The synergetics principles described in this work are
Principles are entirely and only intellectually discernible.
generalized mathematical principles govern subjective
comprehension and objective
realization by man of his conscious participation in
evolutionary events of the Universe.
Pure principles are usable. They are reducible from
theory to practice.
A generalized principle holds true in every case.
If there is one single
exception, then it is no longer a generalized principle.
No one generalization ever
contradicts another generalization in any respect. They
are all interaccommodating .
The physical Universe is a self-regenerative process.
interrelationships and intertransformings are governed
by a complex code of weightless,
generalized principles. The principles are metaphysical.
The complex code of eternal
metaphysical principles is omni-interaccommodative;
that is, it has no intercontradiction.
To be classifiable as “generalized,” principles cannot
terminate or go on vacation. If
indeed they are generalized, they are eternal, timeless.
What the mathematicians have been calling abstraction
is reality. When they
are inadequate in their abstraction, then they are irrelevant
to reality. The mathematicians
feel that they can do anything they want with their
abstraction because they don’t relate it
to reality. And, of course, they can really do anything
they want with their abstractions,
even though, like masturbation, it is irrelevant to
the propagation of life.
The only reality is the abstraction of principles,
the eternal generalized
principles. Most people talk of reality as just the
afterimage effects__the realization lags
that register superficially and are asymmetric and off
center and thereby induce the
awareness called life. The principles themselves have
different lag rates and different
interferences. When we get to reality, it's absolutely
The inherent inaccuracy is what people call the reality.
Man’s way of
apprehending is always slow: ergo, the superficial and
erroneous impressions of solids and
things that can be explained only in principle.
Synergetics constitutes the original disclosure of
a hierarchy of rational
quantation and topological interrelationships of all
experiential phenomena which is
omnirationally accounted when we assume the volume of
the tetrahedron and its six
vectors to constitute both metaphysical and physical
unity. (See chart at 223.64.)
(See Sec. 620.12.)
Omnidirectional Closest Packing of Spheres
Definition: The omnidirectional concentric closest
packing of equal radius
spheres about a nuclear sphere forms a matrix of vector
equilibria of progressively higher
frequencies. The number of vertexes or spheres in any
given shell or layer is edge
frequency (F) to the second power times ten plus two.
|10F2 + 2 = the number of vertexes or spheres in any
|F = edge frequency, i.e., the number of outer-layer
The frequency can be considered as the number of layers
or radius) or the number of edge modules of the vector
equilibrium. The number of layers
and the number of edge modules is the same. The frequency,
that is the number of edge
modules, is the number of spaces between the spheres,
and not the number of spheres, in
the outer layer edge.
Equation for Cumulative Number of Spheres: The equation
for the total
number of vertexes, or sphere centers, in all symmetrically
concentric vector equilibria
10(F12 + F22
+ F32 + · · ·
+ Fn2) + 2Fn + 1
Characteristics of Closest Packing of Spheres: The
closest packing of
spheres begins with two spheres tangent to each other,
rather than omnidirectionally. A
third sphere may become closest packed by becoming tangent
to both of the first two,
while causing each of the first two also to be tangent
to the two others: this is inherently a
A fourth sphere may become closest packed by becoming
tangent to all three
of the first three, while causing each of the others
to be tangent to all three others of the
four-sphere group: this is inherently a tetrahedron.
Further closest packing of spheres is accomplished
by the omniequiangular,
intertriangulating, and omnitangential aggregating of
identical-radius spheres. In
omnidirectional closest-packing arrays, each single
sphere finds itself surrounded by, and
tangent to, at most, 12 other spheres. Any center sphere
and the surrounding 12 spheres
altogether describe four planar hexagons, symmetrically
surrounding the center sphere.
Excess of Two in Each Layer: The first layer consists
of 12 spheres
tangentially surrounding a nuclear sphere; the second
omnisurrounding tangential layer
consists of 42 spheres; the third 92, and the order
of successively enclosing layers will be
162 spheres, 252 spheres, and so forth. Each layer has
an excess of two diametrically
positioned spheres which describe the successive poles
of the 25 alternative neutral axes of
spin of the nuclear group.
(See illustrations 450.11a
Three Layers Unique to Each Nucleus: In closest packing
of spheres, the
third layer of 92 spheres contains eight new potential
nuclei which do not, however,
become active nuclei until each has three more layers
surrounding it__three layers being
unique to each nucleus.
