982.21 Synergetics has discovered that the vectorially most economical control line of nature is in the diagonal of the cube's face and not in its edge; that this diagonal connects two spheres of the isotropicvectormatrix field; and that those spherical centers are congruent with the two onlydiagonallyinterconnected corners of the cube. Recognizing that those cubediagonalconnected spheres are members of the closest packed, allspacecoordinating, unit radius spheres field, whose radii = 1 (unity), we see that the isotropicvectormatrix's fieldoccurringcube's diagonal edge has the value. of 2, being the line interconnecting the centers of the two spheres, with each half of the line being the radius of one sphere, and each of the whole radii perpendicular to the same points of intersphere tangency. 
982.30 Diagonal of Cube as Control Length: We have learned elsewhere that the sum of the second powers of the two edges of a right triangle equals the second power of the right triangle's hypotenuse; and since the hypotenuse of the two similar equiedged right triangles formed on the square face of the cube by the spherecenterconnecting diagonal has a value of two, its second power is four; therefore, half of that four is the second power of each of the equiedges of the right triangle of the cube's diagonaled face: half of four is two. 
982.31 The square root of 2 = 1.414214, ergo, the length of each of the cube's edges is 1.414214. The sqrt(2)happens to be one of those extraordinary relationships of Universe discovered by mathematics. The relationship is: the number one is to the second root of two as the second root of two is to two: 1:sqrt(2) = sqrt(2):2, which, solved, reads out as 1 : 1.414214 = 1.414214 : 2. 
982.32 The cube formed by a uniform width, breadth, and height of sqrt(2) is sqrt(2^{3}), which = 2.828428. Therefore, the cube occurring in nature with the isotropic vector matrix, when conventionally calculated, has a volume of 2.828428. 
982.33 This is exploratorily noteworthy because this cube, when calculated in terms of man's conventional mensuration techniques, would have had a volume of one, being the first cube to appear in the omnigeometrycoordinate isotropic vector matrix; its edge length would have been identified as the prime dimensional input with an obvious length value of one^{__}ergo, its volume would be one: 1 × 1 × 1 = 1. Conventionally calculated, this cube with a volume of one, and an edge length of one, would have had a face diagonal length of sqrt(2), which equals 1.414214. Obviously, the use of the diagonal of the cube's face as the control length results in a much higher volume than when conventionally evaluated. 
982.40 Tetrahedron and Synergetics Constant: And now comes the big surprise, for we find that the cube as coordinately reoccurring in the isotropic vector matrix^{__}as most economically structured by nature^{__}has a volume of three in synergetics' vector edged, structuralsystemevaluated geometry, wherein the basic structural system of Universe, the tetrahedron, has a volume of one. 
982.41 A necklaceedged cube has no structural integrity. A tensionlinked, edge strutted cube collapses. 
982.42 To have its cubical conformation structurally (triangulated) guaranteed (see Secs. 615 and 740), the regular equiangled tetrahedron must be inserted into the cube, with the tetrahedron's six edges congruent with each of the six vacant but omnitriangulatable diagonals of the cube's six square faces. 
982.43 As we learn elsewhere (Secs. 415.22 and 990), the tetrahedron is not only the basic structural system of Universe, ergo, of synergetic geometry, but it is also the quantum of nuclear physics and is, ipso facto, exclusively identifiable as the unit of volume; ergo, tetrahedron volume equals one. We also learned in the sections referred to above that the volume of the octahedron is exactly four when the volume of the tetrahedron of the unitvector edges of the isotropicvectormatrix edge is one, and that four EighthOctahedra are asymmetrical tetrahedra with an equiangular triangular base, three apex angles of 90 degrees, and six lowercomer angles of 45 degrees each; each of the 1/8th octahedron's asymmetric tetrahedra has a volumetric value of onehalf unity (the regular tetrahedron). When four of the EighthOctahedrons are equiangleface added to the equiangled, equiedged faces of the tetrahedra, they produce the minimum cube, which, having the tetrahedron at its heart with a volume of one, has in addition four onehalf unity volumed EighthOctahedra, which add two volumetric units on its corners. Therefore, 2 + 1 = 3 = the volume of the cube. The cube is volume three where the tetrahedron's volume is one, and the octahedron's volume is four, and the cube's diagonally structured faces have a diagonal length of one basic system vector of the isotropic vector matrix. (See Illus. 463.01.) 
