[1053.801053.85 Growth and Decay Scenario]
1053.802 Topology enumerates the criticalproximitybonded pairs of "points" as constituting only one point and not as an almost tangent two. Topological accounting is confined to only superficially visible characteristics of systems. (See Sec. 262.02.) 
1053.803 We learn experientially that lines are trajectories (Sec. 521.20), that two events and their trajectories cannot pass through the same point at the same time (Secs. 517.0106), and that when we have such conflict or transit interference, they result in smashes (always separating each of the intersmashing bodies into a plurality of smaller systems, not dirt or dust), plungeins such as meteors plunging into Earth (to form more complex systems), refractions, reflections, or criticalproximity interrattractiveness cotravelings (Earth and Moon). When we do not have interference conflicts but we have two independent event trajectories converging to pass "near" one another only at a precessionally criticalcourserefracting, massinterattractive distance, they may converge and diverge in a twist vertex exit (see Secs. 921.15 and 942.12). The term vertex embraces all of the foregoing systemfurnished, localfocal, event cases. 
1053.810 The vector equilibrium consists of eight tetrahedra each of which is edge bonded; i.e., vertexially doubleinterbonded with three others, with each of their pretime size internal vertexes theoretically congruent as eightinone. Each of the pretimesize vector equilibrium's eight tetrahedra has six vector edges (6 × 8 = 48). (There are 24 internal and 24 external vector edges, 48 vector edges in all.) Each of the eight tetrahedra has four vertexes (4 × 8 = 32), and in each of the tetrahedra three of these vertexes are external (3 × 8 = 24): There are thus 12 externally paired sets (24/2 = 12) of visible vertexes. Three of each of the eight tetrahedra's vector edges (3 × 8 = 24) are displayed on the outside of the vector equilibrium. (Compare Sec. 1033.020.) 
1053.811 There are 24 external vector edges of the vector equilibrium (8 × 3 = 24). The other three vector edges of each of the eight tetrahedra are arrayed inwardly as 24 internal edges (8 × 3 = 24), but these inwardly arrayed vector edges of the eight tetrahedra, being doublebonded or hinged together, appear as only 12 radial spokes of the vector equilibrium, which has 24 separate vectors in its four closed chordal rims of the four greatcircle planes of the tetrahedra's four dimensionality; these four great circles produce the zerovolume tetrahedron. (See Sec. 441.) 
1053.812 Nature never stops or even pauses at dead center. Nature contracts convergently to the center of its nuclear sphere, where each of its frequencytuned integrities selfinterfere convergently and react reflectively^{__}ergo, omnidivergently^{__}from their own terminally convergent selffrequency interferings. Unity is plural and at minimum two. (See Secs. 905.11 and 1070.) 
1053.82 Life and Death 
1053.824 Apprehension is the physical brain's coordinate storing of all the special case, physically sensed information of otherness, integral (the child's thumb sucked by its mouth) or separate (the mother's udder sucked by the child's mouth.) Comprehension is the metaphysical mind's discovery of the meaningful interrelationship between the special case information data that are neither implicit in, nor inferred by, any of the specialcase information data when taken only separately^{__}the meaning discovered by mind being the generalized principles manifest exclusively by the interrelationship variables and constants. Awareness means apprehending while also intuitively comprehending that the excitement over the novelty of the incoming information is significant because possibly pregnant with meaningful principles. (Compare Sec. 526.18.) 
1053.83 Positive Visible and Integral Invisible 
1053.84 Cay and Decay 
1053.845 In the generalized (subfrequency) nucleusembracing, convergentdivergent, bivalent tetravolume vector equilibrium of frequency one, its tetravolume is 20. VE^{1} = 20. 
1053.846 In the generalized (subfrequency) nucleusembracing, convergentdivergent vector equilibrium of frequency two, the tetravolume is 160. VE^{2} = 160. (See Sec. 966.05 and Fig. 966.05B.) 
1053.847 What must be remembered in considering all the foregoing is that unity is plural and at minimum two, as elucidated in Secs. 905.11 and 1070; wherefore the zero frequency vector equilibrium, the VE^{0} of "apparent" tetravolume 2 1/2, has an inherent but invisible double value that will have an operational resource effectiveness of 5, 2 1/2 of which is convergently effective and 2 1/2 divergently effective. This produces the state of equilibrium whose untenability induces cosmic resonance. 
1053.848 In the symmetrical doubling of linear (radial) dimension the surface area increases four times and the volume eight times their original magnitude. In the case of the nuclear (one sphere) vector equilibrium with radius = 1 and volume = 2 1/2, when surrounded with 12 closestpacked, uniradius spheres and when the center of the nuclear sphere is connected to the respective centers of the 12 surrounding spheres, the distance between the center of the nuclear sphere and the center of any one of its 12 surrounding spheres is equal to 2 radii, or one diameter of the uniradius spheres. With radius 2, 2 1/2 × 8 = 20. (Compare Sec. 1033.63. ) 
1053.849
Table: Initial Frequencies of Vector Equilibrium:

