List of Figures — Cosmography

2.5 A seemingly independently existent triangle is always a four-cornered tetrahedron of minimagnitude altitude. A is a four-flex-cornered tetrahedron; B, a prism; C, a flat piece of paper cut out as a triangle (in reality a prism of meager but geometrically significant altitude).

2.6 Pulsing of a tetrahedron as it turns itself inside out.

2.7 The three great-circle-spun square planes exactly bisecting the tetrahedron in three symmetrical ways. The three triangular-system-formed subdivision-aspect squares are ACBD, AEBF, DECF. Note the primary tetrahedron and the secondary internal octahedron, and only then are the implied square cross sections of the octahedron apparent as tertiary derivations of the primary structural system, the tetrahedron. There is no single-plane, omni-equal-angle, equal-edge ‘‘square’’ structural integrity in Universe. Squares and cubes are always and only tertiary derivations of prime vectorial structuring systems.

2.8 The six great-circle-spun subdivisions of the tetrahedron—what I call the A and B Quanta Modules. All regular polyhedra (other than the icosahedron and the pentagonal odecahedron) are composed of fractional elements of the tetrahedron and octahedron. These elements are known in synergetics as the A and B Quanta Modules. They each have a volume of 1 ⁄ 24 of a tetrahedron (see synergetics [synergetics], Secs. 910--916). This illustration shows the six great-circle-spun subdivisions of the regular primitive tetrahedron into its twenty-four A Quanta Modules and of the contained octahedron into its forty-eight A and forty-eight B Quanta Modules by the further symmetrically spun four great circles of unique spinnability of the four axes of the eight opposite regular triangles of the tetrahedron-contained octahedron (see synergetics2 [synergetics2], Sec. 987).

2.9 Spheres closest packed twelve around one.

2.10 Rhombic dodecahedron.

2.11 The coupler.

3.1 Graph of Newton’s norm of ‘‘no change.’’

3.2 Graph of Einstein’s norm of 186,000 mps.

3.3 Gravity is inherently integrated as a closed system with no ends and ergo is an inherently closed system having twice the coherence integrity of equally energy-vectored radiation.

6.1 Interference phenomena: lines cannot go through the same point.

6.2 A stone transforms to a tetrahedron.

6.3 Macro-micro systems diagram.

6.4 The minimum system.

6.5 Synergetics’ Constants of the Hierarchy of Primitive

6.5 New identification of polyvertexia. ¹Tetravertexion (plural, tetravertexia) is also used in this book.

6.6 Underlying order in superficially seeming randomness law. The number of interrelationships X of a given number N of ‘‘something’’ is

N2-−-N-- X = 2

(1)

When we look at the stars, they appear to be quite randomly scattered throughout the sky. We can say, however, that the number of direct and unique interrelationships among the stars is always given by this equation. Further, we are mathematically justified in assuming order always to be present despite the appearance of disorder. Looking at the starry skies gives us a personal sense of the order-discovering power of weightless mind and at the same time a sense of our physical body’s negligible size in Universe when compared to the vast reaches of visible stars arrayed across the nighttime sky.

6.7 The minimum system. The human-senses-tunable, differentially apprehending minimum system configuration of Universe has insideness and outsideness and is defined by four infra-human-senses-tunable, microsystem somethings. Each of the latter have four micro-macro something corners. Up to three relationships, as pictured above, does not constitute a system.

6.8 System outsideness. Systems always have potentiality to be (1) discovered, (2) tuned-in microsystems inside and macrosystems outside the considered (i.e., tuned-in) system.

6.9 Additive twoness and multiplicative twoness.

6.10 Yin-yang.

6.11 Basic dichotomy of all living phenomena.

6.12 The three ways of physically demonstrating the simplest system in the Universe —the four-vertexion.

6.13 Tetrahedron and truncated tetrahedron.

6.14 Wood-frame-mounted canvas showing all its dimensions.

6.15 Drawing of a ‘‘flat plane’’ revealing its thickness.

6.16 Six-dimensionality of both the tetravertexion and its contained hexavertexion. Diamonds are the minimum physical material system. Thought of tetra-octa systems are the minimum metaphysical (conceptual) system.

6.17 Flexible-corner cube.

6.18 Two four-ball tetravertexion systems.

6.19 The right triangle.

6.20 Snyder-Fuller³ interattraction law. ³ Jaime Snyder [Fuller’s grandson], a student of physics, consulted on the formulation of this law.

