The top diagram is mine, entitled 'Trident Earth' (Earth as nuclear sphere surrounded by partially stellated duo-cuboctahedron with incidental owl imagery due to unfolding the top and bottom tets like flower petals opening), the next image is Bucky Fullers 'Cosmic Hierarchy' without the relative volume data.
"The term cybernetics stems from the Greek (kybernetes, steersman, governor,
pilot, or rudder — the same root as government)."
Does that mean Trim-Tab?
"Systems theory is an interdisciplinary theory about the nature of complex
systems in nature, society, and science, and is a framework by which one can
investigate and/or describe any group of objects that work together to produce
Is that Synergetics?
Synarctics: supplement to Synergetics by Bucky Fuller
Synergetics, tet cube
Fig. 986.210 Diagonal of Cube as Unity in Synergetic Geometry: In synergetic geometry mensural unity commences with the tetra edge as prime vector. Unity is taken not from the cube edge but from the edge of one of the two tetra that structure it. (Compare Fig. 463.01.) Proportionality exactly known to us is not required in nature's structuring. Parts have no existence independent of the polyhedra they constitute.
Synergetics, chefs' hat
416.02 If you next take two triangles, each made of three balls in closest packing, and twist one of the triangles 60 degrees around its center hole axis, the two triangular groups now may be nested into one another with the three spheres of one nesting in the three intersphere tangency valleys of the other. We now have six spheres in symmetrical closest packing, and they form the six vertexes of the octahedron. This twisting of one set to register it closepackedly with the other, is the first instance of two pairs internested to form the tetrahedron, and in the next case of the two triangles twisted to internestability as an octahedron, is called interprecessing of one set by its complementary set.
416.03 Two pairs of two-layer, seven-ball triangular sets of closestpacked spheres precess in a 60-degree twist to associate as the cube. (See Fig. A, illustration 416.01.) This 14-sphere cube is the minimum cube that may be stably produced by closest-packed spheres. While eight spheres temporarily may be tangentially glued into a cubical array with six square hole facades, they are not triangulated; ergo, are unstructured; ergo, as a cube are utterly unstable and will collapse; ergo, no eight-ball cube can be included in a structural hierarchy.
416.04 The two-frequency (three spheres to an edge), two-layer tetrahedron may also be formed into a cube through 90-degree interprecessional effect. (See Fig. A.)
417.00 Precession of Two Sets of 60 Closest-Packed Spheres
Fig. 417.01 417.01 Two identical sets of 60 spheres in closest packing precess in 90 degree action to form a seven-frequency, eight-ball-to-the-edge tetrahedron with a total of 120 spheres; exactly 100 spheres are on the outer shell, exactly 20 spheres are in theinner shell, and there is no sphere at the nucleus. This is the largest possible double-shelled tetrahedral aggregation of closest-packed spheres having no nuclear sphere. As long as it has the 20- sphere tetrahedron of the inner shell, it will never acquire a nucleus at any frequency.
417.02 The 120 spheres of this non-nuclear tetrahedron correspond to the 120 basic triangles that describe unity on a sphere. They correspond to the 120 identical right- spherical triangles that result from symmetrical subdividing of the 20 identical, equilateral, equiangular triangles of either the spherical or planar-faceted icosahedron accomplished by the most economical connectors from the icosahedron's 12 vertexes to the mid-edges of the opposite edges of their respective triangles, which connectors are inherently perpendicular to the edges and pass through one another at the equitriangles' center and divide each of the equilaterals into six similar right triangles. These 120 triangles constitute the highest common multiple of system surface division by a single module unit area, as these 30º , 60º , 90º triangles are not further divisible into identical parts.
417.03 When we first look at the two unprecessed 60-ball halves of the 120-sphere tetrahedron, our eyes tend to be deceived. We tend to look at them "three-dimensionally," i.e., in the terms of exclusively rectilinear and perpendicular symmetry of potential associability and closure upon one another. Thus we do not immediately see how we could bring two oblong quadrangular facets together with their long axes crossing one another at right angles.
417.04 Our sense of exclusively perpendicular approach to one another precludes our recognition that in 60-degree (versus 90-degree) coordination, these two sets precess in 60-degree angular convergence and not in parallel-edged congruence. This 60-degree convergence and divergence of mass-attracted associabilities is characteristic of the four- dimensional system.
418.00 Analogy of Closest Packing, Periodic Table, and Atomic Structure
418.01 The number of closest-packed spheres in any complete layer around any nuclear group of layers always terminates with the digit 2. First layer, 12; second, 42; third, 92 . . . 162, 252, 362, and so on. The digit 2 is always preceded by a number that corresponds to the second power of the number of layers surrounding the nucleus. The third layer's number of 92 is comprised of the 3 multiplied by itself (i.e., 3 to the second power), which is 9, with the digit 2 as a suffix.
418.02 This third layer is the outermost of the symmetrically unique, nuclear-system patterns and may be identified with the 92 unique, selfregenerative, chemical-element systems, and with the 92nd such element__ uranium.
418.03 The closest-sphere-packing system's first three layers of 12, 42, and 92 add to 146, which is the number of neutrons in uranium__which has the highest nucleon population of all the self-regenerative chemical elements; these 146 neutrons, plus the 92 unengaged mass-attracting protons of the outer layer, give the predominant uranium of 238 nucleons, from whose outer layer the excess two of each layer (which functions as a neutral axis of spin) can be disengaged without distorting the structural integrity of the symmetrical aggregate, which leaves the chain-reacting Uranium 236.
418.04 All the first 92 chemical elements are the finitely comprehensive set of purely abstract physical principles governing all the fundamental cases of dynamically symmetrical, vectorial geometries and their systematically self-knotting, i.e., precessionally self-interfered, regenerative, inwardly shunting events.
