0.11 Fig. 222.01 Equation for
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
one-frequency sphere packing system has 12 spheres on the outer layer (A) and
a one-frequency vector equilibrium has 12 vertexes. If another layer of spheres
are packed around the one-frequency system, exactly 42 additional spheres are
required to make this a two-frequency system (B). If still another layer of spheres
is added to the two-frequency system, exactly 92 additional spheres are required
to make the three-frequency system (C). A four-frequency system will have 162
spheres on its outer layer. A five-frequency system will have 252 spheres on its
outer layer, etc.
0.12 Fig. 222.30 Volume of Vector Equilibrium: The volume of the
vector equilibrium consists of eight tetrahedra and six half-octahedra. Therefore,
the volume of the vector equilibrium is exactly twenty.
0.13 Fig. 400.30 Topological
relationships of faces, vertexes, and edges of various polyhedra.
0.14 Figure
401.01 Four Vectors of Restraint Define Minimum System:
0.15 Fig. 401.05
The six compression members are the acceleration vectors trying to escape
from Universe at either end by action and reaction, while the ends of each
would-be escapee are restrained by three tensors; while the ball at the center is
restrained from local torque and twist by three triangulated tensors from each
of the four corners tangentially affixed.
0.16 Fig. 411.05 Four Spheres Lock as
Tetrahedron
0.17 Fig. 412.01 Closest Packing of Rods
0.18 Fig. 413.01 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 packings 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 of equal spheres
around a nucleus (C) gives rise to the vector equilibrium (D).
0.19 Fig. 415.17
Nucleated Cube: The ‘‘Externa’’ Octahedron: ABC show that eight additional
closest-packed spheres are required to form the minimum allspace-filling
nuclear cube to augment the nuclear vector equilibrium. DEF show the eight
Eighth-Octa required to complete the polyhedral transformation. (Compare Fig.
1006.32.)
0.20 Fig. 415.22 Rational Volumes of Tetrahedroning:
0.21 Fig. 470.02D
Reciprocity of Vector Equilibrium and Octahedra in Space-Filling Jitterbug: In
the space-filling ‘‘jitterbug’’ transformation, the vector equilibria contract to
become octahedra, and, because in space filling array there are equal numbers of
octahedra and vector equilibria, the original octahedra expand and ultimately
become vector equilibria. There is a complete change of the two figures.
0.22 Fig.
901.03 Basic Right Triangle of Geodesic Sphere: Shown here is the basic data for
the 31 great circles of the spherical icosahedron, which is the basis for all geodesic
dome calculations. The basic right triangle as the lowest common denominator of
a sphere’s surface includes all the data for the entire sphere. It is precisely 1 ∕ 20th
of the sphere’s surface and is shown as shaded on the 31-great-circle-sphere (A).
An enlarged view of the same triangle is shown (B) with all of the basic data
denoted. There are three different external edges and three different internal
edges for a total of six different edges. There are six different internal angles other
than 60°or 90°. Note that all data given is spherical data, i.e. edges are given as
central angles and face angles are for spherical triangles. No chord factors are
shown. Those not already indicated elsewhere are given by the equation2sin 
,
where 𝜃 is the central angle. Solid lines denote the set of 15 great circles. Dashed
lines denote the set of 10 great circles. Dotted lines denote the set of 6 great
circles.
0.23 Fig. 902.01
0.24 Fig. 902.10.
0.25 Fig. 902.20.
0.26 Fig. 902.30
0.27 Fig.
913.01 Division of the Quarter-Tetrahedron into Six Parts:
0.28 Fig. 931.10
Tetrahedral Characteristics of Chemical Bonding: Tetrahelix: Chemical bonds as
demonstrated by arrangements of tetrahedra:
0.29 Fig. 935.23 Proton and Neutron
Three-vector Teams: The proton and neutron always and only coexist as action
vectors of half-quanta associable as quantum.
0.30 Fig. 936.12 Octahedron as
Conservation and Annihilation Model: If we think of the octahedron as defined by
the interconnections of six closest-packed spheres, gravitational pull can make one
of the four equatorial vectors disengage from its two adjacent equatorial vertexes
to rotate 90 degrees and rejoin the north and south vertexes in the transformation
completed as at I. (See also color plate 6.)