982.44 Therefore the edge of the cube = sqrt(1/2). 
982.45
Humanity's conventional mensuration cube with a volume
of one turns out
in energetic reality to have a conventionally calculated
volume of 2.828428, but this same
cube in the relativeenergy volume hierarchy of synergetics
has a volume of 3.

982.46 To correct 2.828428 to read 3, we multiply 2.828428 by the synergetics conversion constant 1.06066. (See Chart 963.10.) 
982.47 Next we discover, as the charts at Secs. 963.10 and 223.64 show, that of the inventory of wellknown symmetrical polyhedra of geometry, all but the cube have irrational values as calculated in the XYZ rectilinearcoordinate system^{__}"cubism" is a convenient term^{__}in which the cube's edge and volume are both given the prime mensuration initiating value of one. When, however, we multiply all these irrational values of the Platonic polyhedra by the synergetic conversion constant, 1.06066, all these values become unitarily or combinedly rational, and their low firstfourprimenumber accommodation values correspond exactly with those of the synergetic hierarchy of geometric polyhedra, based on the tetrahedron as constituting volumetric unity. 
982.48 All but the icosahedron and its "wife," the pentagonal dodecahedron, prove to be volumetrically rational. However, as the tables show, the icosahedron and the vectoredged cube are combiningly rational and together have the rational value of three to the third power, i.e., 27. We speak of the pentagonal dodecahedron as the icosahedron's wife because it simply outlines the surfacearea domains of the 12 vertexes of the icosahedron by joining together the centers of area of the icosahedron's 20 faces. When the pentagonal dodecahedron is vectorially constructed with flexible tendon joints connecting its 30 edge struts, it collapses, for, having no triangles, it has no structural integrity. This is the same behavior as that of a cube constructed in the same flexible tendonvertex manner. Neither the cube nor the pentagonal dodecahedron is scientifically classifiable as a structure or as a structural system (see Sec. 604). 
982.50
Initial FourDimensional Modelability: The modelability
of the XYZ
coordinate system is limited to rectilinearframeofreference
definition of all specialcase
experience patternings, and it is dimensionally sized
by arbitrary, e.g., c.g_{t}.s.system,
subdivisioning increments. The initial increments are
taken locally along infinitely
extensible lines always parallel to the three sets of
rectilinearly interrelated edges of the
cube. Any one of the cube's edges may become the onedimensional
module starting
reference for initiating the mensuration of experience
in the conventional, elementary,
energetical^{7} school curriculum.
(Footnote 7: Energetical is in contradistinction to synergetical. Energetics employs isolation of special cases of our total experience, the better to discern unique behaviors of parts undiscernible and unmeasurable in total experience.) 
982.51 The XYZ cube has no initially moduled, vertexdefined nucleus; nor has it any inherent, common, mosteconomicallydistanced, uniform, inoutand circumferentiallyaround, cornercutting operational interlinkage, uniformly moduled coordinatability. Nor has it any initial, ergo inherent, timeweightenergy(as mass charge or EMF) expressibility. Nor has it any omniintertransformability other than that of vari sized cubism. The XYZ exploratory coordination inherently commences differentially, i.e., with partial system consideration. Consider the threedimensional, weightless, timeless, temperatureless volume often manifest in irrational fraction increments, the general reality impoverishments of which required the marriage of the XYZ system with the c.g_{t}.s. system in what resembles more of an added partnership than an integration of the two. 
