1005.24 Seen in their skyreturning functioning as recirculators of water, the ecological patterning of the trees is very much like a slowmotion tornado: an evoluting involuting pattern fountaining into the sky, while the roots reversefountain reaching outwardly, downwardly, and inwardly into the Earth again once more to recirculate and once more again^{__}like the pattern of atomic bombs or electromagnetic lines of force. The magnetic fields relate to this polarization as visually witnessed in the Aurora Borealis. (Illus. 505.41) 
1005.51 The very word comprehending is omniinterprecessionally synergetic. 
1005.52 The eternal is omniembracing and permeative; and the temporal is linear. This opens up a very high order of generalizations of generalizations. The truth could not be more omniimportant, although it is often manifestly operative only as a linear identification of a specialcase experience on a specialized subject. Verities are semi specialcase. The metaphor is linear. (See Secs. 217.03 and 529.07.) 
1005.55 The dictionarylabel, special cases seem to go racing by because we are now having in a brief lifetime experiences that took aeons to be differentially recognized in the past. 
1005.56 The highest of generalizations is the synergetic integration of truth and love. 
1005.612 When a person dies, all the chemistry remains, and we see that the human organism's same aggregate quantity of the same chemistries persists from the "live" to the "dead" state. This aggregate of chemistries has no metaphysical interpreter to communicate to self or to others the aggregate of chemical rates of interacting associative or disassociative proclivities, the integrated effects of which humans speak of as "hunger" or as the need to "go to the toilet." Though the associative intake "hunger" is unspoken metaphysically after death, the disassociative discard proclivities speak for themselves as these chemicalproclivity discard behaviors continue and reach selfbalancing rates of progressive disassociation. What happens physically at death is that the importing ceases while exporting persists, which produces a locally unbalanced^{__}thereafter exclusively exporting^{__}system. (See Sec. 1052.59.) 
1006.10 Omnitopology Defined 
1006.12 The closestpacked symmetry of uniradius spheres is the mathematical limit case that inadvertently "captures" all the previously unidentifiable otherness of Universe whose inscrutability we call "space." The closestpacked symmetry of uniradius spheres permits the symmetrically discrete differentiation into the individually isolated domains as sensorially comprehensible concave octahedra and concave vector equilibria, which exactly and complementingly intersperse eternally the convex "individualizable phase" of comprehensibility as closestpacked spheres and their exact, individually proportioned, concaveinbetweenness domains as both closest packed around a nuclear uniradius sphere or as closest packed around a nucleusfree prime volume domain. (See illustrations 1032.30 and 1032.31.) 
1006.23 In omnitopology, each of the lines and vertexes of polyhedrally defined conceptual systems have their respective unique areal domains and volumetric domains. (See Sec. 536.) 
1006.30 Vector Equilibrium Involvement Domain 
Fig. 1006.32 
1006.32 We learn from the complex jitterbugging of the VE and octahedra that as each sphere of closestpacked spheres becomes a space and each space becomes a sphere, each intertransformative component requires a tetravolume12 "cubical" space, while both require 24 tetravolumes. The total internalexternal closestpackedspheresandtheir interstitialspaces involvement domains of the unfrequenced 20tetravolume VE is tetravolume24. This equals either eight of the nuclear cube's (unstable) tetravolume3 or two of the rhombic dodecahedron's (stable) tetravolume6. The two tetravolume12 cubes or four tetravolume6 dodecahedra are intertransformable aspects of the nuclear VE's localinvolvement domain. (See Fig. 1006.32.) 
1006.33 The vector equilibrium at initial frequency, which is frequency^{2}, manifests the fifthpowering of nature's energy behaviors. Frequency begins at two. The vector equilibrium of frequency^{2} has a prefrequency inherent tetravolume of 160 (5 × 2^{5} = 160) and a quantamodule volume of 120 × 24 = 1 × 3 × 5 × 2^{8} nuclearcentered system as the integrated product of the first four prime numbers: 1, 2, 3, 5. Whereas a cube at the same frequency accommodates only eight cubes around a nonnucleated center. (Compare Sec. 1033.632) 
1006.35
With reference to our operational definition of a sphere
(Sec.
224.07), we
find that in an aggregation of closestpacked uniradius
spheres:

1006.37 For other manifestations of the vector equilibrium involvement domain, review Sections 415.17 (Nucleated Cube) and 1033 (Intertransformability Models and Limits), passim. 