Isotropic Vector Matrix: The closest packing of spheres
crystalline assemblages of atoms. All the crystals coincide
with the set of all the polyhedra
permitted by the complex configurations of the isotropic
(see Sec. 420),
a multidimensional matrix in which the vertexes are everywhere
the same and equidistant
from one another. Each vertex can be the center of an
identical-diameter sphere whose
diameter is equal to the uniform vector’s length. Each
sphere will be tangent to the
spheres surrounding it. The points of tangency are always
at the mid-vectors.
The polyhedral shape of these nuclear assemblages
spheres__reliably interdefined by the isotropic vector
matrix’s vertexes__is always that of
the vector equilibrium, having always six square openings
(“faces”) and eight triangular
Volume of Vector Equilibrium: If the geometric volume
of one of the
uniform tetrahedra, as delineated internally by the
lines of the isotropic vector matrix
system, is taken as volumetric unity, then the volume
of the vector equilibrium will be 20.
The volume of any series of vector equilibria of progressively
frequencies is always frequency to the third power times 20.
Equation for Volume of Vector Equilibrium:
Volume of vector equilibrium = 20F3,
F = frequency.
Mathematical Evolution of Formula for Omnidirectional
Packing of Spheres: If we take an inventory of the number
of balls in successive vector
equilibria layers in omnidirectional closest packing
of spheres, we find that there are 12
balls in the first layer, 42 balls in the second layer,
and 92 balls in the third. If we add a
fourth layer, we will need 162 balls, and a fifth layer
will require 252 balls. The number of
balls in each layer always comes out with the number
two as a suffix. We know that this
system is a decimal system of notation. Therefore, we
are counting in what the
mathematician calls congruence in modulo ten__a modulus
of ten units__and there is a
constant excess of two.
In algebraic work, if you use a constant suffix__where
you always have, say,
33 and 53__you could treat them as 30 and 50 and come
out with the same algebraic
conditions. Therefore, if all these terminate with the
number two, we can drop off the two
and not affect the algebraic relationships. If we drop
off the number two in the last
column, they will all be zeros. So in the case of omnidirectional
closest packing of spheres,
the sequence will read; 10, 40, 90, 160, 250, 360, and
so forth. Since each one of these is
a multiple of 10, we may divide each of them by 10,
and then we have 1, 4, 9, 16, 25, and
36, which we recognize as a progression of second powering__two
to the second power,
three to the second power, and so forth.
In describing the number of balls in any one layer,
we can use the term
frequency of modular subdivisions of the radii or chords
as defined by the number of
layers around the nuclear ball. In the vector equilibrium,
the number of modular
subdivisions of the radii is exactly the same as the
number of modular subdivisions of the
chords (the "edge units"), so we can say that frequency
to the second power times ten plus
two is the number of balls in any given layer.
This simple formula governing the rate at which balls
around other balls or shells in closest packing is an
elegant manifest of the reliably incisive
transactions, formings, and transformings of Universe.
I made that discovery in the late
1930s and published it in 1944. The molecular biologists
have confirmed and developed
my formula by virtue of which we can predict the number
of nodes in the external protein
shells of all the viruses, within which shells are housed
the DNA-RNA-programmed design
controls of all the biological species and of all the
individuals within those species.
Although the polio virus is quite different from the
common cold virus, and both are
different from other viruses, all of them employ frequency
to the second power times ten
plus two in producing those most powerful structural
enclosures of all the biological
regeneration of life. It is the structural power of
these geodesic-sphere shells that makes so
lethal those viruses unfriendly to man. They are almost
Classes of Closest Packing: There are three classes
of closest packing of
SYSTEMATIC Symmetrical Omnidirectional Closest Packing:
spheres closest pack omnitangentially around one central
nuclear sphere. Further
symmetrical enclosure by closest-packed sphere layers
agglomerate in successive vector
equilibria. The nucleus is inherent.
ASYMMETRICAL Closest Packed Conglomerates: Closest-packed
conglomerates may be linear, planar, or “crocodile.”
Closest packed spheres without
nuclear organization tend to arrange themselves as the
octet truss or the isotropic vector
matrix. The nuclei are incidental.
VOLUMETRIC Symmetrical Closest Packing: These are
symmetrical embracements by an outer layer. The outer
layer may be any frequency, but it
may not be expanded or contracted by the addition inwardly
or outwardly of complete
closest-packed layers. Each single-layer frequency embracement
must be individually
constituted. Volumetric symmetrical closest packing
aggregates in most economical forms
as an icosahedron geodesic network. The nucleus is excluded.
Copyright © 1997 Estate of Buckminster Fuller