A higher-order tensegrity is always the (inner) combo of the 2 lower-order duals from which it is derived (ie: it combines their Faces in itself), and by interchanging the rest-length of Vertex and Face pull-wires any tensegrity is changed into its own Dual (eg: an Icosa with 20 tri-F pulls and 12 penta-V pulls is changed into a Dodecahedron (Do) with 12 penta-F pulls and 20 tri-V pulls). This is like the 90º Edge-precessing of any structure, which changes that structure into its Dual.
The Yin-order^3 R-ido (120 Es: 20-triangle-Fs + 12-penta-Fs + 30-rhombi-Fs; 60 Vs) with 3-rod, 4-rod & 5-rod Fs changes to:
the Yin-order^2 I-Do with 3-rod & 5-rod Fs, or its
Yin-order^2 dual the R30 (Rhombic-Triacon, 60 Es, 30 Rhombic-Fs, 32 Vs) with 4-rod Fs.
Note: Real-world tensegrity models with rigid rods change a form to its dual or back (eg: Icosa to Do) by twisting its rod one way or the other (with the I-Do combo formed in-between). In Struck however, where the pushes also change with the pulls, the Icosa changes to the 'soccer-ball' instead (pentagons and hexagons). Similarly, the Struck TensegriTet changes to the intermediate order^1 combo, the 6cross-rod TensegriOcta (via the intermediate truncTet).
The Struck tensegrity-cuboc changes to a Rhombido-Cuboc with 12 rhombi, 6 squares and 8 triangles:
Note: the dual of the Yin-order^3 R-ido is the Yin-order^3 T60 (ie: 60 Trapezium Faces);
and the dual of the Yang-order^3 R-Cuboc is the Yang-order^3 T24 (ie: 24 Trapezium Fs).
7-Level Prime-volume Based Hierarchy Of Dual-Pairs
| # | Order | Form | Vol | #Vs | el-part | Dual | d-el |
|---|---|---|---|---|---|---|---|
| 1 | Prime | TET | 1 | 4 | fEdge | dTet | fVertex |
| 2 | Yng^1 | Octa | 4 | 6 | aEdge | Cube | eFace |
| 3 | Yin^1 | Icosa | 18+ | 12 | wEdge | Do | fFace |
| 4 | Yang^2 | Cuboc | 20 | 12 | wEdge | Rdo | eVertex |
| 5 | Yang^3 | R-cuboc | - | 24 | fFace | Trap24 | aVertex |
| 6 | Yin^2 | I-Do | - | 30 | aEdge | R30 | wVertex |
| 7 | Yin^3 | R-ido | - | 60 | aFace | Trap60 | wFace |
*The first 4 primary forms, the Tet, Oc, Ic, the only structurally stable polyhedrons (because all Faces are structurally stable triangles), plus the Cuboc, as tensegrities, demo all of the forms but the last and largest 4 (Tr24,R30,Rido,Tr60). All Duals have the same # of Fs as listed in the Primary forms' #Vs column. The Primary structures correspond to the 6 Edges and 1 Face of the ArcheTet. Their Duals correspond to 4 Vertexes and 3 Faces.
For both yang and yin, ie: for all 6 primary forms above the basic PrimeTet level, the order corresponds to the number of Face-types; eg: the order^1 forms (Oc, Ic) have 1 type of Face (trigon, pentagon); the order^2 (Cuboc, I-Do) have 2 types (trigon-cube, trigon-pentagon); and the order^3 (R-cuboc, R-ido) have 3 types (rhombus-trigon-cube, rhombus-trigon-pentagon).
The unique Tet-Prime is the basis of all others, the simplest and the only one with V-F opposites, and also the only one that's its own dual.
Each structure in the primary column of this hierarchy itself has equi-size Edges, and they also all share the same equi-size Edge.
Each of the Dual-alternates itself has equi-sized Edges except for the Tr24 & Tr60 which have 24 and 60 Trapezial-Faces (a trapezium is like a rhombus with one short diagonal and one long, but with 2 adjoining short-Edges opposite 2 adjoining long-Edges, and thus no parallel Edges).
