Superficial and Volumetric Hierarchies
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1053.00
Superficial and Volumetric Hierarchies
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1053.15
Because each of the octahedron's eight faces is subdivided
by its respective
six sets of spherical "right" triangles (three positive__three
negative), whose total of 6 × 8
= 48 triangles are the 48 LCD's vector-equilibrium,
symmetric-phase triangles, and
because 120/48 = 2 1/2, it means that each of the vector
equilibrium's 48 triangles has
superimposed upon it 2 1/2 positively askew and 2 1/2
negatively askew triangles from out
of the total inventory of 120 LCD asymmetric triangles
of each of the two sets,
respectively, of the two alternate phases of the icosahedron's
limit of rotational aberrating
of the vector equilibrium. This 2 1/2 positive superimposed
upon the 2 1/2 negative, 120-
LCD picture is somewhat like a Picasso duo-face painting
with half a front view
superimposed upon half a side view. It is then in transforming
from a positive two-and-
one-halfness to a negative two-and-one-halfness that
the intertransformable vector-
equilibrium-to-icosahedron, icosahedron-to-vector-equilibrium,
equilibrious-to-
disequilibriousness attains sumtotally and only dynamically
a spherical fiveness (see Illus.
982.61 in color section).
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1053.36
Sphere: Volume-surface Ratios: The largest number of
similar triangles
into which the whole surface of a sphere may be divided
is 120. (See Secs.
905
and
986.)
The surface triangles of each of these 120 triangles
consist of one angle of 90 degrees, one
of 60 degrees, and one of 36 degrees. Each of these
120 surface triangles is the fourth face
of a similar tetrahedron whose three other faces are
internal to the sphere. Each of these
tetra has the same volume as have the A or B Quanta
Modules. Where the tetra is 1,
the volume of the rhombic triacontahedron is approximately
5. Dividing 120 by 5 = 24 =
quanta modules per tetra. The division of the rhombic
triacontahedron of approximately
tetravolume-5 by its 120 quanta modules discloses another
unit system behavior of the
number 24 as well as its appearance in the 24 external
vector edges of the VE. (See Sec.
1224.21)
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 Fig. 1053.37 |
1053.37
Since the surface of a sphere exactly equals the internal
area of the four great
circles of the sphere, and since the surface areas of
each of the four triangles of the
spherical tetrahedron also equal exactly one-quarter
of the sphere's surface, we find that
the surface area of one surface triangle of the spherical
tetrahedron exactly equals the
internal area of one great circle of the sphere; wherefore
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1053.50
Volumetric Hierarchy: With a nuclear sphere of radius-1,
the volumetric
hierarchy relationship is in reverse magnitude of the
superficial hierarchy. In the surface
hierarchy, the order of size reverses the volumetric
hierarchy, with the tetrahedron being
the largest and the rhombic dodecahedron the smallest.
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1053.51
Table: Volumetric Hierarchy: The space quantum equals
the space domain
of each closest-packed nuclear sphere:
(Footnote 10: The octahedron is always double, ergo, its fourness of volume is its prime number manifest of two, which synergetics finds to be unique to the octahedron.) |
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1053.51A
Table: Volumetric Hierarchy (revised): The space quantum
equals the
space domain of each closest-packed nuclear sphere:
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1053.60
Reverse Magnitude of Surface vs. Volume: Returning
to our
consideration of the reverse magnitude hierarchy of
the surface vs. volume, we find that
both embrace the same hierarchical sequence and have
the same membership list, with the
icosahedron and vector equilibrium on one end of the
scale and the tetrahedron on the
other. The tetrahedron is the smallest omnisymmetrical
structural system in Universe. It is
structured with three triangles around each vertex;
the octahedron has four, and the
icosahedron has five triangles around each vertex. We
find the octahedron in between,
doubling its prime number twoness into volumetric fourness,
as is manifest in the great-
circle foldability of the octahedron, which always requires
two sets of great circles,
whereas all the other icosahedron and vector equilibrium
31 and 25 great circles are
foldable from single sets of great circles .
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1053.601
Octahedron: The octahedron__both
numerically and geometrically__should
always be considered as quadrivalent; i.e., congruent
with self; i.e., doubly present. In the
volumetric hierarchy of prime-number identities we identify
the octahedron's prime-
number twoness and the inherent volume-fourness (in
tetra terms) as volume 22, which
produces the experiential volume 4.
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1053.61
The reverse magnitudes of the surface vs. volume hierarchy
are completely
logical in the case of the total surface subdivision
starting with system totality. On the
other hand, we begin the volumetric quantation hierarchy
with the tetrahedron as the
volumetric quantum (unit), and in so doing we build
from the most common to the least
common omnisymmetrical systems of Universe. In this
system of biggest systems built of
smaller systems, the tetrahedron is the smallest, ergo,
most universal. Speaking
holistically, the tetrahedron is predominant; all of
this is analogous to the smallest chemical
element, hydrogen, being the most universally present
and plentiful, constituting the
preponderance of the relative abundance of chemical
elements in Universe.
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1053.62
The tetrahedron can be considered as a whole system
or as a constituent of
systems in particular. It is the particulate.
