A knot in a spliced rope consisting successively of
manila, cotton, wool, or
nylon may be progressively slipped along the spliced-together
rope with all the latter's
material changes of thickness, color, and texture along
its length. We agree that the "knot"
is not really any of these locally traversed substances.
They were just so many colors and
tactile experiences whose pattern displacement reported
something moving through as a
locally recurring pattern configuration. The knot is
not the rope; it is a weightless,
mathematical, geometric, metaphysically conceptual,
pattern integrity tied momentarily
into the rope by the knot-conceiving, weightless mind
of the human conceiver__knot-
What we call the rope itself turns out to be wave
phenomena. The fibers
themselves were humanly twisted into a spiral wave phenomenon.
We are beginning to
discover that there is not too much difference between
the tactile superficiality of
apprehension and the real frequency phenomena that we
cannot see in the intervals
between the waves. The actual fact is that the water
wave and the manila wave are
frequencies nontunable within the electromagnetic frequency
range of the human
organism's optical faculties, wherefore human cognition
of the water waves is provided
exclusively by the human brain's afterimage lag and
the brain's successive recall
apprehending of static picture frames of successively
different pattern states as moving
I'll bet a monkey can't invent a knot. If they could,
they would tie the whole
jungle up in knots. What would the behaviorists say?
Mind saw the knot; monkey did not.
The monkeys hold hands, but they have not yet discovered
that the handshake is two
circles knotted through one another.
You cannot have a knot with less than two circles
(two finite unities). The
mind tells the brain to control the muscles in a knot-tying
event scenario as follows: one
hand grasps the rope end and describes the first circle.
When the first circle is complete,
the second hand holds the completed circle as the first
hand continues to lead the rope end
through the center of the first circle in an orbital
plane different from that of the first circle.
(If they were both in the same plane, they would generate
a coil or a spiral and fail to
knot.) The perimeter of the second circle should go
through the center of the first circle.
One has to capture the other in an interference pattern.
The rope with the knot in it is a physical memory
pattern tracery of where
your hands have led its end. The hand-led rope end and
its pulled-through rope section
form a visibly sustained trajectory of the conceptual
patterning employed by mind in
negotiating its visual realization by the brain-coordinated
sensing of self or others. Like the
contrails of jet planes, in the sky, the smoke trails
of skywriting airplanes, or the extruded
plastic threads of spiders, the roped knot represents
a long-lasting memorandum of the
abstract, weightless mind's weightless conceptioning
in pure principle.
Each circle has 360 degrees; the two interference
circles that comprise the
minimum knot always involve 720 degrees of angular change
in the hand-led pattern, just
as the total angles of the four triangles of a tetrahedron
add up to 720 degrees. The hands
describe circles nonsimultaneously; the result is a
progression. The knot is the same 720-
degree angular value of a minimum structural system
in Universe, as is the tetrahedron.
Pulling on the two ends of the knotted rope causes
the knot to contract. This
is a form of interference wave where the wave comes
back on itself, and as a consequence
of any tension in it, the knot gets tighter. This is
one of the ways in which the energy-mass
patterns begin to tighten up. It is self-tightening.
This is the essence of "matter" as a
consequence of two circles of 720 degrees tending to
annihilate or lose one's self.
Tetrahedron creates an insideness. Knot attempts to
annihilate it. The knot is a tetrahedron
or a complex of tetrahedra. Yin-Yang is a picture of
a minimum tetrahedron knot
(See Sec. 505.21.)
At the end of the piece of rope, we make a metaphysical
disconnect and a
new set of observations is inaugurated, each consisting
of finite-quanta integral ingredients
such as the time quality of all finite-energy quanta.
The metabolic flow that passes through a man is not
the man. He is an
abstract pattern integrity that is sustained through
all his physical changes and processing,
a knot through which pass the swift strands of concurrent
transformations of solar energy.