1006.40 Cosmic System Eightdimensionality 
1006.41 We have a cosmically closed system of eightdimensionality: four dimensions of convergent, syntropic conservation + 4, and four dimensions of divergent, entropic radiation  4 intertransformabilities, with the noninsideoutable, symmetric octahedron of tetravolume 4 and the polarized semiasymmetric Coupler of tetravolume 4 always conserved between the interpulsative 1 and the rhombic dodecahedron's maximum involvement 6, (i.e., 1 + 4 + 1); ergo, the always doublevalued^{__}2^{2} ^{__}symmetrically perfect octahedron of tetravolume 4 and the polarized asymmetric Coupler of tetravolume 4 reside between the convergently and divergently pulsative extremes of both maximally aberrated and symmetrically perfect (equilibrious) phases of the generalized cosmic system's always partiallytunedinandtunedout eightdimensionality. 
1007.10 Omnitopology Compared with Euler's Topology 
1007.11 While Euler discovered and developed topology and went on to develop the structural analysis now employed by engineers, he did not integrate in full potential his structural concepts with his topological concepts. This is not surprising as his contributions were as multitudinous as they were magnificent, and each human's work must terminate. As we find more of Euler's fields staked out but as yet unworked, we are ever increasingly inspired by his genius. 
1007.12 In the topological past, we have been considering domains only as surface areas and not as uniquely contained volumes. Speaking in strict concern for always omnidirectionally conformed experience, however, we come upon the primacy of topological domains of systems. Apparently, this significance was not considered by Euler. Euler treated with the surface aspects of forms rather than with their structural integrities, which would have required his triangular subdividing of all polygonal facets other than triangles in order to qualify the polyhedra for generalized consideration as structurally eternal. Euler would have eventually discovered this had he brought to bear upon topology the same structural prescience with which he apprehended and isolated the generalized principles governing structural analysis of all symmetric and asymmetric structural components. 
1007.13 Euler did not treat with the inherent and noninherent nuclear system concept, nor did he treat with totalsystem angle inventory equating, either on the surfaces or internally, which latter have provided powerful insights for further scientific exploration by synergetical analysis. These are some of the differences between synergetics and Euler's generalizations. 
1007.14 Euler did formulate the precepts of structural analysis for engineering and the concept of neutral axes and their relation to axial rotation. He failed, however, to identify the structural axes of his engineering formulations with the "excess twoness" of his generalized identification of the inventory of visual aspects of all experience as the polyhedral vertex, face, and line equating: V + F = L + 2. Synergetics identifies the twoness of the poles of the axis of rotation of all systems and differentiates between polar and nonpolar vertexes. Euler's work, however, provided many of the clues to synergetics' exploration and discovery. 
1007.15 In contradistinction to, and in complementation of, Eulerian topology, omnitopology deals with the generalized equatabilities of a priori generalized omnidirectional domains of vectorially articulated linear interrelationships, their vertexial interference loci, and consequent uniquely differentiated areal and volumetric domains, angles, frequencies, symmetries, asymmetries, polarizations, structuralpattern integrities, associative interbondabilities, intertransformabilities, and transformativesystem limits, simplexes, complexes, nucleations, exportabilities, and omniinteraccommodations. (See Sec. 905.16.) 
1007.16 While the counting logic of topology has provided mathematicians with great historical expansion, it has altogether failed to elucidate the findings of physics in a conceptual manner. Many mathematicians were content to let topology descend to the level of a fascinating game^{__}dealing with such Moebiusstrip nonsense as pretending that strips of paper have no edges. The constancy of topological interrelationships^{__}the formula of relative interabundance of vertexes, edges, and faces^{__}was reliable and had a great potential for a conceptual mathematical strategy, but it was not identified operationally with the intertransformabilities and gaseous, liquid, and solid interbondings of chemistry and physics as described in Gibbs' phase rule. Now, with the advent of vectorial geometry, the congruence of synergetic accounting and vectorial accounting may be brought into elegant agreement. 
1007.20 Invalidity of Plane Geometry 
1007.21 We are dealing with the Universe and the difference between conceptual thought (see Sec. 501.101) and nonunitarily conceptual Universe (see Scenario Universe, Sec. 320). We cannot make a model of the latter, but we can show it as a scenario of meaningfully overlapping conceptual frames. 
1007.22
About 150 years ago Leonhard Euler opened up the great
new field of
mathematics that is topology. He discovered that all
visual experiences could be treated as
conceptual. (But he did not explain it in these words.)
In topology, Euler says in effect, all
visual experiences can be resolved into three unique
and irreducible aspects:

1007.23 In topology, then, we have a unique aspect that we call a line, not a straight line but an event tracery. When two traceries cross one another, we get a fix, which is not to be confused in any way with a noncrossing. Fixes give geographical locations in respect to the system upon which the topological aspects appear. When we have a tracery or a plurality of traceries crossing back upon one another to close a circuit, we surroundingly frame a limited view of the omnidirectional novents. Traceries coming back upon themselves produce windowed views or areas of novents. The areas, the traces, and the fixes of crossings are never to be confused with one another: all visual experiences are resolved into these three conceptual aspects. 
