BUCKYVERSE

1044.00 Minimum Topological Aspects

•[1044.00-1044.13 Minimum Topology Scenario]

1044.01 Euler + Synergetics: The first three topological aspects of all minimum systems—vertexes, faces, and edges—were employed by Euler in his formula V + F =E + 2. (See Table 223.64 and Sec. 505.10.) Since synergetics’ geometry embraces nuclear and angular topology, it adds four more minimum aspects to Euler’s inventory of three:

vertexes faces edges

EULER

angles

insideness & outsideness convexity & concavity axis of spin

SYNERGETICS

1044.02 Euler discovered and developed the principle of modern engineering’s structural analysis (Sec. 1007.04). He recognized that whereas all statically considered objects have a center of gravity, all dynamically considered structural components of buildings and machinery—no matter how symmetrically or asymmetrically conformed—always have a uniquely identifiable neutral axis of gyration. Euler did not think of his topology as either static or dynamic but as a mathematically permitted abstraction that allowed him to consider only the constant relative abundance of vertexes, faces, and edges isolated within a local area of a nonsystem. (The local consideration of the constant relative abundance of vertexes, faces, and edges applies to polyhedra as well as to cored-through polyhedra.)

1044.03 Euler’s analysis failed to achieve the generalization of angles (whose convergence identified his corners), the complementary insideness and outsideness, and the convexity-concavity of all conceptual experience. Being content to play his mathematical game on an unidentified surface, he failed to conceive of systems as the initial, all-Universe separators into the tunably relevant, topologically considered set. Euler’s less-than-system abstraction also occasioned his failure to identify the spin axis of any and all systems with his axis of gyration of physical objects; thus he also failed to realize that the subtraction of two vertexes from all systems for assignment as polar vertexes of the spin axis was a failure that would necessitate the “plus two” of his formula V + F = E + 2.

1044.04 Any and all conceptuality and any and all think-about-ability is inherently systemic (see Secs. 905.01-02). Systemic conceptuality and think-about-ability are always consequent only to consideration. Consideration means bringing stars together so that each star may be then considered integrally as unity or as an infrasystem complex of smaller systems.

1044.05 A system consists at minimum of four star events (vertexes) with four nothingness window facets and six lines of unique four-star interrelationships. As in synergetics’ 14 truncation faces, Euler’s three aspects result in 14 cases:

4 vertexes + 4 faces + 6 edges = 14 cases.

1044.06 Synergetics further augments Euler’s inventory of three topological aspects (14 cases) with six additional and primitively constant topological aspects:

1044.07 The total of nine minimum topological aspects consists of three from Euler (14 cases) plus synergetics’ inventory of six additional aspects, with 12 angular cases and six nuclear cases for a total of 18 synergetics cases. The 14 Euler cases and the 18 synergetics cases provide a total of 32 minimum topological cases.

1044.08 Topological analysis permits the generalization of all structuring in Universe as systemic.

1044.09 What we speak of as substance—a planet, water, steam, a cloud, a speck, or a pile of dust—always has both insideness and outsideness. A substance is a single system or a complex of neighboring interbonded or critical-proximity systems. Substances have inherent insideness “volumes.”

1044.10 An Earthian observer can point in a describable compass direction and a describable angle of elevation toward the location in the sky where the contrails of two differently directioned jet air transports traveling at different altitudes appear to him to cross one another. Because they are flown at different altitudes, the “to-him” crossing does not mean that they touch one another; it is simply a moment when their two separate trajectories are nearest to one another. What the observer points to is a “nearest-to-one-another” moment. The observer points to an interrelationship event, which is not part of either contrail considered only by itself. This directionally identifiable interrelationship event is known as a “fix.” (See Sec. 532.02.)

1044.11 The four comer fixes of an environmental tetrahedron may be pointed toward with adequate communicability to visually inform others of a specific tetrahedral presence. This is accomplished as follows: Two sky fixes must have a most economical linear interrelatedness but no insideness. Three sky fixes define a triangle between whose three edge-defining, interrelationship lines is described a plane that has no altitude—ergo, no insideness. Then the triangle described by the three sky fixes plus the position of the observer on the ground altogether describe the four corners of a tetrahedron that has six lines of observably inductable interrelatedness defining four triangular planes that observably divide all Universe into the included insideness and the excluded outsideness.

1044.12 One fix does not have insideness. Two fixes define a no- insideness linear relationship. Three fixes define a no-insideness plane. Four fixes define an insideness-including and outsideness-excluding tetrahedron, which is the minimum cosmic system and which cannot have less than 32 unique and differentially describably generalized cases of the nine irreducible- in-number unique topological aspects of the minimum system, but which in special frequenced cases may have more.

1044.13 Although not enumerated topologically (because unconsidered and because nonsimultaneously considerable) there are—in addition to the nine aspects and 32 cases—two additional ultimate conceptual aspects of the complementary macro- and microremainder of the physical Universe: all the as-yet-undiscovered—ergo, unconsidered—special cases as an epistemographic complementary to all the as-yet-undiscovered—ergo, unconsidered—generalized principles.