Growth and Decay
[1053.80-1053.85 Growth and Decay Scenario]
In chemical interbonding of atomic systems single-bonded
tetrahedra are only single-vertex-to-single-vertex congruent.
This means that only one of
each of any two tetrahedra's directionally differentiable
four corners__which are as yet
only infra- or ultratunable, only noisy subsystem, vertexial
subcongruent critically intimate; that is, the magnitude
of their mutual interattractiveness is
greater than any other of their cosmic attracters. Singly
interbonded tetrahedra are always
attracted in critically intimate degree by one__and only
one__of their corner-identifying
infratunable systems attractively bonded with a neighboring
tetrahedron's corner vertex
subdifferentiable-system "points." For ages the vertexial
superficially apprehended by humans were experientially
identified visually as "specks,"
audibly as noises, tactilely as prickly points, topologically
as vertexes, and geometrically as
sharp (corner) angles.
Topology enumerates the critical-proximity-bonded
pairs of "points" as
constituting only one point and not as an almost tangent
two. Topological accounting is
confined to only superficially visible characteristics
of systems. (See Sec.
We learn experientially that lines are trajectories
521.20), that two
events and their trajectories cannot pass through the
same point at the same time (Secs.
and that when we have such conflict or transit
interference, they result in
smashes (always separating each of the intersmashing
bodies into a plurality of smaller
systems, not dirt or dust), plunge-ins such as meteors
plunging into Earth (to form more
complex systems), refractions, reflections, or critical-proximity
cotravelings (Earth and Moon). When we do not have interference
conflicts but we have
two independent event trajectories converging to pass
"near" one another only at a
precessionally critical-course-refracting, mass-interattractive
distance, they may converge
and diverge in a twist vertex exit (see Secs.
942.12). The term vertex embraces
all of the foregoing system-furnished, local-focal,
In chemical double-bonding the edge vectors of the
tetrahedra__as well as
the terminal vertexes__are also so critically proximate
as superficially to seem to be
congruent and are topologically accredited numerically
only as "one" because of their
superficial aspect of unity as a single hinge-pin.
The vector equilibrium consists of eight tetrahedra
each of which is edge-
bonded; i.e., vertexially double-interbonded with three
others, with each of their pre-time-
size internal vertexes theoretically congruent as eight-in-one.
Each of the pre-time-size
vector equilibrium's eight tetrahedra has six vector
edges (6 × 8 = 48). (There are 24
internal and 24 external vector edges, 48 vector edges
in all.) Each of the eight tetrahedra
has four vertexes (4 × 8 = 32), and in each of the tetrahedra
three of these vertexes are
external (3 × 8 = 24): There are thus 12 externally
paired sets (24/2 = 12) of visible
vertexes. Three of each of the eight tetrahedra's vector
edges (3 × 8 = 24) are displayed
on the outside of the vector equilibrium. (Compare Sec.
There are 24 external vector edges of the vector equilibrium
(8 × 3 = 24).
The other three vector edges of each of the eight tetrahedra
are arrayed inwardly as 24
internal edges (8 × 3 = 24), but these inwardly arrayed
vector edges of the eight
tetrahedra, being double-bonded or hinged together,
appear as only 12 radial spokes of the
vector equilibrium, which has 24 separate vectors in
its four closed chordal rims of the
four great-circle planes of the tetrahedra's four dimensionality;
these four great circles
produce the zerovolume tetrahedron. (See Sec.
Nature never stops or even pauses at dead center.
convergently to the center of its nuclear sphere, where
each of its frequency-tuned
integrities self-interfere convergently and react reflectively__ergo,
their own terminally convergent self-frequency interferings.
Unity is plural and at minimum
two. (See Secs.
In the vector-diametered VE the convergent 2 l/2 phase
coalesces with the
divergent 2 l/2 phase and produces a univalent 5-ness
whose consequence is also
quadrivalent__producing also the vector-radiused VE's
5 × 4 = 20-ness of the vector
equilibrium's subfrequency embracement of its eight
edge-bonded, bivalent tetra and their
six half-octahedra interstices.
Life and Death
The decaying and growing are complementary. Death
is a cofunction of
birth: the father is dying; the child is being born.
