great circle foldability of the octahedron, which always requires two sets of great circles, whereas all the other icosahedron and vector equilibrium 31 and 25 great circles are foldable from single sets of great circles . 
1053.601 Octahedron: The octahedron^{__}both numerically and geometrically^{__}should always be considered as quadrivalent; i.e., congruent with self; i.e., doubly present. In the volumetric hierarchy of primenumber identities we identify the octahedron's prime number twoness and the inherent volumefourness (in tetra terms) as volume 22, which produces the experiential volume 4. 
1053.61 The reverse magnitudes of the surface vs. volume hierarchy are completely logical in the case of the total surface subdivision starting with system totality. On the other hand, we begin the volumetric quantation hierarchy with the tetrahedron as the volumetric quantum (unit), and in so doing we build from the most common to the least common omnisymmetrical systems of Universe. In this system of biggest systems built of smaller systems, the tetrahedron is the smallest, ergo, most universal. Speaking holistically, the tetrahedron is predominant; all of this is analogous to the smallest chemical element, hydrogen, being the most universally present and plentiful, constituting the preponderance of the relative abundance of chemical elements in Universe. 
1053.62 The tetrahedron can be considered as a whole system or as a constituent of systems in particular. It is the particulate. 
1053.70 Container Structuring: Volumesurface Ratios 
1053.71 When attempting to establish an international metric standard of measure for an integrated volumeweight unit to be known as "one gram" and deemed to consist of one cubic centimeter of water, the scientists overlooked the necessity for establishing a constant condition of temperature for the water. Because of expansion and contraction under changing conditions of temperature a constant condition of 4 degrees centigrade was later established internationally. In much the same way scientists have overlooked and as yet have made no allowance for the inherent variables in entropic and syntropic rates of energy loss or gain unique to various structurally symmetrical shapes and sizes and environmental relationships. (See Sec. 223.80, "Energy Has Shape.") Not only do we have the hierarchy of relative volume containments respectively of equiedged tetra, cube, octa, icosa, "sphere," but we have also the relative surfacetovolume ratios of those geometries and the progressive variance in their relative structuralstrengthtosurface ratios as performed by flat planes vs simple curvature; and as again augmented in strength out of the same amount of the same material when structured in compound curvature. 
1053.72 In addition to all the foregoing structuralcapability differentials we have the tensegrity variables (see Chap. 7), as all these relate to various structural capabilities of various energy patternings as containers to sustain their containment of the variously patterning contained energies occurring, for instance, as vacuum vs crystalline vs liquid vs gaseous vs plasmic vs electromagnetic phases; as well as the many cases of contained explosive and implosive forces. Other structural variables occur in respect to different containercontained relationships, such as those of concentrated vs distributive loadings under varying conditions of heat, vibration, or pressure; as well as in respect to the variable tensile and compressive and sheer strengths of various chemical substances used in the container structuring, and their respective heat treatments; and their sustainable strengthtime limits in respect to the progressive relaxing or annealing behaviors of various alloys and their microconstituents of geometrically variant chemical, crystalline, structural, and interproximity characteristics. There are also external effects of the relative size strength ratio variables that bring about internal interattractiveness values in the various alloys as governed by the secondpower rate, i.e., frequency of recurrence and intimacy of those alloyed substances' atoms. 
1053.73 As geometrical systems are symmetrically doubled in linear dimension, their surfaces increase at a rate of the second power while their volumes increase at a third power rate. Conversely, as we symmetrically halve the linear dimensions of geometrical systems, their surfaces are reduced at a secondroot rate, while their volumes decrease at a thirdroot rate. 
1053.74 A cigarshaped piece of steel six feet (72 inches) long, having a small hole through one end and with a midgirth diameter of six inches, has an engineering slenderness ratio (length divided by diameter) of 12 to 1: It will sink when placed on the surface of a body of water that is more than six inches deep. The sameshaped, endpierced piece of the same steel of the same 12to1 slenderness ratio, when reduced symmetrically in length to three inches, becomes a sewing needle, and it will float when placed on the surface of the same body of water. Diminution of the size brought about so relatively mild a reduction in the amount of surface of the steel cigarneedle's shape in respect to the great change in volume^{__}ergo, of weight^{__}that its shape became so predominantly "surface" and its relative weight so negligible that only the needle's surface and the atomicintimacy produced surface tension of the water were importantly responsible for its interenvironmental relationship behaviors. 
1053.75 For the same reasons, grasshoppers' legs in relation to a human being's legs have so favorable a volumetosurfacetension relationship that the grasshopper can jump to a height of 100 times its own standing height (length) without hurting its delicate legs when landing, while a human can jump and fall from a height of only approximately three times his height (length) without breaking his legs. 
1053.76 This same volumetosurface differential in rate of change with size increase means that every time we double the size of a container, the contained volume increases by eight while the surface increases only fourfold. Therefore, as compared to its previous halfsize state, each interior molecule of the atmosphere of the building whose size has been symmetrically doubled has only half as much building surface through which that interior molecule of atmosphere can gain or lose heat from or to the environmental conditions occurring outside the building as conductively transferable inwardly or outwardly through the building's skin. For this reason icebergs melt very slowly but accelerate progressively in the rate of melting. For the same reason a very different set of variables governs the rates of gain or loss of a system's energy as the system's size relationships are altered in respect to the environments within which they occur. 
1053.77 As oil tankers are doubled in size, their payloads grow eightfold in quantity and monetary value, while their containing hulls grow only fourfold in quantity and cost. Because the surface of the tankers increases only fourfold when their lengths are doubled and their cargo volume increases eightfold, and because the power required to drive them through the sea is proportional to the ship's surface, each time the size of the tankers is doubled, the cost of delivery per cargo ton, barrel, or gallon is halved. The last decade has seen a tenfolding in the size of the transoceanic tankers in which both the cost of the ship and the transoceanic delivery costs have become so negligible that some of the first such shipowners could almost afford to give their ships away at the end of one voyage. As a consequence they have so much wealth with which to corrupt international standards of safety that they now build them approximately without safety factors^{__}ergo, more and more oil tanker wrecks and spills. 
Next Section: 1053.80 