Aside from this primary volume-based Hierarchy, if the Es of any lower-level structure are precessionally crossed 90º; they re-form its Dual-partner, and v.v. If interpenetrated, and their crossing-Edge mid-points are connected, the mid-points become the Vs of a contained 'inner' combo of the next-higher level; and if their crossing-Edge 'outer' end-points are connected they become the Vs of a containing 'outer' combo of the next-higher level (with the 'outer' & 'inner' combos being 'smaller-Edged' and 'larger-Edged' Duals of each other). Thus, the 'inner' and 'outer' Combo-Duals of any level interpenetrate to form the next higher-level Combos. (Or, if either dual is mid-edge truncated it forms the next higher inner-combo). Conversely, the crossed diagonals of the Faces of any higher-level all-quadrilateral-faced 'outer' Dual form the Duals of the next-lower level.
For the Tr24, the yang Inner-Cuboc and outer-Rdo (formed by the precessionally crossed Es of the lower-level Oc-&-Cube), when interpenetrating each other with crossed-Edges, themselves form the highest-yang level's inner R-cuboc and outer Tr24.
Similarly, the yin inner I-Do & outer R30(formed by the precessionally crossed Es of the lower-level Icosa-&-Dodeca), form the highest-yin level's inner R-ido and outer Tr60.
6 Structure-types associated with ArcheTet's 3 opposite yang-yin Edge-Pairs
On the Prime-ArcheTet, there are 3 yang-pushrod Es diverging from a yang-Vertex, and opposite them are 3 Yin-pullwire convergent Es cyclically-connecting the other 3 Yin-Vertexes. The 3 yang-Es have 3 associated yang-structures (Tet, Oc, Cuboc); the Yin-Es have 3 associated 5-sym Yin-structures (Ic, Do, I-Do).
The ArcheTet's 4 Poles & 4 Holes As Yang or Yin
1 yang-Pole (Hades-fire) has 3 yang-Es (2 stable, 1 unstable). Yang-hexagram 07 Melding
1 Yin-Hole (Persephone-water) 3 Yin-Es: 2 unstable.
8 Structure-types associated with ArcheTet's 4 Pole-Hole Pairs
3 yang-Pole structures: dTet (f), RDo (e), Tr24 (a)
2 yang-Hole structures: Cube (e), R-cuboc (f) Yang-hexagrams: 25 Uttermost Enfolding, and 61 Lasting Enduring
Yang-structures form concentric shells within an Isotropic Vector Matrix that becomes more densely packed with higher frequencies. Yang-partial structures represent functions that are additive-subtractive, 2-valued, either-or, reductive-analytic, linear-logical, chrono-logical, cause-effect oriented, radiating wave-dispersive, divergent and fragmentive.
Yin-structures are hole-enclosing single-shell at any frequency (only their sur-face Faces, Vertices and Edges become increasingly numerous and more densely packed). Yin-structures represent functions that are wholistic, multi-valued, synchronic, acausal, convergent, and locally-cycling.
Tensegrity Pole-Hole Vortexes
In all 'floating' or 'islanded-compression' tensegrities (no compressive rods touching each other), with rod-wire alignment (rather than 'cross-rod' tensegrities), where each Vertex-Pole is a Vortex formed by push-rods that come to a Focus with a clockwise-twist, directing the energy outwards through the Vertex, each E-framed window-opening Face-Hole is also a Vortex, but whose push-rod Es twist counter-clockwise, sucking the energy in through the Face-Hole. I'm unaware that anyone else has previously noticed and mentioned this obvious but significant fact about tensegrities.
For those somewhat technophobic who may have inadvertently wandered into these pages, here is one translation of the structure transformations. Of course, this is over-simplified, and only recounts from the Yin-side. The Yang-side (involving the Ic-Cuboc jitterbug transform) is more complex, and together they require repeated study before it all clicks together. This will thus have to be further expanded upon in a subsequent Webpage).
Do...I-Do...Icosa; Sun...Jupiter...Venus; Isolation...Inspiration...Intuition; etc,.
As the Solar Photosphere permeates the Atmosphere and enlivens all bioforms within the Hydrosphere, just so
The vision of Light becomes condensed into the Heart of Love as it moves through the Breath of all Life.
Conversely, the Soul's Compassion expands into Luminous-Clarity as it acts through the provident nurturing of all life.
I-Do...R-ido...R30; Athena modulates between Zeus' Power and Poseidon's Passion.
Thus, Life AEthereally Articulates Wisdom, and the Warrior's action of Athena's triple-jointed aHole, through the Light of Yin Insight, connects Ares' drive for Understanding Power to Luna's lust for sensory Knowing;
..as Persephone (Tr60) falls into Hades' Underworld (dTet)...