982.52 The synergetics coordinate system's initial modelability accommodates four dimensions and is operationally developable by frequency modulation to accommodate fifth and sixthdimensional conceptualmodel accountability. Synergetics is initially nuclearvertexed by the vector equilibrium and has initial inoutandaround, diagonaling, and diametrically opposite, omnishortestdistance interconnections that accommodate commonly uniform wavilinear vectors. The synergetics system expresses divergent radiational and convergent gravitational, omnidirectional wavelength and frequency propagation in one operational field. As an initial operational vector system, its (mass x velocity) vectors possess all the unique, specialcase, time, weight, energy (as mass charge or EMF) expressibilities. Synergetics' isotropic vector matrix inherently accommodates maximally economic, omniuniform intertransformability. 
982.53 In the synergetics' four, five, and sixdimensionally coordinate system's operational field the linear increment modulatability and modelability is the isotropic vector matrix's vector, with which the edges of the cooccurring tetrahedra and octahedra are omnicongruent; while only the face diagonals^{__}and not the edges^{__}of the inherently cooccurring cubes are congruent with the matrix vectors. Synergetics' exploratory coordination inherently commences integrally, i.e., with wholesystems consideration. Consider the onedimensional linear values derived from the initially stated whole system, sixdimensional, omnirational unity; any linear value therefrom derived can be holistically attuned by unlimited frequency and onetoone, coordinated, wavelength modulatability. To convert the XYZ system's cubical values to the synergetics' values, the mathematical constants are linearly derived from the mathematical ratios existing between the tetrahedron's edges and the cube's cornertooppositecorner distance relationships; while the planar area relationships are derived from the mathematical ratios existing between cubicaledged square areas and cubicalfacediagonalededged triangular areas; and the volumetric value mathematical relationships are derived from ratios existing between (a) the cubeedgereferenced third power of theoften oddfractionededge measurements (metric or inches) of cubically shaped volumes and (b) the cubefacediagonalvector referenced third power of exclusively whole number vector, frequency modulated, tetrahedrally shaped volumes. (See Sec. 463 and 464 for exposition of the diagonal of the cube as a wavepropagation model.) 
982.54
The mathematical constants for conversion of the linear,
areal, and
volumetric values of the XYZ system to those of the
synergetics system derive from the
synergetics constant (1.060660). (See Sec.
963.10
and
Chart
963.12.) The conversion
constants are as follows:

982.55 To establish a numerical value for the sphere, we must employ the synergetics constant for cubical thirdpower volumetric value conversion of the vector equilibrium with the sphere of radius 1. Taking the vector equilibrium at the initial phase (zero frequency, which is unitytwo diameter: ergo unityone radius) with the sphere of radius l; i.e., with the external vertexes of the vector equilibrium congruent with the surface of the sphere = 4/3 pi () multiplied by the third power of the radius. Radius = 1. 1^{3} = 1. l × 1.333 × 3.14159 = 4.188. 4.188 times synergetics thirdpower constant 1.192 = 5 = volume of the sphere. The volume of the radius 1 vector equilibrium = 2.5. VE sphere = 2 VE. 
982.56 We can assume that when the sphere radius is 1 (the same as the nuclear vector equilibrium) the Basic Disequilibrium 120 LCD tetrahedral components of mild off sizing are also truly of the same volumetric quanta value as the A and B Quanta Modules; they would be shortened in overall greatest length while being fractionally fattened at their smallesttriangularface end, i.e., at the outer spherical surface end of the 120 LCD asymmetric tetrahedra. This uniform volume can be maintained (as we have seen in Sec. 961.40). 
982.57 Because of the fundamental 120module identity of the nuclear sphere of radius 1 (F = 0), we may now identify the spherical icosahedron of radius 1 as five; or as 40 when frequency is 2F^{2}. Since 40 is also the volume of the F^{2} vectorequilibrium vertexescongruent sphere, the unaberrated vector equilibrium F^{2} = 20 (i.e., 8 × 2 1/2 nuclearsphere's inscribed vector equilibrium). We may thus assume that the spherical icosahedron also subsides by loss of half its volume to a size at which its volume is also 20, as has been manifested by its prime number five, indistinguishable from the vector equilibrium in all of its topological hierarchies characteristics. 