0.31 Fig. 936.16 Iceland Spar Crystal:
Double vector image.
0.32 Fig. 936.19 Tetrahedral Quantum is Lost and Reappears
in Transformation between Octahedron and Three-tetra-arc Tetrahelix: This
transformation has the precessional effect of rearranging the energy vectors from
4-tetravolumes to 3-tetravolumes and reverse. The neutral symmetric octahedron
rearranges itself into an asymmetric embryonic wave pattern. The four-membered
individual-link continuity is a potential electromagnetic-circuitry gap closer. The
furthermost ends of the tetra-arc group are alternatively vacant. (See also color
plate 6.)
0.33 Fig. 937.20 Six-great-circle Spherical Octahedron: The doubleness
of the octahedron is illustrated by the need for two sets of three great circles to
produce its spherical foldable form.
0.34 Fig. 938.13 Six Vectors of Additional
Quantum Vanish and Reappear in Jitterbug Transformation Between Vector
Equilibrium and Icosahedron: The icosahedral stage in accommodated by the
annihilation of the nuclear sphere, which in effect reappears in transformation as
six additional external vectors that structurally stabilize the six ‘‘square’’ faces
of the vector equilibrium and constitute an additional quantum package. (See
also color plate 7.)
0.35 Fig. 938.15 Two Tetrahedra Open Three Petal Faces and
Precess to Rejoin as Octahedron
0.36 Fig. 938.16 Octahedron Produced from
Precessed Edges of Tetrahedron: An octahedron may be produced from a single
tetrahedron by detaching the tetra edges and precessing each of the faces 60
degrees. The sequence begins at A and proceeds through BCD to arrive at E
with an octahedron of four positive triangular facets interspersed symmetrically
with four empty triangular windows. From F through I the sequence returns to
the original tetrahedron.
0.37 Fig. 950.12 Three Self-Packing, Allspace-Filling,
Irregular Tetrahedra: There are three self-packing irregular tetrahedra that will
fill allspace without need of any complementary shape (not even with the need of
right- and left-hand versions of themselves). One, the Mite, has been proposed by
Fuller (A). The other two are described by Coxeter in his book Regular Polytopes,
p. 71 and are called the trirectangular tetrahedron (B) and the tetragonal
disphenoid, or isosceles tetrahedron since it is bounded by four congruent isosceles
triangles (C). All three of these tetrahedra can be subdivided into component A
and A Quanta Modules. The Mite has a population of two Bs and four A and the
tetragonal disphenoid is simply a somewhat different arrangement of the same
components. The trirectangular tetrahedron consists of two A and oneB Modules.
(It is of interest to note that theB Module may be either right- or lefthanded.
See the remarks of Arthur L. Loeb.) Two trirectangular tetrahedra will combine
to form either of the other two tetrahedra. The edge dimensions given above
are all with respect to a ‘‘unity’’-edged regular tetrahedron. The self-packing,
allspace-filling, irregular tetrahedron of synergetics is also known as the Mite.
It is composed of four/ Quanta Modules and two B Quanta Modules.
0.38 Fig.
950.31 Tetrahedra and Octahedra Combine to Fill Space: Regular tetrahedra
alone will not fill space, but when four tetrahedra (A) are grouped to define a
larger tetrahedron (B), the resulting central space is an octahedron (C). Therefore
tetrahedra and octahedra will combine to fill all the space. If the volume of the
smaller tetrahedron is equal to one then the volume of the larger tetrahedron is
eight, i.e. edge length two to the third power (2×2×2). (When we double the linear
dimension of a figure we always increase its volume eight-fold.) If the volume of the
large tetrahedron is eight the central octahedron must have a volume of exactly
four, while the small tetrahedra each equal one. The volume of a pyramid is */s
the base area times the height. Therefore the 'A-octahedron (D) has exactly the
same volume as its corresponding tetrahedron, further proof that the regular
octahedron has exactly four times the volume of a regular tetrahedron of the
same edge length.
0.39 Fig. 986.052 Robot Camera Photograph of Tetrahedra
on Mars: On their correct but awkward description of these gigantic polyhedra
as ‘‘three-sided pyramids’’ the NASA scientists revealed their unfamiliarity with
tetrahedra.
