1041.00 Superficial Poles of Internal Axes
1041.01 There are only three topological axes of crystallography. They are:
| Spin of diametrically opposite vertexes | |
| Spin of diametrically opposite mid-edges | = Three topological types of axes |
| Spin of diametrically opposite centers of face ares |
1041.10 Seven Axes of Truncated Tetrahedron
1041.11
The prime generation of the seven axes of symmetry are the seven unique perpendiculars to the faces of the seven possible truncations of the tetrahedron:
| 4 | original faces |
| 4 | triangular truncated vertexes |
| 6 | quadrilateral truncated edges |
| ----- | |
| 14 | faces of the truncated tetrahedron, which produce seven unique pairs of parallel faces whose axes, perpendicular to their respective centers of area, generate the seven axes of symmetry. (See Secs. 100.103-.05 and Fig. 1041.11.) |
1041.12
The seven unique axes of the three unique sets (4 + 4 + 6) producing the 14 planes of the truncated tetrahedron are also identifiable with:
- the 14 planes that bound and enclosingly separate all biological cells;
- the 14 facets interbonding all bubbles in the bubble complexes; and
- the 25 and 31 unique planes generated by the seven sets of foldable great circles, which are the only such foldably unbroken sets (i.e., the 3, 4, 6, and 12 sets of the vector equilibrium and the 6, 10, and 15 sets of the icosahedron).
1041.13
Various high frequencies of modular subdividings of the tetrahedron produce a variety of asymmetrical truncatabilities of the tetrahedron. The dynamics of symmetry may employ any seven sets of the 56 foldable-greatcircle variations of planar orientation. Thus it follows that both the biological cell arrays and the bubble arrays display vast varieties of asymmetries in their 14 enclosing planes, so much so that this set of interidentifiability with the 14 topological characteristics of the tetrahedron, the prime structural system of Universe, has gone unnoticed until now. (See Sec. 1025.14)
1042.00 Seven Axes of Symmetry
1042.01
Whatever subdivisions we may make of the tetrahedra, octahedra, and icosahedra, as long as there is cutting on the axes of symmetry, the components always come apart in whole rational numbers, for this is the way in which nature chops herself up.
1042.02
The four sets of unique axes of symmetry of the vector equilibrium, that is, the 12 vertexes with six axes; the 24 mid-edges with 12 axes; and the two different centers of area (a) the eight centers of the eight triangular areas with four axes, and (b) the six centers of the six square areas with three axes—25 axes in all—generate the 25 great circles of the vector equilibrium. These are the first four of the only seven cosmically unique axes of symmetry. All the great circles of rotation of all four of these seven different cosmic axes of symmetry which occur in the vector equilibrium go through all the same 12 vertexes of the vector equilibrium (see Sec. 450).
1042.03
The set of 15 great circles of rotation of the 30 mid-edge-polared axes of the icosahedron, and the set of 10 great circles of rotation of the icosahedron’s mid-faces, total 25, which 25 altogether constitute two of the three other cosmic axes of symmetry of the seven-in-all axes of symmetry that go through the 12 vertexes of the icosahedron, which 12 represent the askewedly unique icosahedral rearrangement of the 12 spheres of the vector equilibrium. Only the set of the seventh axis of symmetry, i.e., the 12-vertex- polared set of the icosahedron, go through neither the 12 vertexes of the icosahedron’s 12 corner sphere arrangement nor the 12 of the vector equilibrium phase 12-ball arrangement. The set of three axes (that is 12 vertexes, 30 mid-edges, and 20 centers of area) of the icosahedron produce three sets of the total of seven axes of symmetry. They generate the 25 twelve-icosa-vertex-transiting great circles and the six nontransiting great circles for a total of the 31 great circles of the icosahedron. These are the last three of the seven axes of symmetry.
