Closest Packing of Spheres
410.00
Closest Packing of Spheres
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410.01
Nature's Coordination
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410.011
About 1917, I decided that nature did not have separate,
independently
operating departments of physics, chemistry, biology,
mathematics, ethics, etc. Nature did
not call a department heads' meeting when I threw a
green apple into the pond, with the
department heads having to make a decision about how
to handle this biological encounter
with chemistry's water and the unauthorized use of the
physics department's waves. I
decided that it didn't require a Ph.D. to discern that
nature probably had only one
department and only one coordinate, omnirational, mensuration
system.
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410.02
I determined then and there to seek out the comprehensive
coordinate
system employed by nature. The omnirational associating
and disassociating of
chemistry__always joining in whole low-order numbers,
as for instance H2O and never
HpiO__persuaded me that if I could discover nature's comprehensive
coordination, it
would prove to be omnirational despite academic geometry's
fortuitous development and
employment of transcendental irrational numbers and
other "pure," nonexperimentally
demonstrable, incommensurable integer relationships.
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410.03
I was dissatisfied with abstract, weightless, unstable,
ageless,
temperatureless, straight-line-defined squares and cubes
as models for calculating our
omnicurvilinear experience. I was an early rebel against
blockheads and squares.
Reviewing the history of chemical science, I became
intuitively aware that the clue to
vectorial, volumetric, geometrical coordination with
physical reality and all the
fundamental energetic experiences of reality, such as
temperature, time, and force, might
be found in Avogadro's experimental proof of his earlier
hypothesis, which stated that all
gases under identical conditions of heat and pressure
will always disclose the same
number of molecules per given volume. Here was disclosed
a "Grand Central Station"
accommodating all comers; despite "fundamental" or elementarily
unique differences of
identity, all accommodated on a common volume (space)-to-number
basis. One molecule
of any element: One space. A cosmic democracy.
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410.04
I felt intuitively that inasmuch as the variety of
gases experimented with
often consisted of only one unique chemical element,
such as hydrogen or oxygen, and
that inasmuch as these gases also could be liquefied,
and also inasmuch as most of the
elements are susceptible to some heat- or pressure-produced
transformation between their
liquid, crystalline, and vapor, or incandescent, states,
it might also be hypothetically
reasonable to further generalize Avogadro's hypothesis
by assuming that, under identical
energy conditions, all elements may disclose the same
number of "somethings" per given
volume. Such a generalized concept is not limited to
pressure and heat: we wanted to be
much more inclusive, so we said we assume that all the
conditions of energy are identical;
this includes not only the pressure and heat conditions
of thermodynamics, which
developed before electromagnetics became an applied
realization, but also the conditions
of electromagnetics as constant.
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410.05
I went on from there to reason that vectors, being
the product of physical
energy constituents, are "real," having velocity multiplied
by mass operating in a specific
direction; velocity being a product of time and size;
and mass being a volume-weight
relationship. On impact, mass at velocity transforrns
into heat and work. These energy
factors can be translated not only into work, but into
heat or into time as well.
Furthermore, electromagnetic scientists had found that
all their EMF (electromotive force)
problems could be graphed vectorially; the fact that
"graphable" or "modelable" vectors
can interact modelably in real Universe space seemed
to promise that the equations of
nature's omnicoordinate transactions, expressed in omni-space-intruding
vectorial models,
might produce real models of reality of nature's Grand
Central Station of
omnicoordination.
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410.06
So I then went on to say that, if all the energy conditions
were everywhere
the same, then all the vectors would be the same length
and all of them would interact at
the same angle. I then explored experimentally to discover
whether this "isotropic vector
matrix," as so employed in matrix calculus, played with
empty sets of symbols on flat
sheets of paper, could be realized in actual modeling.
Employing equilength toothpicks
and semi-dried peas, as I had been encouraged to do
in kindergarten at the age of four
(before receiving powerful eyeglasses and when I was
unfamiliar with the rightangled
structuring of buildings as were the children with normal
vision), I fumbled tactilely with
the toothpicks and peas until I could feel a stable
structure, and thus assembled an
omnitriangulated complex and so surprised the teachers
that their exclamations made me
remember the event in detail. I thus rediscovered the
octet truss whose vertexes, or
convergent foci, were all sixty-degree-angle interconnections,
ergo omniequilateral,
omniequiangled, and omni-intertriangulated; ergo, omnistructured.