As curves__lines__cannot reenter or "join back into
themselves," the circling
line can only wrap around or pass over or under another
"part" of its continuity self, as the
knot-making sailors says it. Because of a line's inability
to reenter itself, when circles are
followed around and around upon themselves, the result
is a coil__which is a mildly
asymmetric spiral wave accumulation that may be piled
upon its micro-diameter self only
as long as intellect wishes to pursue such an experiential
Each lobe of a baseball is simply a precessed triangle
of a tetrahedron. The
baseball is yin-yang, not in a plane but in Universe:
it is telling us that complementarities
interprecess omnidirectionally and not just in a plane,
as the planar yin-yang suggests.
The spherical tetrahedron can be demonstrated by placing
a light inside a
translucent plastic sphere. The light at the system
center casts the shadow lines of the
tetrahedron's four vertexes and their six interconnecting
edges outwardly and
symmetrically onto the plastic sphere to produce the
outlines of a spherical tetrahedron.
We may then inscribe four circles around each of the
four vertexes of the spherical
tetrahedron of such a unit radius so that each of the
four circles is tangent to each of the
three others. We can take a sharp-edged cutting tool
and severingly trace around the
perimeter of one circle to its point of tangency with
the next adjacent circle, and there we
can inflect the cutting tool to cut around the next
tangent circle to its next point of
tangency, where once more we can inflect the cutting
tool's severance trace to follow
around the next circle to reach the next tangent point,
repeating the procedure until we
finally return to the point of original cutting. Upon
completion of the severance tracing we
find we have cut apart the surface of the spherical
tetrahedron into two similar, equiarea
sections, each of which corresponds to the two similar,
dumbbell-profiled, skin sections of
a baseball. With these two similar half-sphere surface
sections precessingly aimed toward
one another in such a manner that the bulge of one section
registers symmetrically with the
half-circle opening on the other, we find that we can
sew the edges of the sections
together around a core to produce a baseball.
When you look at the baseball with the inflection
point of its S-pattern
stitching, located at the center of the visible hemisphere's
circular profile, aimed directly at
you, you will see that the baseball's surface pattern
is the same inflection pattern as that of
the most profound symbol of the orient: yin-yang. Long
ago human minds of the orient
must have discovered precession, tetrahedra, and symmetry.
(See Sec. 1056.12.)
The rubber glove, with its red exterior and green
interior, when stripped
inside-outingly from off the left hand as red, now fits
the right hand as green. First the left
hand was conceptual and the right hand was
the process of stripping
off inside-outingly created the right hand. And then
vice versa as the next strip-off occurs.
Strip it off the right hand and there it is left again.
That is the way our Universe is. There are the visibles
and the invisibles of
the inside-outing nonsimultaneity. What we call thinkable
is always outside out. What we
call space is just exactly as real, but it is inside
out. There is no such thing as right and left.
The always and only coexisting convex and concave
demonstrates that unity
is plural and at minimum two, in which only one is spontaneously
accounted as obvious.
The positive (right) spiral and negative (left) spiral
make one tetrahedron:
1+1=4: therefore no parity. But there is parity in the
internal complementary macrocosmic
tetrahedron of the sum of the angles around all the
convex vertexes of the system, and an
internal complementary microcosmic tetrahedron of the
sum of the angles around all the
concave vertexes of the system.
When physics finds experimentally that a unique energy
erroneously referred to in archaic terms as a particle__is
annihilated, that annihilation is
only of the inside-outing rubber-glove kind. The positive
becomes the negative and the
positive only seems to have been annihilated. We begin
to realize conceptually the finite,
yet nonsensorial, outness continuum integrity that can
be converted into sensorial inness
by the inside-outing process, but only at the expense
of losing afterimage of the previous
sense-experienced conceptual fixation.
The complementary of parity is disparity and not a
Numbers are experiences. You have one experience and
which, when reviewed, are composited. Numbers have unique
experiential meaning. The
minimum structural systems of Universe, the tetrahedron
and the thinkable set, both
consist of four points and their six unique interrelatednesses.