There never has been a real negative
except as a positively complementary function of the
oppositely directioned positive.
We do not have two Universes__"this world" and "the
next world." Death is
only the nonresonant, between-frequency silence of our
passages through the Grand Central Station of the vector
equilibrium's equilibrious center,
as the lags in our cognition "realizations" time us
into life's inherently aberrated
imperfection aspects__ somewhere off center.
As we learn through experience to identify and comprehend
inclusively and precisely the generalized principles
manifest in our experiences, and as we
learn to communicate and share our recognitions of these
manifests, we gradually reduce
the lag rates in human cognition and come ever nearer
to realization of the perfection.
Apprehension is the physical brain's coordinate storing
of all the special
case, physically sensed information of otherness, integral
(the child's thumb sucked by its
mouth) or separate (the mother's udder sucked by the
child's mouth.) Comprehension is
the metaphysical mind's discovery of the meaningful
interrelationship between the special-
case information data that are neither implicit in,
nor inferred by, any of the special-case
information data when taken only separately__the meaning
discovered by mind being the
generalized principles manifest exclusively by the interrelationship
variables and constants.
Awareness means apprehending while also intuitively
comprehending that the excitement
over the novelty of the incoming information is significant
because possibly pregnant with
meaningful principles. (Compare Sec.
Since "life" is experientially demonstrable to be
metaphysical__its awareness and comprehension of meanings
synchronize exclusively with
the nonphysical intervals concentrically occurring between
the only physically sensed
frequencies of exclusively inanimate, radiantly propagated,
Both death and life are complementary metaphysical
and embracing our electromagnetic physical experience.
Life's physical reality is
constituted by the unique frequency identifications
of the chemical elements and their
atomic components as well as the humanly tune-in-able
"color" frequencies of the
electromagnetic spectrum's concentrically interpositioned
occurrences (usually published
by humans as a chart of positions along any one radius
of the omnidirectional
comprehensive concentric system). The metaphysical cognition
of life-death reality is
constituted exclusively by all the intervals between
and beyond__inwardly and
outwardly__ all of the comprehensive electromagnetic
phenomena sensed by human
The music of John Cage is preoccupied with the silent
intervals; his growing
audience constitutes the dawning of the transition of
all humanity into synchronization
with the metaphysical rather than the physical. The
decibel amplification of youth's "rock"
music has switched its physical beat into the old silent
intervals and is inducing
metaphysical preoccupation in its listeners.
Positive Visible and Integral Invisible
To free ourselves from our preconditioned ill-chosen
words of plus-minus
and positive-negative, we may say operationally that
there never has been a minus
Universe to cofunction with Universe. There has always
been cosmically integral, visible
and invisible experience, which we have learned only
in the past 100 years to be the
consequence of whether or not we are integrally equipped
organically with receiving sets
having frequency tunability under the particular electromagnetic-waveband
Radiation outcasts. Radiation does not broadcast;
broadcast is a planar
statement; there are no planes. Out is inherently omnidivergent.
Radiation omnicasts but
does not and cannot incast; it can only go-in-to-go-out.
In is gravity.
If radiation "goes through" a system and comes out
on the other side, it does
so because (I) there was no frequency interference__it
just occurred between the system's
occurrence frequencies__or (2) there was tangential interference
and deflection thereby of
the angle of travel, wherefore it did not go through;
it went by.
Cay and Decay
In Webster's dictionary cay is an "emergent reef of
coral or sand." We
deduce that its earlier etymological meaning is a "growth,"
a coming together of parts (of
sand or coral creatures)__ergo, we have cay and de-cay.
Cay is convergently associative
and syntropically cumulative. Decay is divergently disassociative
The nuclear vector equilibrium with a frequency of
one has a double intensity
(quadrivalent) tetravolume of 5 with a convergent cay
volume of 2 1/2 and a divergent
decay volume of 2 1/2; a congruent double 2 1/2 whose
energy involvement potential is 5.
In the generalized (subfrequency) case of the nuclear
(pulsatively impotent), either convergent or divergent
(not both) quadrivalent tetravolume
where frequency is half-zero, the tetravolume of the
VE0 = 2 l/2.