6.21 Square face ADWX.

6.22 Intraposed tetrahedra ABCDWXY Z. Internal octahedron PQRSTU.

6.23 Alternating red and blue windows. Red alternates in this illustration are left open for simplification of conceptualization.

6.24 The blue alternates.

6.25 Square face XAWD.

6.26 Star octahedron.

6.27 Quasicube.

6.28 Earth with apparent perpendiculars on surface shown to diverge. Tops of long suspension-bridge masts, being exactly perpendicular to Earth, are measurably farther apart from each other than are their bases. Cubes fill only all cubical space.

6.29 The spherical ‘‘cube.’’ It is impossible to ‘‘square’’ or ‘‘cube’’ a sphere. Since we live on a sphere in an omnicurvilinear operative Universe, it is futile to mensurate squarely and cubically. All we do are ‘‘squares.’’

6.30 Earth surface considerations around the world.

6.31 Four spheres lock as a tetrahedron. Four unit-radius spheric ‘‘somethings’’ (microsystems) when closest interpacked form a tetrahedron.

6.32 Vector equilibrium: omnidirectional closest packing around a nucleus. Triangles can be subdivided into greater and greater numbers of similar units. The number of modular subdivisions along any edge can be referred to as the frequency of a given triangle. In triangular grids each vertex may be expanded to become a circle or sphere showing the inherent relationship between closest-packed spheres and triangulation. The frequency of triangular arrays of spheres in the plane is determined by counting the number of intervals (A) rather than the number of spheres on a given edge. In the case of concentric packages or spheres around a nucleus the frequency of a given system can either be the edge subdivision or the number of concentric shells or layers. Concentric packings in the plane give rise to hexagonal arrays (B), and omnidirectional closest packing or an equal sphere around a nucleus (C) gives rise to the vector equilibrium (D).

6.33 Equation/or omnidirectional closest packing of spheres. Omnidirectional concentric closest packings of equal spheres about a nuclear sphere form series of vector equilibria of progressively higher frequencies. The number of spheres or vertexes on any symmetrically concentric shell or layer is given by the equation 10F2+2, where F = frequency. The frequency can be considered as the number of layers (concentric shells or radius) or the number of edge modules on the vector equilibrium. A 1-frequency sphere-packing system has 12 spheres on the outer layer (A) and a 1-frequency vector equilibrium has 12 vertexes. If another layer of spheres is packed around the 1-frequency system, exactly 42 additional spheres are required to make this a 2-frequency system (B). If still another layer of spheres is added to the 2−frequency system, exactly 92 additional spheres are required to make the 3−frequency system (C). A 4−frequency system will have 162 spheres on its outer layer. A 5−frequency system will have 252 spheres on its outer layer, etc.

6.34 Realized nucleus appears at fifth shell layer. In concentric closest packing of successive shell layers, potential nuclei appear at the third shell layer, but they are not realized until surrounded by two shells at the fifth layer.

6.35 Tetrahedral closest packing of spheres: nucleus and nestable configurations.

6.36 Trivalent bonding of vertexial spheres forms rigid structures. At C gases are monovalent, single-bonded, omniflexible, with inadequate interattraction, separatist, compressible. At B liquids are bivalent, double-bonded, hinged, flexible, with viscous integrity. At A rigids are trivalent, triple-bonded, rigid, with highest tension coherence.

6.37 Frequency pictured as equatorial layer through nuclear sphere. The modular frequency of the spheric, omnidirectionally, omni-closest-packed uniform-radius spheres is determined by the number of spaces between the spheres along one edge of the closest-packed system. This is a three-frequency, four-dimensional system of closest-packed-together unit-radius spheres, pictured here as an equatorial layer through the aggregate at the nuclear sphere level.

6.38 Nuclear structural systems. Nuclear structural systems consist entirely of tetrahedra having a common interior vertex. They may be interiorly truncated by introducing special-case frequency, which provides chordal as well as radial modular subdivisioning of the isotropic-vector-matrix intertriangulation, while sustaining the structural rigidity of the system.