0.40 986.061 Truncation of Tetrahedra: Only vertexes and edges may
be truncated. (Compare Figs. 987.241 and 1041.11.)
0.41 986.062 Truncated
Tetrahedron within Five-frequency Tetra Grid: Truncating the vertexes of the
tetrahedron results in a polyhedron with four triangular faces and our hexagonal
faces. (Compare Figs. 1041.11 and 1074.13.)
0.42 Fig. 986.076 Diagram of
Verrazano Bridge: The two towers are not parallel to each other.
0.43 986.096
4-D Symbol: Adopted by the author in 1928 to characterize his fourthdimensional
mathematical explorings.
0.44 Fig. 986.419 T Quanta Modules within Rhombic
Triacontahedron: The 120 T Quanta Modules can be grouped two different ways
within the rhombic triacontahedron to produce two different sets of 60 tetrahedra
each: 60 BAAO and 60 BBAO.
0.45 Fig. 986.421 A and B Quanta Modules:
The top drawings present plane nets for the modules with edge lengths of the
A Modules ratioed to the tetra edge and edge lengths of the B Modules ratioed
to the octa edge. The middle drawings illustrate the angles and foldability. The
bottom drawings show the folded assembly and their relation to each other. Tetra
edge = octa edge. (Compare Figs. 913.01 and 916.01.)
0.46 BITE (See color plate
17) & RITE (See color plate 19)
0.47 LITE (See color plate 18)
0.48 Fig. 986.427
Bite, Rite, Lite:
0.49 Fig. 986.429 Kate, Kat.
0.50 Fig. 986.40 OCTET (See color
plate 22)
0.51 Fig. 986.431 COUPLER (See color plate 23)
0.52 Fig. 986.432
CUBE (See color plate 24)
0.53 Fig. 986.433 RHOMBIC DODECAHEDRON
(See color plate 25)
0.54 Fig. 986.816 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.
0.55 Fig. 987.081 Trivalent Bonding of
Vertexial Spheres Form Rigids:
0.56 Fig. 987.132E Composite of Primary and
Secondary Icosahedron Great Circle Sets : This is a black-and-white version of
color plate 30. The Basic Disequilibrium 120 LCD triangle as presented at Fig.
901.03 appears here shaded in the spherical grid. In this composite icosahedron
spherical matrix all of the 31 primary great circles appear together with the three
sets of secondary great circles. (The three sets of secondary icosahedron great
circles are shown successively at color pla tes 27--29.)
0.57 Fig. 987.132F Net
Diagram of Angles and Edges for Basic Disequilibrium 120 LCD Triangle: This
is a detail of the basic spherical triangle shown shaded in Fig.
0.58 Fig. 987.242
Evolution of Duo-Tet Cube and Hourglass Polyhedron:A. One positive regular
tetrahedron and one negative regular tetrahedron are intersym-metri cally arrayed
within the common nuclear-vertexed location. Their internal trussing permits
their equi-inter-distanced vertexes to define a stable eight-cornered structure, a
‘‘cube.’’ The cube is tetravolume-3; as shown here we observe and AIH at Fig.
987.21 ON. The 48 similar triangles (24 plus, 24 minus) are the surface-system
set of the 48 similar asymmetric tetrahedra whose 48 central vertexes are
congruent in the one—V E’s—nuclear vertex’s center of volume.
0.59 Fig. 987.312
Rhombic Dodecahedron: A. The 25 great circle system of the vector equilibrium
with the four great circles shown in dotted lines. (Compare Fig. 454.06D, third
printing.) B. Spherical rhombic dodecahedron great circle system generated
from six-great-circle system of vector equilibrium, in which the two systems are
partially congruent. The 12 rhombuses of the spherical rhombic dodecahedron
are shown in heavy outline. In the interrelationship between the spherical and
planar rhombic dodecahedron it is seen that the planar rhombus comes into
contact with the sphere at the mid-face point.