In the generalized (subfrequency case) of the nuclear
(potentially pulsative), congruently one-half-convergent
quadrivalent tetravolume where frequency is zero, the
half-convergent tetravolume of 2
2 compounded with the half-divergent tetravolume of
2 1/2 produces a double intensity
two-and-a-halfness which has__an only potential__quadrivalent
tetravolume of 5; ergo,
VE0 = 5, one-half of which is alternatively invisible;
ergo, VE0 appears deceptively to
have a tetravolume of 2 1/2.
In the generalized (subfrequency) nucleus-embracing,
bivalent tetravolume vector equilibrium of frequency
one, its tetravolume is 20. VE1 = 20.
In the generalized (subfrequency) nucleus-embracing,
vector equilibrium of frequency two, the tetravolume
is 160. VE2 = 160. (See Sec.
What must be remembered in considering all the foregoing
is that unity is
plural and at minimum two, as elucidated in Secs.
1070; wherefore the zero-
frequency vector equilibrium, the VE0 of "apparent"
tetravolume 2 1/2, has an inherent
but invisible double value that will have an operational
resource effectiveness of 5, 2 1/2 of
which is convergently effective and 2 1/2 divergently
effective. This produces the state of
equilibrium whose untenability induces cosmic resonance.
In the symmetrical doubling of linear (radial) dimension
the surface area
increases four times and the volume eight times their
original magnitude. In the case of the
nuclear (one sphere) vector equilibrium with radius
= 1 and volume = 2 1/2, when
surrounded with 12 closest-packed, uniradius spheres
and when the center of the nuclear
sphere is connected to the respective centers of the
12 surrounding spheres, the distance
between the center of the nuclear sphere and the center
of any one of its 12 surrounding
spheres is equal to 2 radii, or one diameter of the
uniradius spheres. With radius 2, 2 1/2 ×
8 = 20. (Compare Sec.
Table: Initial Frequencies of Vector Equilibrium:
|Radius 1||VE0/2|| 2 1/2
|Radius 1|| VE0|| 5
|Radius 2|| VE1|| 20
|Radius 4|| VE2|| 160
Inventory of Alternatives to Positive
What are the relative frequency ranges involved? (Compare
|| range-reachable, frequency dense, ergo interferable,
ergo "solidly" or firmly touchable vs out-of-reach untunable
|| cold-warm; also frequency conditions
||infra- or ultratunable
||sound and noise; we say "noise" when the frequencies
are not differentiable but altogether overlap the frequency limit of our equipment
|| frequency; again, electromagnetic
|| infra- and ultratunable
|| distance factor not a matter of resolution but of wavelength
||you can't differentiate the untunable
||sweet vs obnoxious
||decay; the divergent, the coming apart; decaying tends to be malodorous
||cay (growth); the convergent freshness tends to be olfactorily welcome
||attractions and repulsions.
Relationship of Gibbs to Euler
Synergetic Analysis: Euler's topology and Willard Gibbs'
phase rule give
us synergetic-analysis capability. Euler differentiated
all physical Universe into lines,
crossings, and areas: the fundamental visual aspects
of our experiences having to do with
our eyes, radiation frequencies, and conceptual images.
Gibbs' phase rule differentiated
the physical Universe into liquid, crystalline, and
gaseous phases, which are not so much
visual as thermal, which is tactile, and which are always
characterized by unique whole-
number interattractions, i.e., restraints. Conversely,
with successive whole-number degrees
of freedom, thermal, sonic, or viscosity frequencies
are differentiated in respect to their
condition within their respective states as well as
between those states.