1007.24 Look at any picture, point your finger at any part of the picture, and ask yourself: Which aspect is that, and that, and that? That's an area; or it's a line; or it's a crossing (a fix, a point). Crossings are loci. You may say, "That is too big to be a point"; if so, you make it into an area by truncating the corner that the point had represented. You will now have two more vertexes but one more area and three more lines than before. Euler's equation will remain unviolated. 
1007.25 A circle is a loop in the same line with no crossing and no additional vertexes, areas, or lines. 
1007.26 Operationally speaking, a plane exists only as a facet of a polyhedral system. Because I am experiential I must say that a line is a consequence of energy: an event, a tracery upon what system? A polyhedron is an event system separated out of Universe. Systems have an inside and an outside. A picture in a frame has also the sides and the back of the frame, which is in the form of an asymmetrical polyhedron. 
1007.27 In polyhedra the number of V's (crossings) plus the number of F's, areas (noventsfaces) is always equal to the number of L's lines (continuities) plus the number 2. If you put a hole through the system^{__}as one cores an apple making a doughnutshaped polyhedron^{__}you find that V + F = L. Euler apparently did not realize that in putting the hole through it, he had removed the axis and its two poles. Having removed two axial terminal (or polar) points from the inventory of "fixes" (locivertexes) of the system, the V + F = L + 2 equation now reads V + F = L, because two V's have been deducted from the inventory on the left side of the equation. 
1007.28 Another very powerful mathematician was Brouwer. His theorem demonstrates that if a number of points on a plane are stirred around, it will be found after all the stirring that one of the points did not move relative to all the others. One point is always the center of the total movement of all the points. But the mathematicians oversimplified the planar concept. In synergetics the plane has to be the surface of a system that not only has insideness and outsideness but also has an obverse and re exterior. Therefore, in view of Brouwer, there must also always be another point on the opposite side of the system stirring that also does not move. Every fluidly bestirred system has two opposed polar points that do not move. These two polar points identify the system's neutral axis. (See Sec. 703.12.) 
1007.29 Every system has a neutral axis with two polar points (vertexesfixes). In synergetics topology these two polar points of every system become constants of topological inventorying. Every system has two polar vertexes that function as the spin axis of the system. In synergetics the two polar vertexes terminating the axis identify conceptually the abstract^{__}supposedly nonconceptual^{__}function of nuclear physics' "spin" in quantum theory. The neutral axis of the equatorially rotating jitterbug VE proves Brouwer's theorem polyhedrally. 
Fig. 1007.30 
1007.30
When you look at a tetrahedron from above, one of its
vertexes looks like
this: (See Fig. 1007.30)
You see only three triangles, but there is a fourth underneath that is implicit as the base of the tetrahedron, with the Central vertex D being the apex of the tetrahedron. The crossing point (vertexfix) in the middle only superficially appears to be in the same plane as ABC. The outer edges of the three triangles you see, ACD, CDB, ADB, are congruent with the hidden base triangle, ABC. Euler assumed the three triangles ACD, CDB, ADB to be absolutely congruent with triangle ABC. Looking at it from the bird'seye view, unoperationally, Euler misassumed that there could be a nonexperienceable, nothickness plane, though no such phenomenon can be experientially demonstrated. Putting three points on a piece of paper, interconnecting them, and saying that this "proves" that a no thickness, nonexperiential planar triangle exists is operationally false. The paper has thickness; the points have thickness; the lines are atoms of lead strewn in linear piles upon the paper. 
1007.31 You cannot have a somethingnothingness, or a plane with no thickness. Any experimental event must have an insideness and an outsideness. Euler did not count on the fourth triangle: he thought he was dealing with a plane, and this is why he said that on a plane we have V + F = L + 1 . When Euler deals with polyhedra, he says "plus 2." In dealing with the false plane he says "plus 1." He left out "1" from the righthand side of the polyhedral equation because he could only see three faces. Three points define a minimum polyhedral facet. The point where the triangles meet in the center is a polyhedral vertex; no matter how minimal the altitude of its apex may be, it can never be in the base plane. Planes as nondemonstrably defined by academic mathematicians have no insideness in which to get: ABCD is inherently a tetrahedron. Operationally the fourth point, D, is identified or fixed subsequent to the fixing of A, B, and C. The "laterness" of D involves a time lag within which the constant motion of all Universe will have so disturbed the atoms of paper on which A, B, and C had been fixed that no exquisite degree of measuring technique could demonstrate that A, B, C, and D are all in an exact, socalled flatplane alignment demonstrating ABCD to be a zeroaltitude, nothicknessedged tetrahedron. 
Next Section: 1008.10 