6.39 Tetravertexion, one-quarter tetravertexion, and one-twenty-fourth tetravertexion, or A module. A, tetravertexion; B, one-quarter tetravertexion; C, one-twenty-fourth tetravertexion, which we call an A module; D, six equiangled asymmetric tetravertexia. Since one-quarter of a regular tetravertexion has been further subdivided into six similar equiangled, asymmetric tetrahedra, each of these asymmetries is one-twenty-fourth of the regular tetravertexion. Each of these twenty-fourth subdivision tetravertexia is called an A module.

6.40 Angles are angles independent of the length of their edges. Lines are ‘‘size’’ phenomena and unlimited in length. Angle is only a fraction of one cycle.

6.41 Six-frequency tensegrity icosahedron.

6.42 Single and double bonding of members in tensegrity spheres. A, negatively rotating triangles on a 270-strut tensegrity geodesic sphere with double-bonded triangles; B, a 270-strut isotropic tensegrity geodesic sphere, with single-bonded turbo triangles forming a complex six-frequency triacontahedron tensegrity; C, complex of basic three-strut tensegrities, with axial alignment whose exterior terminals are to be joined in single bond as 90-strut tensegrity; D, complex of basic three-strut tensegrity units with exterior terminals now joined.

6.43 Basic tensegrities. A, the four-strut, twelve-tendoned, outside-in (negative) tetrahedron, showing the four outer vertex turbining. B, the six-strut tensegrity, 18-tendoned, outside-out (positive) tetrahedron, showing central-angle turbining. C, the three-strut, twelve-tendoned tensegrity octahedron. The three compression struts do not touch each other as they pass at the center of the octahedron; they are held together only at their terminals by the comprehensive, triangular tension net. It is the simplest form of tensegrity. D, the twelve-strut, 48-tensioned tensegrity cube, which is unstable.

6.44 Chordal ricochet pattern in stretch action of a balloon net. A gas balloon’s exterior tension ‘‘net’’ has the shape that it has because some of the molecules are too large to escape and, crowded by the other molecules, are hitting the balloon. But the molecules do not huddle together at the center and then simultaneously explode outward to hit the balloon skin in one omnidirectionally outbound wave. The molecules near the surface are coursing in chordally ricocheting patterns all around the inner net’s surface. I therefore saw that because every action has its reaction, it would be possible to pair all the molecules so that they would behave like two swimmers who dive into a swimming tank from opposite ends, meet in the middle, and then, employing each other’s inertia, shove off from each other’s feet in opposite directions.

6.45 A system within a system: tensegrity tetrahedron with a tensionally positioned central ball suspended at its center of volume. Central ball completely restrained in terms of all twelve degrees of freedom of all individual systems. Note that the six ‘‘solid,’’ push-pull compression members are the acceleration vectors trying to escape from the system at either end by action and reaction, whereas both ends of each would-be escapee are restrained by four tensional wires, two long and two short, while the ball at the center is restrained from local displacement, torque, and twist by three triangulated tension wires, each also tangentially affixed to each of the four outer corner balls.

6.46 Functions of positive and negative tetrahedra in tensegrity stacked cubes. Every cube has six faces (A). Every tetrahedron has six edges (B). Every cube has eight corners and every tetrahedron has four corners. Every cube contains two tetrahedra (ABCD and WXY Z) because each of its six faces has two diagonals, the positive and negative set. These may be called the symmetrically juxtaposed positive and negative tetrahedra whose centers of gravity are congruent with one another as well as congruent with the center of gravity of the cube (C). It is possible to stack cubes (D) into two columns. One column contains the positive tetrahedra (E), and the other contains the negative tetrahedra (F).

6.47 Stabilization of tension in tensegrity column. We put a steel sphere at the center of gravity of a cube which is also the center of gravity of a tetrahedron and then run steel tubes from the center of gravity to four corners, W, X, Y , and Z, of negative tetrahedron (A). Every tetrahedron’s center of gravity has four radials from the center of gravity to the four vertexes of the tetrahedron (B). In the juncture between the two tetrahedra (D), ball joints at the center of gravity are pulled toward one another by a vertical tension stay, thus thrusting universally jointed legs outward, and their outward thrust is stably restrained by finite sling closure WXY Z. This system is nonredundant: a basic discontinuous-compression continuous-tension, or ‘‘tensegrity,’’ construction. It is possible to have a stack (column or mast) of center-of-gravity radial tube tetrahedra struts (C) with horizontal (approximate) tension slings and vertical tension guys and diagonal tension edges of the four superimposed tetrahedra, which, because of the (approximate) horizontal slings, cannot come any closer to one another and, because of their vertical guys, cannot get any farther away from one another and therefore compose a stable relationship: a structure.