Being
omnidirectionally equally interspaced from one another,
this ornni-intertriangulation
produced the isotropic matrix of foci for omni-closest-packed
sphere centers. This opened
the way to a combinatorial geometry of closest-packed
spheres and equilength vectors.
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410.07
Over and over again, we are confronted by nature
obviously formulating her
structures with beautiful spherical agglomerations.
The piling of oranges, coconuts, and
cannonballs in tetrahedral or half-octahedral pyramids
has been used for centuries and
possibly for ages. Almost a half-century ago, F. W.
Aston, the British scientist first
identified for physics the most economical uniradius
spherical interagglomerations as the
"closest packing of spheres," which had fresh interest
for the physicist and
crystallographer because of the then recently discovered
microscopic realization that
nature frequently employed omni-intertriangulated systems,
which hold mathematical clues
to the principles of symmetrical coordination governing
natural structure, the dynamic
vectorial geometry of the atomic nucleus, as well as
of the atoms themselves.
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410.10
Omnitriangulation of Sixty Degrees
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410.11
The closest packing of unit radius spheres always
associates in
omnitriangulations of 60 degrees, whether in planar
or omnidirectional arrays. Six unit
radius spheres pack most tightly around one sphere on
a billiard table. Twelve unit radius
spheres compact around one sphere in omnidirectional
closest packing.
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410.12
If we take three billiard balls on a flat table, we
find that they compact
beautifully into a triangle. If we arrange four of them
on a billiard table in a square, they
tend to be restless and roll around each other. If compacted
into a condition of stability,
the four form a 60-degree-angled diamond shape made
of two stable triangles.
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411.00
Four Spheres as Minimum System
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411.01
The South Seas islander piling his coconuts, the fruit
dealer selling oranges,
and the cannoneer stacking equiradius cannonballs, or
the much earlier round-rock-
slinging soldiers were probably the first to learn about
the closest packing of spheres. The
stacking of balls in symmetrical rows and layers leads
inevitably to a stable pyramidal
aggregate.
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411.02
Closest packing of spheres does not begin with a nucleus.
Closest packing
begins with two balls coming together.
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411.03
One ball cannot zoom around alone in Universe. Without
otherness, there is
no consciousness and no direction. If there were only
one entity__say it is a sphere called
"me"__there would be no Universe: no otherness: no awareness:
no consciousness: no
direction. Once another entity__let's say a sphere__is
sighted, there is awareness and
direction. There is no way to tell how far away the
other sphere may be or what its size
may be. Size sense comes only with a plurality of comparative
experiences.
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411.04
As a single sphere, now aware of an otherness sphere
somewhere out there,
"me" has to rotate about without restraint and can observe
its rotation in relation to the
otherness, but could misassume the otherness sphere
to be zooming around it, as there are
no third, or more, othernesses by which to judge.
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 Fig. 411.05 |
411.05
When moved unknowingly toward another by mass attraction,
the "other"
ball and the "me" ball, either or both of which could
have been rotating as they approach
one another, each misassumes the other to be growing
bigger, until finally they touch each
other. Now they can roll around upon one another, and
they might be cotraveling
together, but there is not as yet anywhere to go because
there is no otherness than their
joint selves relative to which to travel. They can't
go through each other and they can't
get away from each other. They are only free to rotate
upon each other, and because of
friction they must do so cooperatively.
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411.06
When a third ball looms into sight, providing a sense
of direction for the
tangently rolling-upon-each-other first couple, the
third one and the first two are mass-
attracted toward each other and finally make contact.
The newcomer third sphere may roll
around on one of the first couple until it rolls into
the valley between the first two. The
third ball then gets locked into the valley between
the first two by double mass attraction,
and now becomes tangent to both of the first two.