On the yang side, Moon-Mars-Merc are the lower earthy 1-2-3 places of a hexagram, and the 1st, 2nd and 3rd phases of all 4 Stages of the Yang-Cycle:
Luna Lava Lust Abyss (yang^2) connects
Ares Thundercrack (yang^1) connects
The 6 Edges of the ArcheTet can be viewed as A): '3x2', ie a triad of paired yin-yang water-air-or-fire elemental Edge-Crosses, with each Cross having non-touching arms, one yang and one yin, whose arms are precessional to each other at 90º angles, as opposite Edges of the Tet.
Or, B): as '2x3', a 'pair-of-triads', ie a triple-yang-partial tripod-'Pole' of divergent water-air-fire Edges opposite a triple-Yin cyclo-holistic triangular 'Hole' of convergent water-air-fire Edges. Of course, since there are 4 Pole-Hole oppositions of the Tet, this 'pair-of-triads' grouping can be seen from the 3 other viewpoints of the other 3 Poles, each having 1-yang and 2-yin Edges.
Incidentally, since there are 14 basic parts to the ArcheTet (excluding the various Face and dihedral angles and open-triangle 'coils' made from 2 or 3 Edges), each of the 14 can be seen as related to all the others; thus there are (7x14), or 98 such possible direct relationships, all of which together permit the 14 parts to mutually define each other in terms of such relationships. This gives a quick glimpse of the hidden complexity which is nevertheless easily mastered and memorized via the simplicity of the unique, simplest possible, and Prime structure of the universe, from which all such complex relationships can easily be worked out in one's head, along with their applications, not only intra-system, but also inter-system, such as the mythological archetypes, or the astrological signs and functions, or the 22 major arcana Tarot signs and images, or the cabalistic sephiroth, or the body chakras, or the 64 hexagrams of the I-Ching, etc,.
A simplistic-vs-complicated dichotomy is NOT the same as an elegantly simple yet complex, richly multifold wholistic comprehension. To see the simple in the complex, and the complex in the simple, is not only an art of soulful feeling, but also a philosophy, science and ritual of mind, body and spirit, all integrated with thinking, sensing and acting.
3 Yin-Poles: (Saturn-earth) 2 Yin-Es, 1 yang-E; all unstable. Yin-hexagram 16 Unfolding Increase
(Uranus-air) 2 Yin-Es: 1 unstable; 1 yang-E stable. Yin hexagram 52 Almost
(Neptune-water) 2 Yin-Es: 1 unstable; 1 yang-E stable. Yin hexagram 34 Wooing Inductive Influence
3 yang-Holes (Hephaestus-fire) 3 yang-Es: all stable
(Athena-air) 2 yang-Es: 1 stable; 1 Yin-E unstable
(Geae-earth) 2 yang-Es (1 stable); 1 Yin-E (unstable).
(5-sym structures are Yin; others are yang)
1 Yin-Pole structure: R30 (w)
2 Yin-Hole structures: R-ido (a), Tr60 (w) Yin-hexagrams: 43 Cutting, and 70 Parting.
(1: sudra-serf-peasant-pawn; 2: warrior-knight; 3: merchant-builder rook).
Hades Infernal fiery Underworld (dPrime) of Mind-Possession with
the earthy-aethereal Time-binding Cronos Unfolding Outerworld (dyang^2) of Sensory-Perception,
whose hexagram is aethereal-acting Zeus Sky Ruler (yin^2) above earthy-acting Ares (yang^1).
Hades (dPrime) with
Uranus'' Supernal Overworld (dyang^3) of Spirit-Power,
whose hexagram is aethereal fiery Mind-Body Sol-Apollo Seer-Sage (dYin^1) above earth-liquid Luna-Lava (yang^2), thus outlining Athena's Face.
Mercurial Hermes Volcanic earthy fiery-thinking Mt. Keep Still (yang-Prime) connects
Hades (dPrime) with
Neptune's watery Innerworld (dyin^2) of Soul Passion,
whose hexagram is aethereal-watery Venus Sea Minister (yin^1) above Mercurial-Hermes Head-Mind itself (self-reflexive yang-Prime).
The 12 Hexagrams with 2-yang lines and 12 with 2-yin lines are order^2 relationships between order^1 relationship Edges; ie: 12 yang and 12 yin Face-angles.
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