Fig. 982.58 
982.58 Neither the planarfaceted exterior edges of the icosahedron nor its radius remain the same as that of the vector equilibrium, which, in transforming from the vector equilibrium conformation to the icosahedral state^{__}as witnessed in the jitterbugging (see Sec. 465) ^{__}did so by transforming its outer edge lengths as well as its radius. This phenomenon could be analagous the disappearance of the nuclear sphere, which is apparently permitted by the export of its volume equally to the 12 surrounding spheres whose increased diameters would occasion the increased sizing of the icosahedron to maintain the volume 20ness of the vector equilibrium. This supports the working assumption that the 120 LCD asymmetric tetrahedral volumes are quantitatively equal to the A or B Quanta Modules, being only a mild variation of shape. This effect is confirmed by the discovery that 15 of the 120 LCD Spherical Triangles equally and interiorly subdivide each of the eight spherical octahedron's triangular surfaces, which spherical octahedron is described by the threegreatcircle set of the 25 great circles of the spherical vector equilibrium. 
982.59 We may also assume that the pentagonalfaced dodecahedron, which is developed on exactly the same spherical icosahedron, is also another transformation of the same module quantation as that of the icosahedron's and the vector equilibrium's prime number five topological identity. 
982.60
Without any further developmental use of pi () we
may now state in
relation to the isotropic vector matrix synergetic system,
that:

Fig. 982.61 
982.61 There is realized herewith a succession of concentric, 12aroundone, closestpacked spheres, each of a tetra volume of five; i.e., of 120 A and B Quanta Modules omniembracing our hierarchy of nuclear event patternings. See Illus. 982.61 in the color section, which depicts the synergetics isometric of the isotropic vector matrix and its omnirational, loworder whole number, equilibrious state of the micromacro cosmic limits of nuclearly unique, symmetrical morphological relativity in their interquantation, intertransformative, intertransactive, expansivecontractive, axially rotative, operational field. This may come to be identified as the unified field, which, as an operationally transformable complex, is conceptualizable only in its equilibrious state. 
982.61A Cosmic Hierarchy of Omnidirectionallyphased Nuclearcentered, Convergentlydivergently Intertransformable Systems: There is realized herewith a succession of concentric, 12aroundone, closestpacked spheres omniembracing our hierarchy of nuclear event patternings. The synergetics poster in color plate 9 depicts the synergetics isometric of the isotropic vector matrix and its omnirational, loworderwhole number, equilibrious state of the macromicro cosmic limits of nuclearly unique, symmetrical morphological relativity in their interquantation, intertransformative, intertransactive, expansivecontractive, axially rotative, operational field. This may come to be identified as the unified field, which, as an operationally transformable complex, is conceptualized only in its equilibrious state. 
982.62
Table of Concentric, 12AroundOne, ClosestPacked
Spheres, Each of a
Tetra Volume of Five, i.e., 120 A and B Quanta Modules,
Omniembracing Our Hierarchy
of Nuclear Event Patternings. (See also Illus.
982.61
in drawings section.)

982.62A
Table of Concentric, 12aroundone, Closestpacked
Spheres
Omniembracing Our Hierarchy of Nuclear Event Patternings
(Revised):
* The spheric spin domain of the rhombic triacontahedron "sphere." 
982.63 Sphere and Vector Equilibrium: Sphere = vector equilibrium in combined fourdimensional orbit and axial spin. Its 12 vertexes describing six great circles and six axes. All 25 great circles circling while spinning on one axis produce a spinprofiling of a superficially perfect sphere. 