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411.07
The three balls, each one tangent to both of the others,
now form an
equiangular triangular group with a small opening at
their center. The friction of each of
their double contacts with the other two gives them
a geartooth interaction effect. With
two gears, one can turn clockwise and the other can
tum counterclockwise. Even numbers
of gears reciprocate; odd numbers of gears block one
another. Thus our three balls can no
longer roll circumferentially around each other; they
can only rotate cooperatively on the
three axes formed between each of their two tangent
contact points. The friction between
their surface contacts forces all three to rotate unidirectionally
about their respective
contact axes, which are parallel to the edges of the
equilateral triangle defined by the three
sphere centers; i.e., the three spheres can now only
co-rotate over and into the hole at the
center of the triangle, and out and away upwardly again
from the bottom of the center
hole. Thus the three balls can involute or evolute axially,
like a rubber doughnut in respect
to the hole at their triangular center, but they cannot
rotate circumferentially.
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411.08
Finally, a fourth ball appears in Universe and is
mass-attracted to tangency
with one of the three previously triangulated spheres,
then rolls on one of the joined balls
until it falls into the valley formed by the hole at
the top center of the triangulated three;
but being of equal diameter with the other three balls,
it cannot fall through that hole
whose radius is less than those of the associated spheres.
Thus nested in the central valley,
the fourth sphere now touches each of the other three
and vice versa. The four closest-
packed spheres make a closest-packing array. They are
mass-attractively locked together
as a tetrahedron.1 No further interrotation is possible.
As a tetrahedron, they form the
minimum stable structural system and provide the nuclear
matrix for further mass
attraction and closest-packed growth of additional spheres
falling into their four-surface
triangular nests. This produces increasing numbers of
closest-packed nests. Thus do atoms
agglomerate in closest packing in tetrahedrally conformed
arrays, often truncated
asymmetrically at corners and along edges to obscure
the tetrahedral origin of the
collection.
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(Footnote 1: There is an alternative sequence, which is perhaps more likely, in which the balls would first join as two pairs__like dumbbells "docking" in space. Then our old friend precession would cause them to form a momentary square, only to have one pair revolve until its axis is at 90 degrees to the other couple's axis, whereafter they dock to interlock as a tetrahedron.) |
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411.10
Unpredicted Degrees of Freedom: Reviewing the history
of self-discoveries
of restraints progressively and mutually interimposed
upon one another by the arrival of
and association with successive othernesses, self may
discover through progressive
retrospections and appreciate the significance of theretofore
unrecognized and unrealized
degrees of cosmic freedom successively and inadvertently
deducted from the original total
inventory of unexpended self-potentials with which the
individual is initially, always and
only, endowed. Only with the progressive retrospection
inventoryings induced only by
otherness-developed experience of awareness does the
loss of another degree of freedom
become consciously subtracted from the previous experience
inventory of now
consciously multiplying rememberable events.
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411.11
With the discovery of principles through progressive
deprivations dawns
new awareness of the elective employability of the principles
initially separated out from
their special-case experiencing to self-control more
and more of the pattern of events.
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411.12
Only through relationships with otherness can self
learn of principles; only by
discovery of the relationships existing between self
and othernesses does inspiration to
employ principles objectively occur. There is nothing
in self per se, or in otherness per se,
that predicts the interrelatedness behaviors and their
successively unique characteristics.
Only from realization of the significance of otherness
can it be learned further that only by
earnest commitment to others does self become inadvertently
behaviorally advantaged to
effect even greater commitment to others, while on the
other hand all selfseeking induces
only ever greater self-loss. There is an alternate sequence,
which is perhaps more likely, in
which the balls would first join as two pairs__like dumbbells
"docking" in space. Then our
old friend precession would cause them to form a momentary
square, only to have one
pair revolve until its axis is at 90 degrees to the
other couple's axis, whereafter they dock
to interlock as a tetrahedron.
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411.20
Discovery as a Function of Loss: It is a basic principle
that you only
discover what you had had by virtue of losing it. Due
to our subconscious organic
coordination, you don't know what you are losing until
you lose it. Naught can be so
advantageous as thoughtfully considered loss and resolve
to employ the principles thereby
discovered for others. You don't know how much you have
to give until you start trying
to give. The more you try to give effectively to advantage
others, the more you will
possess to give, and vice versa.