0.60 987.326 Stellated Rhombic
Dodecahedron: (A) Rhombic dodecahedron with diamond faces subdivided
into quadrants to describe mid-face centers. Interior lines with arrows show us
mid-face centers. This is the initial rhombic dodeca of tetravolume-6. (B) The
rhombic dodecahedron system is ‘‘pumped out’’ with radii doubled from unit
0.61 987.412 Rational Fraction Edge Increments of 60-degree Great-circle
Subdivid-ings of Vector Equilibrium: When these secondary V E great-circle sets
are projected
0.62 988.00 Polyhedral Evolution: S Quanta Module: Comparisons
of skew polyhedra
0.63 Fig. 988.100 Octa-Icosa Matrix: Emergence of S Quanta
Module: A. Vector equilibrium inscribed in four-frequency tetrahedral grid. B.
Octahedron inscribed in four-frequency tetrahedral grid. C. Partial removal of
grid reveals icosahedron inscribed within octahedron. D. Further subdivision
defines modular spaces between octahedron and icosahedron. E. Exploded view
of six pairs of asymmetric tetrahedra that make up the space intervening between
octa and icosa. Each pair is further subdivided into 24 S Quanta Modules.
Twenty-four S Quanta Modules are added to the icosahedron to produce the
octahedron.
0.64 Fig. 988.12 Icosahedron Inscribed Within Octahedron: The
four-frequency tetrahedron inscribes an internal octahedron within which may
be inscribed a skew icosahedron. Of the icosahedron’s 20 equiangular triangle
faces, four are congruent with the plane of the tetra s faces (and with four external
faces of the inscribed octahedron). Four of the icosahedron’s other faces are
congruent with the remaining four internal faces of the icosahedron. Two-fifths of
the icosa faces are congruent with the octa faces. It requires 24 S Quanta Modules
to fill in the void between the octa and the icosa.
0.65 Fig. 988.13A S Quanta
Module Edge Lengths: This plane net for the S Quanta Module shows the edge
lengths ratioed to the unit octa edge (octa edge = tetra edge.)
0.66 988.13B S
Quanta Module Angles: This plane net shows the angles and foldability of the S
Quanta Module.
0.67 Fig. 988.13C S Quanta Module in Context of Icosahedron
and Octahedron
0.68 990.01.
0.69 995.03 Vector Models of Atomic Nuclei:
Magic Numbers: In the structure of atomic nuclei there are certain numbers of
neutrons and protons which correspond to states of increased stability. These
numbers are known as the magic numbers and have the following values: 2, 8,
20, 50, 82, and 126. A vector model is proposed to account for these numbers
based on combinations of the three fundamental omnitriangulated structures:
the tetrahedron, octahedron, and icosahedron. In this system all vectors have a
value of one-third. The magic numbers are accounted for by summing the total
number of vectors in each set and multiplying the total by Vs. Note that although
the tetrahedra are shown as opaque, nevertheless all the internal vectors defined
by the isotropic vector matrix are counted in addition to the vectors visible on
the faces of the tetrahedra.
0.70 Fig. 995.03A Vector Models of Atomic Nuclei:
Magic Numbers: In the structure of atomic nuclei there are certain numbers of
neutrons and protons which correspond to states of increased stability. These
numbers are known as the magic numbers and have the following values: 2, 8,
20, 28, 50, 82, and 126. A vector model is proposed to account for these numbers
based on combinations of the three fundamental omnitriangulated structures:
the tetrahedron, octahedron, and icosahedron. In this system all vectors have a
value of one-third. The magic numbers are accounted for by summing the total
number of vectors in each set and multiplying the total by 1 ∕ 3. Note that although
the tetrahedra are shown as opaque, nevertheless all the internal vectors defined
by the isotropic vector matrix are counted in addition to the vectors visible on
all the faces of the tetrahedra.
0.71 Fig. 995.31A Reverse Peaks in Descending
Isotope Curve: Magic Numbers
0.72 Fig. 985.31
0.73 401.00 Tensegrity
Tetrahedron with ‘‘Me’’ Ball Suspended at Center of Volume of the Tetrahedron:
Note that the six solid compression members are the acceleration vectors trying
to escape from Universe at either end, by action and reaction; whereas the ends
of each would-be escapee are restrained by three tensors, one long and two
short; while the ball at the center is restrained from local torque and twist by
three triangulated tensors tangentially affixed from each of the four corners.
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