Euler's synergetic differentiation and equatingly accomplished
of Universe deals with energy disassociative as radiation;
Gibbs deals with energy
associative as matter at various thermal stages. Euler's
and Gibbs' are two different
system aspects or behaviors of Universe. Euler deals
with the static, geometrical field
aspects of Universe. Gibbs deals with energy associative
as matter, and what the degrees
of energetic freedom may be within a local physical
complex, and what amounts of energy
would have to be added locally to bring about other
Relationship of Gibbs to Euler
|Energy as radiation||Energy as matter
|They come together
|in vertexial bonding
|which is always|| = Mass attraction = ||which is kinetically
|potentially directional|| active in GIBBS
|and descriptive in EULER
Synergetic Integration of Topology and Quanta: Synergetics'
"breakthrough" integration of Euler's topology and Willard
Gibbs' phase rule is explained
by the number of intertetrahedral bonds:
|Phases:|| Bonds:|| States:
|energies present in||=||Medium +2||S
|the eccentric phases||phase||I
The rigid ice stage is characterized by load concentration,
no degrees of
freedom, and slow creep. The flexible, fluid stage is
characterized by hinge-bonding, load
distribution, one degree of freedom, and noncompressibility.
The flexible, fluid vapor stage
is characterized by universal jointing, load distribution,
six degrees of freedom, and
Median unity is two, therefore unity plus two equals
|Median state|| = Unity + 2
|Frozen state|| = Median unity - 1 = 1
|Vapor state|| = Median unity + 1 = 3
(3+ 1 =4; 1 +3=4; 2+2=4)
|Ice|| = Median freedom minus one freedom
|Water|| = Median freedoms
|Vapor|| = Median freedoms plus one freedom
Polyhedral Bonding: Willard Gibbs' phase rule treats
with the states of the
environment you can sense with your eyes closed: crystallines,
liquids, gases, and vapors.
Euler's points, lines, and areas are visually described,
but they too could be tactilely
detected (with or without fingers).
The mathematicians get along synergetically using Euler's
topology alone. It
is the chemists and physicists who cannot predict synergetically
without using Gibbs'
Euler deals with the superficial aspects of polyhedra:
of visual conceptuality.
He deals only with the convex surfaces of polyhedral
systems. Euler deals with unit,
integral, single polyhedra, or with their subaspects.
He is not concerned with the modus
operandi of the associabilities or disassociabilities
of a plurality of polyhedra.
But Gibbs unknowingly deals with polyhedra that are
composited of many
polyhedra, i.e., compounds. He does not think or talk
about them as polyhedra, but we
find the connection between Euler and Gibbs through
the polyhedral bonding in respect to
Euler's aspects. Euler's lines are double bonds, i.e.,
hinges. Euler's vertexes are single
bonds. Euler's areas are triple bonds. Gibbs accommodates
the omnidirectional system
complementations of the other senses__thermal, tactile,
aural, and olfactory__not just
associatively, but radiationally. Gibbs brings in time.
Time is tactile. Time is frequency.
Our pulses measure its passing.
People see things move only relative to other things
and feel small vibrations
when they cannot see motion. The tactile feels angular
promontories or sinuses with the
fingers or body. Sinus means "without"__ "nothing," invisible,
ergo, nonidentified by
Euler. The frequencies we call heat are tactilely sensed.
We have radiation-frequency
tunability range. Our skin structuring is tuned to frequencies
beyond the eye-tunable
range, i.e., to ultraviolet and infrared.
Euler did not anticipate Gibbs. Gibbs complements Euler__as
synergetics' identification of the two excess vertexes
as constituting the axis of conceptual
observation in respect to all independent, individual
orientations of all systems and
subsystems; i.e., quantum mechanics' abstract, nonspinnable
We find Euler and Gibbs coming together in the vertexial
polyhedral "corners," or point convergency of polyhedral
lines. The bonds have nothing to
do with the "faces" and "edges" they terminally define.
Two bonds provide the hinge,
which is an edge bonding. One bond gives a universal
joint. Triple or areal bonding gives
Mass-interattraction is always involved in bonding.
You may not have a
bond without interattraction, mass or magnetic (integral
or induced), all of which are
precessional effects. As Sun's pull on Earth produces
Earth orbiting, orbiting electrons
produce directional field pulls. This was not considered
by Euler because he was dealing
only with aspects of a single system.
Gibbs requires the mass-interattraction without saying
interattraction is necessary to produce a bond. Gases
may be tetrahedrally bonded singly,
corner to corner, or as a universal joint. Gibbs does
not say this. But I do.
Copyright © 1997 Estate of R. Buckminster Fuller