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411.21
Retrospective awareness of losses can bring preoccupation
with self,
blinding self to recognition of the synergetic gains
that, by virtue of the second-power law,
have brought group advantage gains in which the individual
has attained fourth-power
continuance potential often way offsetting individual
freedom losses, particularly in view
of the group's discovery that as a group it can enjoy
all the original freedoms individually
lost but never realized by the individual to exist__ergo,
unemployable consciously by the
individual, who was more of a victim of the unknown
freedoms than an enjoyer.
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411.22
Only as group structuring occurs do the discovered
cosmic freedoms
become consciously employable__employable effectively
only for all and not for self. It is
when this retrospective discovering is made by the grouped-in
individual and he tries to
employ the freedoms exclusively for self or exclusive
subdivisions of the group, that his
attempts become inherently unfulfillable and scheduled
for ultimate failure.
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411.23
Self-seeking brings a potential loss that engenders
first caution, then fear:
fear of change; change being inexorable, fear increases
and freezes. Self-seeking always
eventuates in self-destruction through inability to
adapt.
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411.30
Intergeared Mobility Freedoms: Only with the arrival
of the second
otherness do individuals become aware of the loss of
mutual anywherearound-one-another
rollability, and then discover that they have also lost
the ability to go in all directions, for
they cannot go through each other. It is inferred that
they haven't lost mutually
accomplishable omnidirectional mobility. (Here is an
example of one of those
comprehendings from an apprehending.) With the acquisition
of the second otherness, self
discovers what it has lost which self didn't know it
had, until the loss brought
retrospective awareness of the lost freedom: an inter-anywhere-roll-aroundness
with the
first other.
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411.31
With the mutually interattracted threeness, each having
two contacts with
their two otherness partners, they learn, as a fourth
otherness nests into their triangular
opening, that they have now lost a frictionally intergeared
mutually evoluting-involuting
rotational freedom (torus). Now blocked by the frictionally
intergeared fourth otherness,
the mutual omnidirectionality of the structural system
so produced by that structural
system can be discovered only by the self-observation
of the realization of another
structural system's cumulative repetition repeating
the evolutionary accumulation of its
own fourfolding; observing that the other structural
system can move omnidirectionally,
their observed rotations and magnitude changes can be
explained only by the
omnidirectional freedom only mutually experienceable
by the whole individual structural
system as it had been originally and only subconsciously
experienced by the individual self.
Naught has been lost. Much has been mutually gained.
Each can take off from and return
to the others.
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411.32
The variations of the features of second-structural-system
otherness can be
explained only by the self-structural system assumption
of increasing distance of travel of
the otherness to and away, or by the other system's
experiencing a freedom theretofore
unself-realized: that of individual expansion and contraction.
For any one of the four
members of a structural system team can expand and contract
coordinatedly at individual
rates, mutual rates, or interpaired rates, provided
one does not become so small as to "fall
through" the triangular opening at its nest bottom.
(If it fell through, one otherness would
start rotating hingewise around the axis of tangent
contact with the next-largest, without
touching the fourth and next-smallest-to-self.)
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411.33
Thus the self-structural system discloses by observation
of otherness's
system changing features that its own system had been
enjoying, as with freedoms of
which it was previously unconscious.
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411.34
Our inventory of intergeared mobility freedoms is
fourfold. It is four-
dimensional:
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411.35
The inward-outward expandability is the basis of convergence-divergence
and
radiation-gravitation pulsation__which seems furthest
from man's awareness. This is what
science has discovered: a world of waves in which waves
are interpenetrated by waves in
frequency modulation. There is a systemic interrelationship
of basic fourness always
accompanied by a sixness of alternatives of freedoms.
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411.36
When a sphere gets so small that it can roll through
a hole between other
close packed spheres, the omnidirectionality of any
one individual would not be impeded
under the following circumstances:
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411.37
Rate occurs only when there is terminal. Rate is a
modulation between
terminals. With termination, a system's integrity is
brought about by the individually
covarying magnitudes and the omnidirectional experience
pulls on the system.
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411.38
The degree of freedom that is lost is discoverable
only retrospectively by the
very fact of the loss. It is an inverse synergetic behavior
wherein no feature of the self part
predicted the successive behaviors of the whole and
where the individual part freedoms
were only mutually disclosed by their subsequently realized
loss.
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