982.64 The vector equilibrium also has 25 great circles (see Sec. 450.10), of which 12 circles have 12 axes of spin, four great circles have four axes of spin, six great circles have six axes of spin, and three great circles have three axes of spin. (12 + 4 + 6 + 3 = 25) 
982.65 Vector equilibrium = sphere at equilibrious, ergo zero energized, ergo unorbited and unspun state. 
982.70 Hierarchy of Concentric Symmetrical Geometries: It being experimentally demonstrable that the number of A and B Quanta Modules per tetrahedron is 24 (see Sec. 942.10); that the number of quanta modules of all the symmetric polyhedra congruently cooccurring within the isotropic vector matrix is always 24 times their whole regulartetrahedralvolume values; that we find the volume of the nuclear sphere to be five (it has a volumetric equivalence of 120 A and B Quanta Modules); that the common prime number five topological and quantamodule value identifies both the vector equilibrium and icosahedron (despite their exclusively unique morphologies^{__}see Sec. 905, especially 905.55; that the icosahedron is one of the threeandonly prime structural systems of Universe (see Secs. 610.20 and 1011.30) while the vector equilibrium is unstable^{__}because equilibrious^{__}and is not a structure; that their quanta modules are of equal value though dissimilar in shape; and that though the vector equilibrium may be allspacefillingly associated with tetrahedra and octahedra, the icosahedron can never be allspacefillingly compounded either with itself nor with any other polyhedron: these considerations all suggest the relationship of the neutron and the proton for, as with the latter, the icosahedron and vector equilibrium are interexchangingly transformable through their common sphericalstate omnicongruence, quantitatively as well as morphologically. 
982.71
The significance of this unified field as defining
and embracing the minimum
maximum limits of the inherent nuclear domain limits
is demonstrated by the nucleus
concentric, symmetrical, geometrical hierarchy wherein
the rhombic dodecahedron
represents the smallest, omnisymmetrical, selfpacking,
allspacefilling, sixtetravolume,
uniquely exclusive, cosmic domain of each and every
closestpacked, unitradius sphere.
Any of the closestpacked, unitradius spheres, when
surrounded in closest packing by 12
other such spheres, becomes the nuclear sphere, to become
uniquely embraced by four
successive layers of surrounding, closestpacked, unitradius
spheres^{__}each of which four
layers is uniquely related to that nucleus^{__}with each
additional layer beyond four
becoming duplicatingly repetitive of the pattern of
unique surroundment of the originally
unique, first four, concentriclayered, nuclear set.
It is impressive that the unique nuclear
domain of the rhombic dodecahedron with a volume of
six contains within itself and in
nuclear concentric array:
(Footnote 8: For further suggestions of the relationship between the rhombic dodecahedron and the degrees of freedom see Sec. 426 537.10 954.47.) 
982.72 The domain limits of the hierarchy of concentric, symmetrical geometries also suggests the synergetic surprise of two balls having only one interrelationship; while three balls have three^{__}easily predictable^{__}relationships; whereas the simplest, ergo prime, structural system of Universe defined exclusively by four balls has an unpredictable (based on previous experience) sixness of fundamental interrelationships represented by the six edge vectors of the tetrahedron. 
982.73 The onequantum "leap" is also manifest when one vector edge of the volume 4 octahedron is rotated 90 degrees by disconnecting two of its ends and reconnecting them with the next set of vertexes occurring at 90 degrees from the previously interconnectedwith vertexes, transforming the same unitlength, 12vector structuring from the octahedron to the first threetriplebondedtogether (facetoface) tetrahedra of the tetrahelix of the DNARNA formulation. One 90degree vector reorientation in the complex alters the volume from exactly 4 to exactly 3. This relationship of one quantum disappearance coincident to the transformation of the nuclear symmetrical octahedron into the asymmetrical initiation of the DNARNA helix is a reminder of the disappearingquanta behavior of the always integrally endcohered jitterbugging transformational stages from the 20 tetrahedral volumes of the vector equilibrium to the octahedron's 4 and thence to the tetrahedron's 1 volume. All of these stages are rationally concentric in our unified operational field of 12aroundone closest packed spheres that is only conceptual as equilibrious. We note also that per each sphere space between closestpacked spheres is a volume of exactly one tetrahedron: 6  5 = 1. 
Next Section: 982.80 