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1053.70
Container Structuring: Volume-surface Ratios
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1053.71
When attempting to establish an international metric
standard of measure for
an integrated volume-weight unit to be known as "one
gram" and deemed to consist of
one cubic centimeter of water, the scientists overlooked
the necessity for establishing a
constant condition of temperature for the water. Because
of expansion and contraction
under changing conditions of temperature a constant
condition of 4 degrees centigrade
was later established internationally. In much the same
way scientists have overlooked and
as yet have made no allowance for the inherent variables
in entropic and syntropic rates of
energy loss or gain unique to various structurally symmetrical
shapes and sizes and
environmental relationships. (See Sec.
223.80, "Energy
Has Shape.") Not only do we have
the hierarchy of relative volume containments respectively
of equiedged tetra, cube, octa,
icosa, "sphere," but we have also the relative surface-to-volume
ratios of those geometries
and the progressive variance in their relative structural-strength-to-surface
ratios as
performed by flat planes vs simple curvature; and as
again augmented in strength out of
the same amount of the same material when structured
in compound curvature.
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1053.72
In addition to all the foregoing structural-capability
differentials we have the
tensegrity variables
(see Chap. 7),
as all these relate
to various structural capabilities of
various energy patternings as containers to sustain
their containment of the variously
patterning contained energies occurring, for instance,
as vacuum vs crystalline vs liquid vs
gaseous vs plasmic vs electromagnetic phases; as well
as the many cases of contained
explosive and implosive forces. Other structural variables
occur in respect to different
container-contained relationships, such as those of
concentrated vs distributive loadings
under varying conditions of heat, vibration, or pressure;
as well as in respect to the
variable tensile and compressive and sheer strengths
of various chemical substances used
in the container structuring, and their respective heat
treatments; and their sustainable
strength-time limits in respect to the progressive relaxing
or annealing behaviors of various
alloys and their microconstituents of geometrically
variant chemical, crystalline, structural,
and interproximity characteristics. There are also external
effects of the relative size-
strength ratio variables that bring about internal interattractiveness
values in the various
alloys as governed by the second-power rate, i.e., frequency
of recurrence and intimacy of
those alloyed substances' atoms.
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1053.73
As geometrical systems are symmetrically doubled in
linear dimension, their
surfaces increase at a rate of the second power while
their volumes increase at a third-
power rate. Conversely, as we symmetrically halve the
linear dimensions of geometrical
systems, their surfaces are reduced at a second-root
rate, while their volumes decrease at a
third-root rate.
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1053.74
A cigar-shaped piece of steel six feet (72 inches)
long, having a small hole
through one end and with a midgirth diameter of six
inches, has an engineering slenderness
ratio (length divided by diameter) of 12 to 1: It will
sink when placed on the surface of a
body of water that is more than six inches deep. The
same-shaped, end-pierced piece of
the same steel of the same 12-to-1 slenderness ratio,
when reduced symmetrically in length
to three inches, becomes a sewing needle, and it will
float when placed on the surface of
the same body of water. Diminution of the size brought
about so relatively mild a
reduction in the amount of surface of the steel cigar-needle's
shape in respect to the great
change in volume__ergo, of weight__that its shape became
so predominantly "surface"
and its relative weight so negligible that only the
needle's surface and the atomic-intimacy-
produced surface tension of the water were importantly
responsible for its
interenvironmental relationship behaviors.
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1053.75
For the same reasons, grasshoppers' legs in relation
to a human being's legs
have so favorable a volume-to-surface-tension relationship
that the grasshopper can jump
to a height of 100 times its own standing height (length)
without hurting its delicate legs
when landing, while a human can jump and fall from a
height of only approximately three
times his height (length) without breaking his legs.
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1053.76
This same volume-to-surface differential in rate of
change with size increase
means that every time we double the size of a container,
the contained volume increases
by eight while the surface increases only fourfold.
Therefore, as compared to its previous
half-size state, each interior molecule of the atmosphere
of the building whose size has
been symmetrically doubled has only half as much building
surface through which that
interior molecule of atmosphere can gain or lose heat
from or to the environmental
conditions occurring outside the building as conductively
transferable inwardly or
outwardly through the building's skin. For this reason
icebergs melt very slowly but
accelerate progressively in the rate of melting. For
the same reason a very different set of
variables governs the rates of gain or loss of a system's
energy as the system's size
relationships are altered in respect to the environments
within which they occur.
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1053.77
As oil tankers are doubled in size, their payloads
grow eightfold in quantity
and monetary value, while their containing hulls grow
only fourfold in quantity and cost.
Because the surface of the tankers increases only fourfold
when their lengths are doubled
and their cargo volume increases eightfold, and because
the power required to drive them
through the sea is proportional to the ship's surface,
each time the size of the tankers is
doubled, the cost of delivery per cargo ton, barrel,
or gallon is halved. The last decade has
seen a tenfolding in the size of the transoceanic tankers
in which both the cost of the ship
and the transoceanic delivery costs have become so negligible
that some of the first such
shipowners could almost afford to give their ships away
at the end of one voyage. As a
consequence they have so much wealth with which to corrupt
international standards of
safety that they now build them approximately without
safety factors__ergo, more and
more oil tanker wrecks and spills.
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