418.05 The chemical elements are each unique pattern integrities formed by their self-knotting, inwardly precessing, periodically synchronized selfinterferences. Unique pattern evolvement constitutes elementality. What is unique about each of the 92 self- regenerative chemical elements is their nonrepetitive pattern evolvement, which terminates with the third layer of 92.
Small duotet 1.5tv, small int octa .5tv, small cube 3tv
Large duotet 12tv, large int octa 4tv, large cube 24tv
Double Duo-tet = includes VE w/o 1/2-octas, in IVM and cubic matrix
4F tet (slightly offset)
1 F tet, volume 1, surface area 4 (triangle faces).
2 F tet, volume 8, surface area 16.
3 F tet, volume 27, surface area 36.
4 F tet, volume 64, surface area 64.
5 F tet, volume 125, surface area 100.
6 F tet, volume 216, surface area 144.
What about the oddball relative volumes of the icosa and vector edge cube? (cont'd)
18.51 + 8.49 = 27. 8 icosas in 2F cube pattern has central vector edge cube
27 = 3 x 3 x 3 (3 Freq small cube). 8 = 2 x 2 x 2 (2 Freq small cube)
3 layers of a cube of 9 spheres/cubes
01.02.03. + 04.05.06. + 07.08.09.
10.11.12. + 13.14.15. + 16.17.18.
19.20.21. + 22.23.24. + 25.26.27.
27 + 27 = 54 (2 cubes)
54 - 42 = 12 (2 cubes - 2nd freq VE vertices = 1st freq VE 12 vertices)
54 - 12 = 42 (2 cubes - 1st freq icosa vertices = 2nd freq icosa vertices)
92 = 90 + 2 where 2 is suffix, 90 is 9 x 10, 9 is the full set.
Wonky watercube: foam has 14 sided cells of minimal surface area
involution, revolution, evolution A geodesic hemispheric dome of aluminium tubing and fabric with congruent torus with central funnel column as supporting mast, wind turbine, tracking solar collector, chimney, water spout reservoir, air conditioning.
. In single symmetrical systems, all the vertexes are equidistant radially from their common volumetric centers, and the centers of area of all their triangular facets are also equidistant from the system's common volumetric center.
400.41 The minimum single symmetrical system is the regular tetrahedron, which contains the least volume with the most surface as compared to all other symmetrical single systems. There are only three single symmetrical systems: the regular tetrahedron, with a "unit" volume-to-skin ratio of 1 to l; the regular octahedron, with a volume-to- surface ratio of 2 to 1; and the regular icosahedron, with a volume-to-surface ratio of 3.7 to 1. Single asymmetrical systems contain less volume per surface area of containment than do symmetrical or regular tetrahedra. The more asymmetrical, the less the volume-to- surface ratio. Since the structural strength is expressed by the vector edges, the more asymmetrical, the greater is the containment strength per unit of volumetric content.
VE to octa showing icosa phases
Flexible polyhedra, VE, golden icosa/tetra, the Arc: red curve at upper left at page 12 includes the Synergetics jitterbug transformation from VE cuboctahedron through the various icosahedral phases to the octahedron. The Synarctics full jitterbug actually continues through to the tetrahedron and triangle without stopping, though it can reverse at any click-stop phase.
The Synarctic version of structural dimensionality of the jitterbug transformation, with decrease in bond distribution:
|Single Jitterbug||Dual Jitterbugs||Int plane polygons|
|6th Dimension||VE non-nuclear cuboctahedron||Dual non-nuc cuboctahedra||hexagon|
|5th Dimension||fluid icosahedra||rigid icosahedron||pentagon|
|4th Dimension||octahedron||cube faced w/ 6 octamids||square|
|3rd Dimension||tetrahedron||tri prism w/ 3 octamids & 2 tetramids||triangle|
|2nd Dimension||2 Freq triangle||"winged" octahedron||triangle|
|1st Dimension||equil triangle 8 layer non-Z "prism"||octa? tet? SoD? triangle? non-Z "anti-prism"||triangle|
internal flat planar polygon defines dimension:
cuboctahedron = 6 dimension, hexagon, dynamic cycle
icosahedron = 5 dimension, pentagon, static shell
octahedron/cube/triprism = 4 dimension, square, matrix
tetrahedron = 3 dimension, triangle, structure, crystal
2 Freq triangle = 2 dimension, triangle, system, tile
1 Freq triangle = 1 dimension, triangle, unit, cell
Note on Synarctic Jitterbug:
VE vertex single bond pin hinged, (compress able)
octa double bond edge hinged, (torque able)
tetra tri bond cirumferentially hinged (squish able)
triangle 8 plane full face hinged (coplanar polar layerable)
SoD nuclear plane face hinged (noncoplanar neutral layerable)
12, 42, 92
12 = 6 x 2, 3 x 4
42 = 6 x 7, 6 is non-nuc hexagon, 7 is nuc hexagon
42 = 3 x 14, 3 is prime triangle, 14 is VE & foam faces
92 = 90 degrees/surface spheres + polarity suffix
am: A regular octahedron can be cumulated so that it is either a rhombic
dodecahedron or a stella octangula.
Math World won't admit this, but if you cumulate an octahedron, then
you can get the rhombic dodecahedron, just as if you cumulate the cube.
A stella octangula and rhombic dodecahedron are identical on a spherical surface AFAICT.
VE as 5tv
VE**5 = 160tv
Table: Initial Frequencies of Vector Equilibrium
Spheres Freq Tetravolumes
Radius 1 VE0/2 2 1/2
Radius 1 VE0 5
Radius 2 VE1 20
Radius 4 VE2 160