987.070 Topological Minima

987.071 In synergetics all topological characteristics are interconformationally conceptual independent of size; for instance, a vertex is one of the convergence loci of a system’s inherent plurality of conceptual interrelationships.

987.072 Since vertexes are omnidimensional, system topology deals with the loci of interrelationship convergences at anyone of the system set of defining lociwith a closest-packing-of-spheres-imposed maximum of 12 unit-radius convergences around any one unit-radius locus sphere. In the latter case vertexes may be predominantly identified as spheres of unit radius an may identify a prime nucleated system.

987.073 The minimum conceptual system in Universe is the regular tetrahedron, which consists of a minimum of four vertexes that can be represented as four approximately intertangent, equiradius spheres. Vertex-representing spheres do not occur in Universe or become conceptually considerable in sets of less than four. (This process is described at Secs. 100.331 and 411. minimum of four successive events and three intervals is required to define a frequency cycle; see Sec. 526.23.) In the same way linesor edges, as they are spoken of in topologyoccur only in sets of six, as the most economic interrelationships of vertexes of polyhedral systems.

987.074 The minimum system in Universe is the tetrahedron; its unitradius spheres at each of the four vertexes have a minimum of six intersystem vertexial relationships. We have learned that topological system vertex interrelationships always occur in sets of six. The formula for the number of system interrelationships is\sfrac{n^2 - n}{2}whereinnis the number of system vertexes (or unit radius spheres). A tetrahedron has four vertexes: 4²=16, minus n4 = 12, divided by 2 = 6i.e., the number of unique vertex interrelationship lines of the minimum structural systemthe tetra hedronis six.

987.075 Although Alfred North Whitehead and Bertrand Russell did not recognize the full conceptual implications, their “new mathematics” of set theory and empty sets were tour de force attempts by the leading abstract non-conceptual mathematicians of their day to anticipate the inevitable historical convergence of their mathematics with the inherently conceptual topology of Euler, as well as with the phase rule of Gibbs in chemistry, the simplified quantum mechanics of Dirac in physics, and the homogenizing biochemistry a nd physics of virology’s DNA-RNA design programmingall remotely but inexorably rendezvousing with Boltzmann’s, Einstein’s, and Hubble’s astrophysics and cosmology to constitute unitary science’s unitary self-regenerative, untenably equilibrious, cosmic-coordinate system to be embraced and accommodated by the epistemography of synergetics.

987.076 What are known in the terminology of topology as faces the polyhedron’s hedrasides or facetsare known in synergetics as windows, being the consequences of system-vertex interrelationship lines framing or viewing “windows of nothingness”windows opening to a nonconvergence, to nonrelatedness, to the untuned-in. Nothingness is the at-present-untuned-ininformation of each special case individual’s special local-in-Universe, momentary, tuned-in, preoccupying consideration. Vertexes are tuned-in; hedraare untuned-in, ergo out. Hedrafaces are system outs.

987.077 Unit radius spheres are unit-wavelength, tuned-in, event loci; topological faces (Greek *hedra)*are all the windows looking out upon all the rest of the Universe’s presently-untuned-in information in respect to the considered or tuned-in system.

987.078 Since two system-interrelationship lines (vectorially energetic in pure principle) cannot pass through the s ame point at the same time, the windows’ “corners” are always superimposed time-crossing aspectsone crossing behind or in front of, but not touching, the other. The topological windows of synergetics are polygonal aspects of the system’s interrelationships and not of physical lines.

987.079 Synergetics’ experimentally produced, minimum-structural-system subdivisions of Universe have four tuned-in vertexial loci, four windows looking toward all the untuned-in complementary balance of Universe, and six vertexial interrelationship vector lines, with all the latter occurring as outermost system features. The minimum nonnucleated structural system does not require internal vertexes.

At c: Gasesare monovalent, single-bonded, omniflexible, inadequate-interattraction, separatist, compressible. At b:Liquidsare bivalent, double-bonded, hinged, flexible, viscous integrity. At a:Rigidsare trivalent, triple-bonded, rigid, highest tension coherence.