1053.85
Inventory of Alternatives to Positive

1054.00 Relationship of Gibbs to Euler 
1054.30
Synergetic Integration of Topology and Quanta: Synergetics'
"breakthrough" integration of Euler's topology and Willard
Gibbs' phase rule is explained
by the number of intertetrahedral bonds:

1054.31 The rigid ice stage is characterized by load concentration, no degrees of freedom, and slow creep. The flexible, fluid stage is characterized by hingebonding, load distribution, one degree of freedom, and noncompressibility. The flexible, fluid vapor stage is characterized by universal jointing, load distribution, six degrees of freedom, and compressibility. 
1054.32
Median unity is two, therefore unity plus two equals
four.

Fig. 1054.40 
1054.40 Topology and Phase (see Table 1054.40) 
1054.50 Polyhedral Bonding: Willard Gibbs' phase rule treats with the states of the environment you can sense with your eyes closed: crystallines, liquids, gases, and vapors. Euler's points, lines, and areas are visually described, but they too could be tactilely detected (with or without fingers). 
1054.51 The mathematicians get along synergetically using Euler's topology alone. It is the chemists and physicists who cannot predict synergetically without using Gibbs' phase rule. 
1054.52 Euler deals with the superficial aspects of polyhedra: of visual conceptuality. He deals only with the convex surfaces of polyhedral systems. Euler deals with unit, integral, single polyhedra, or with their subaspects. He is not concerned with the modus operandi of the associabilities or disassociabilities of a plurality of polyhedra. 
1054.53 But Gibbs unknowingly deals with polyhedra that are composited of many polyhedra, i.e., compounds. He does not think or talk about them as polyhedra, but we find the connection between Euler and Gibbs through the polyhedral bonding in respect to Euler's aspects. Euler's lines are double bonds, i.e., hinges. Euler's vertexes are single bonds. Euler's areas are triple bonds. Gibbs accommodates the omnidirectional system complementations of the other senses^{__}thermal, tactile, aural, and olfactory^{__}not just associatively, but radiationally. Gibbs brings in time. Time is tactile. Time is frequency. Our pulses measure its passing. 
1054.54 People see things move only relative to other things and feel small vibrations when they cannot see motion. The tactile feels angular promontories or sinuses with the fingers or body. Sinus means "without"^{__} "nothing," invisible, ergo, nonidentified by Euler. The frequencies we call heat are tactilely sensed. We have radiationfrequency tunability range. Our skin structuring is tuned to frequencies beyond the eyetunable range, i.e., to ultraviolet and infrared. 
1054.55 Euler did not anticipate Gibbs. Gibbs complements Euler^{__}as does synergetics' identification of the two excess vertexes as constituting the axis of conceptual observation in respect to all independent, individual orientations of all systems and subsystems; i.e., quantum mechanics' abstract, nonspinnable "spin." 
1054.56 We find Euler and Gibbs coming together in the vertexial bonds, or polyhedral "corners," or point convergency of polyhedral lines. The bonds have nothing to do with the "faces" and "edges" they terminally define. Two bonds provide the hinge, which is an edge bonding. One bond gives a universal joint. Triple or areal bonding gives rigidity. 
1054.57 Massinterattraction is always involved in bonding. You may not have a bond without interattraction, mass or magnetic (integral or induced), all of which are precessional effects. As Sun's pull on Earth produces Earth orbiting, orbiting electrons produce directional field pulls. This was not considered by Euler because he was dealing only with aspects of a single system. 
1054.58 Gibbs requires the massinterattraction without saying so. Mass interattraction is necessary to produce a bond. Gases may be tetrahedrally bonded singly, corner to corner, or as a universal joint. Gibbs does not say this. But I do. 
Next Section: 1054.60 