Even the development of sets
derives from experience. Mathematics is generalization,
a third-degree generalization that
is a generalization of generalizations. But generalization
itself is sequitur to experience
where intuition and mind discover the synergetic interbehavior
that is not implicit in any
single item of the empirical data of the past.
Intuition and mind apprehend that which is comprehensively
not of, the parts.
The mathematician talks of "pure imaginary numbers"
on the false
assumption that mathematics could cerebrate a priori
to experience. "Lines" are definitions
of experiences__of graven traceries, or of erosively
deposited tracks, or of gaseous fallout
along a trajectory__and the symbols for number extractions,
such as X and Y, are always
and only experientially conceived devices.
All number awareness is discovered through experiences,
which are all
special cases. Every time you write a number__every time
you say, write, or read a
number__you see resolvable clusters of light differentiation.
And clusters are an
experience. Conscious thoughts of numbers, either subjective
or objective, are always
Before topology, mathematicians erroneously thought
that they had attained
utter abstraction or utter nonconceptuality__ergo, "pure"
nonsensoriality__by employing a
series of algebraic symbols substituted for calculus
symbols and substituted for again by
"empty-set" symbols. They overlooked the fact that even
their symbols themselves were
conceptual patterns and only recognizable that way.
For instance, numbers or phonetic
letters consist of physical ingredients and physical-experience
recalls, else they would not
have become employable by the deluding, experience-immersed
(N2 - N) / 2 is always a triangular number as, for instance, the
balls in the rack on a pool table. A telephone connection
is a circuit; a circuit is a circle;
two people need one circuit and three people need three
circles, which make a triangle.
Four people need six circuits, and six circuits cluster
most economically and symmetrically
in a triangle. Five people need 10 private circuits,
six people need 15, and seven people
need 21, and so on: all are triangular numbers. (See
Sec. 227, Order Underlying
Randomness, and illustration 227.01.)
Successive stackings of the number of relationships
of our experiences are a
stacking of triangles. The number of balls in the longest
row of any triangular cluster will
always be the same number as the number of rows of balls
in the triangle, each row always
having one more than the preceding row. The number of
balls in any triangle will always
(where R = the number of rows (or the number of balls
in the longest row).
(See Sec. 230, Tetrahedral Number.)
|(R + 1)2 - (R + 1)
The conceptual process is never static. Thinking does
not consist of the
insertion of invented images into an otherwise empty
vacuum-tube chamber called brain.
Thinking is the self-disciplined process of preoccupied
consideration of special-case sets of
feedback answers selected out of the multitude of high-frequency
brain traffic. This traffic consists of omniexperienced
and processed answers to present or
past questions, formulated either by the conscious or
subconscious coordinating initiative
of the individual or possibly the individual's overlapping
generation of group memory.
A considerable set is a locally definitive system
within Universe that returns
upon its considerability in all circumferential directions
and therefore has an inherent
withoutness and withinness; the latter two differentiable
functions inherently subdivide all
Universe into the two unique extremes of macro- and
For instance, we find that all irrelevancies fall
into two main categories, or
bits. One set embraces all the events that are irrelevant
because they are too large in
magnitude and too delayed in rate of reoccurrence to
have any effect on the set of
relationships we are considering. The other set of irrelevancies
embraces all the events that
are too small and too frequent to be differentially
resolved at the wavelength to which we
are tuned, ergo, in any discernible way to alter the
interrelationship values of the set of
experience relationships we are considering. Having
dismissed the two classes of
irrelevancies, there remains the lucidly relevant set
to be studied.