6.48 Tensegrity masts as struts: miniaturization approaches atomic structure. The tensegrity masts can be substituted for the individual (so-called solid) struts in the tensegrity spheres. In each one of the separate tensegrity masts acting as struts in the tensegrity spheres, it can be seen that there are little (so-called) solid struts. The subminiature tensegrity mast may be substituted for each of those solid struts, and so on to sub-sub-subminiature tensegrities until we finally get down to the size of the atom, and this becomes completely compatible with the atom, for the atom is tensegrity and there are no ‘‘solids’’ left in the entire structural system. There are no solids in structures and ergo no solids in Universe. There is nothing incompatible with what we may see as solid at the visual level and what we are finding out to be the structural relationships in nuclear physics.

6.49 Model of adjustable spherical triangle made of stainless steel straps.

6.50 The spherical triangle. The sum of the angles of a triangle is never 180^∘.

6.51 Triangles on surface of sphere, several views.

6.52 The four great circles of a sphere. The spherical tetrahedron divides area of sphere into four triangulated areas (base X altitude), eliminating need for pi.

6.53 Tetrahedral mensuration applied to spheres.

6.54 Angular topology independent of size.

6.55 Tetrahedral mensuration applied to well-known polyhedra. We discover that the sum of the angles around all vertexes of all solids is evenly divisible by the sum of the angles of a tetrahedron. The volumes of all solids may be expressed in tetrahedra.

6.56 Equivector investments with opposite results. (See also gravity radiation model, Fig. 3.3.)

6.57 Falling sticks. Six vectors provide minimum stability.

6.58 Four vectors of restraint define minimum system. Music: wind instruments, string instruments, drums, gongs. Exclusively tensional investigation of the means of providing a minimum weight, structurally stable system.

6.59 Axes of rotation of icosahedron.

6.60 Minimum of twelve spokes oppose torque.

6.61 Implicit inside-outing of triangle. This illustrates the inside-outing of a triangle.

6.62 Inside-outing of glove.

6.63 Reality is spiro-orbital. All terrestrial critical path developments inherently orbit the Sun. No path can be linear. All paths are precessionally modulated by remotely operative forces producing spiralinear paths.

6.64 Four axes of vector equilibrium with rotating wheels or triangular cams.

6.65 Involution and evolution.

6.66 An experiment in angular acceleration.

6.67 Tetrahedral precession of closest-packed spheres.

6.68 Precession of two sets of 60 closest-packed spheres as seven-frequency tetrahedron. Two identical sets of 60 spheres in closest packing precess in 90^∘ action to form a seven-frequency, eight-ball-edged tetrahedron with 120 spheres, of which exactly 100 spheres are on the surface of the tetrahedron and 20 are inside. The 120-sphere nonnucleated tetrahedron is the largest possible double-shelled tetrahedral aggregation of closest-packed spheres having no nuclear sphere.

6.69 Tetrahedral precession of closest-packed spheres.

6.70 Flexible cube and octahedron.

6.71 Octahedron’s three axes cross each other at 90^∘ at octahedron’s center.

6.72 Cube stabilized with tetrahedron.

6.73 Square = 2N².

6.74 Quadrangular accounting, squaring and triangling, cubing and tetrahedroning.

6.75 The cheese tetrahedron. If you slice parallel to one of the faces of all the symmetrical geometries (i.e., all the Platonic and Archimedean ‘‘solids’’), each made of cheese, what is left after the parallel slice is removed is no longer the same symmetrical polyhedron—with but one exception, the tetrahedron.

6.76 Symmetrical contraction of vector equilibrium:jitterbug system.

6.77 Vector equilibrium constructed of four foldable great circles. As with the other polyhedra, a vector equilibrium may be constructed of great circles cut from paper.

6.78 Constant-unit-volume progressions of asymmetric tetrahedra. In this progression of ever-more-asymmetric tetrahedra, only the sixth edge remains constant. Tetrahedral wavelength and tuning permit any two points in Universe to connect with any other two points in Universe.

6.79 Constant properties of the tetrahedron.

6.80 Four different ways in one, i.e., four congruent tetrahedra. This omnicongruence of atomic nuclei is also demonstrated in the chemical bonding of diamonds and alloying of metals.

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