1042.04
We note that the set of four unique axes of symmetry of the vector equilibrium and the fifth and sixth sets of axes of the icosahedron all go through the 12 vertexes representing the 12 spheres either (a) closest-packed around a nuclear sphere in the vector equilibrium, or (b) in their rearrangement without a nuclear sphere in the icosahedron. The six sets of unique cosmic symmetry transit these 12 spherical center corner vertexes of the vector equilibrium and icosahedron; four when the tangential switches of the energy railway tracks of Universe are closed to accommodate that Universe traveling; and two sets of symmetry when the switches are open and the traveling must be confined to cycling the same local icosahedron sphere. This leaves only the seventh symmetry as the one never going through any of those 12 possible sphere-to- sphere tangency railway bridges and can only accommodate local recycling or orbiting of the icosahedron sphere.
1042.05
The seven unique cosmic axes of symmetry describe all of crystallography. They describe the all and only great circles foldable into bow ties, which may be reassembled to produce the seven, great-circle, spherical sets (see Secs. 455 and 457).
| Vector Equilibrium | Axes of Symmetry | ||
|---|---|---|---|
| (squares) 3 | #1 | ||
| (triangles) 4 | #2 | ||
| (vertexes) 6 | #3 | ||
| (midedges) 12 | #4 | ||
| --------- | |||
| 25* | all go through the same 12 vertexes of vector equilibrium and icosahedron | ||
| Icosahedron | |||
| (faces) 10 | #5 | ||
| 25* | |||
| (midedges) 15 | #6 | ||
| (vertexes) 6 | #7 | ||
| --------- | |||
| 31 | |||
| 25 | |||
| 31 | |||
| ------- | |||
| 56 |
1043.00 Transformative Spherical Triangle Grid System
1043.01
All the great circles of all the seven axes of symmetry together with all great- circle-trajectory interactions can be reflectively confined and trigonometrically equated with only one of the icosahedral system’s 120 similar right-spherical triangles (of 90, 60, and 36 degrees, in contradistinction to the right-planar triangle of 90-, 60-, and 30-degree corners). (See Sec. 905.60.) The rational spherical excess of six degrees (of the icosahedron’s 120—60 plus and 60 minus—similar tetrahedral components) is symmetrically distributed to each of the three central and three surface angles of each of the 120 tetrahedral components of the spherical icosahedron.
1043.02
This sixness phenomenon tantalizingly suggests its being the same transformative sixness as that which is manifest in the cosmically constant sixfoldedness of vectors of all the topological accountings (see Secs. 621.10 and 721); and in the sixness of equieconomical alternative degrees of freedom inherent in every event (see Sec. 537.10); as well as in the minimum of six unique interrelationships always extant between the minimum of four “star events” requisite to the definitive differentiation of a conceptual and thinkable system from out of the nonunitarily conceptual but inherently finite Universe, because of the latter’s being the aggregate of locally finite, conceptually differentiable, minimum-system events (see Secs. 510 and 1051.20).
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 | EULER |
| edges | |
| angles | |
| insideness & outsideness | SYNERGETICS |
| convexity & concavity | |
| axis of spin |
1044.02
Euler discovered and developed the principle of modern engineering’s structural analysis. 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:
- 4th aspect (12 cases): the 12 unique, trigonometrically integral, intercovariant vertex angles of the minimum system.
- 5th aspect (two cases): ultravisible macrocosmic rest-of-Universe outsideness and infravisible rest-of-Universe insideness separated by the considered system; the insideness is all the integral otherness, and the outsideness is the as-yet-unconsidered irrelevance otherness.
- 6th aspect (two cases): the multiplicative twoness of the divergent convexity and convergent concavity; there are two manifestations of multiplicative twoness, (a) and (b) (see Secs. 223.05-09), both of which make unity plural and at minimum two: (a) the always and only inseparable and co-occurring concavity and convexity of all systems, and (b) the always and only inseparable convergence to and divergence from system center.
- 7th aspect (two cases): the additive twoness of the two vertexes always extracted from the system’s total inventory of vertexes to serve as the poles of the system’s neutral axis of spin.
- 8th aspect: the sum of the angles externally surrounding the vertexes of any system will always equal 720 degrees less than the number of external vertexes of the system multiplied by 360 degrees.
- 9th aspect: the sum of the angles around all the external vertexes of any system will always be evenly divided by 720 degrees, which is the angular description of one tetrahedron.
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 corner 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.