Because of the varying depths of storage of past experiences,
come back swiftly, some slowly. The recollectibility
rates are unpredictable. Ergo, the
returning-answers traffic is heterogeneous. Many answers
come to questions we have
forgotten that we asked ourselves. Conceptually systematic
tuning of questions and
feedback answers, comparatively considered in the brain,
results in temporary, tunably
valved exclusion of all other incoming signals. Discrete
tuning admits consideration of
only those recollections that are clearly relevant to
the omnidirectional rounding out of
systematic comprehension of the special-case set of
events intuitively selected for
momentary focal consideration. Thinking consists, then,
of a self-disciplined deferment of
conscious consideration of any incoming information
traffic other than that which is
lucidly relevant to the experience-intuited quest for
comprehension of the significance of
the vividly emergent pattern under immediate priority
Neither the set of all experiences, nor the set of
all words that describe them,
nor the set of all the generalized conceptual principles
harvested from the total of
experiences is either instantly or simultaneously reviewable.
"What was that man's name?"
Our answering service may take five seconds, five hours,
five days, or five generations to
reply. Our conscious, orderly reconsideration of our
variable-lag experiences discloses
subconsciously coordinated regularities of feedback
rates governing the recall phenomena.
What we do when we think is to dismiss momentarily
all the irrelevant
thoughts as we would part the grass to right and left
in order to find a path. Thinking is
high-frequency interception and very temporary diversion
to a local holding pattern
outside our consideration of all the irrelevant inbound
feedback__just as inbound airplanes
are "stacked up" in the sky near airports by the ground
control when too many come in at
about the same time and may interfere with each discretely
safe landing operation. Landing
is a slowing operation and an exact timing operation.
Having isolated a finite set of
experiences__spontaneously grouped for comprehensive
consideration__by dismissing the
irrelevancies, we may proceed to comprehend or "land"
the isolated system by applying
the theory of bits, which breaks up finite wholes into
We may now say that what we do in thinking, after
the irrelevancies and thereby inadvertently isolating
the considered set, is to further
subdivide Universe into four parts:
Part 1 is the untuned, macrocosmic, long-wavelength,
low-frequency, high-energy set.
Part 2 is the untuned, microcosmic, short-wavelength,
high-frequency, low-energy set.
Parts 3 and 4 are the tuned, plus (+) and minus (-),
- All of the parts of Universe that are externally
irrelevant because too large and
- all the events of Universe that are internally irrelevant
because too small and
too frequent to be resolvable and discretely differentiated
out for inclusion in
our interrelationship considerations;
- all of the lucidly relevant remainder of Universe,
which constitutes the
considered and reconsidered set of experiences as viewed
from outside the set;
- the lucidly relevant set as viewed from inside the
The thinking process results in varying degrees of
lucidity of the arrayed
residue of focal-event patterns uniquely consequent
to the disciplined deferment of
irrelevancies. Thinking is a putting-aside, rather than
a putting-in, discipline. Thinking is
FM__frequency modulation__for it results in the tuning
out of irrelevancies (static) as a
result of definitive resolution of the exclusively tuned-in
or accepted feedback messages'
pattern differentiability. And as the exploring navigator
picks his channel between the
look-out-detected rocks, the intellect picks its way
between irrelevancies of feedback
messages. Static and irrelevancies are the same.
There are two inherent twilight zones of "tantalizingly
recollections" spontaneously fed back in contiguous
frequency bands: the macro-twilight
and the micro-twilight. They inherently subdivide all
Universe into the two unique
extremes of macro- and micro-frequencies.
So I find that you and I and the lamppost and its
lamp are basic subdivisions
of Universe. You and I and complex it are either all
of the Universe that is inside, all of the
Universe that is outside, or all the remaining Universe,
which comprises a given
recognizable system or set. The residual constellation
to be reconsidered constitutes a
local conceptual system.
You cannot program the unknowns you are looking for
because they are the
relationship connections and not the things. The only
thing you can program is the
dismissal of irrelevancies.
When we say "we think," our feedback has variable
lags that may take
overnight or months of time, for all we know. Because
we want to understand__that is, to
know the interrelationships of clusters of experiences__our
first great discovery is
dismissing irrelevancies, the macro-micro characteristics.
Add: forgotten questions;
different rates of feedback; persons' names; random
questionings; the challenging set you
would like to understand; our friend intuition.
Copyright © 1997 Estate of R. Buckminster Fuller