A star is an exquisitely concentrated coordination
of events that your optical
tuning facilities are unable to resolve differentially
into separately identifiable events. We
may call a star a point. Playing Euler's game, "stars,"
or "points," are "crossings," or
"fixes" as navigators would say it. As so considered,
a "star-point-crossing" does not have
an outsideness and an insideness. It is the point of
superimposed crossing of trajectories or
of their interferences. A point fix is a potential embryo
consideration, a potential thought,
a potential system.
Thinking is the consideration of different experiences
and of inherently
separate sets of events, and trying to find out what
their relatedness is. Each one is a star.
How many stars does it take to develop a geometry of
outwardness and inwardness? What
is the minimum number of stars needed to divide the
Universe into outwardness and
inwardness? I find it takes a minimum of four; you can't
do it with three. Four very clearly
has an outsideness and an insideness. This is what we
call the tetrahedron, which has these
four stars and six sets of interrelatedness. This comes
in very interestingly in mathematics
with the generalization that you don't have to worry
too much about the shape, but the
four stars are the minimum we can have for a thought.
If I can at first discover only three
stars in a thought challenge, there must be at least
a fourth star lurking somewhere in the
critical neighborhood. In fact, I discover that the
total number of stars that could possibly
be related is always subdivisible by four. The mathematics
shows this up very clearly as
complexes of tetrahedra. Tetrahedron becomes the minimum
thinkable set, the minimum
reconsiderable set, and it turns out to be the fundamental
increment out of which all
thoughts are constructed.
The minimum set that may form a system to divide Universe
into macro- and
micro-cosmos is a set of four items of consideration.
Four nonsimultaneously bursting
rockets in a unitarily considerable set of overlapping
The stars of the four rocket bursts constitute the
four vertexes of a
tetrahedron__the fundamental quantum of Universe's structuring.
There is a tetrahedral
structuring interrelationship between (a) the day before
yesterday, (b) yesterday, (c)
today, and (d) tomorrow. Though we speak of them as
"the four balls in the air"
__maintained there successively by a juggler using five
balls to do his trick__they are not
the same balls, and the four are never in the same positions;
nonetheless, there are always
and only six fundamental interrelationships between
"the four balls in the air" __i.e., ab,
ac, ad, bc, bd, and cd, although a, b, c, and d are
nonsimultaneous events. Universe
structures most frequently consist of the physical interrelationship
A star is the focal point of an as yet undifferentiated
concentration of events,
ergo, considerable or constellar patterning: an exploratory
grouping of stars or complex-
idea entities that seem to man's limited tunability
to stand out together. The word
consideration comes from con, together, and sidus, the
Latin "star." When we have found
all the relationships between the number of items of
our consideration, we have what we
speak of as "understanding." When we understand, we
have all the fundamental
connections between the star events of our consideration.
They then become a
constellation. They stand clearly together.
When n stands for the number of stars or items of
consideration, the number
of connections necessary to understanding is always
Four is the minimum number of stars having an inherent
withoutness and withinness. The minimum conceptually
experiences-set affording macro-micro separation of
Universe is a set of four local-event
foci. Between the four stars that form the vertexes
of the tetrahedron, there are six edges
that constitute all the possible relationships between
those four stars. The four stars have
an inherent sixness of interrelationship. The four-foci,
six-relationship set is definable as
the tetrahedron. This minimum fourness of relevant frequency__ergo,
coincides with quantum mechanics' requirement of four
unique quanta numbers per each
uniquely considerable "particle."
The regenerative patterns of structural events may
be described as constellar
because their component events interinterfere tensively
in high-frequency, dynamic, self-
regenerative patternings which only superficially seem
to stand together as "static"
structures. Star groupings "fly" in celestial formation,
though seeming to hang motionless
in the celestial theater. Any event patternings that
become locally regenerative are
constellar patterns. They are momentarily conceptual.
Until the present age, people thought that all of
their faculties were
simultaneously and instantly coordinate and operating
at equal velocities. Einstein showed
that neither simultaneous nor instant are valid, i.e.,
experimentally demonstrable. Observe
that when we send up four rockets one-half second apart,
their afterimages are
approximately simultaneous. So we say that we see four
rockets "at the same time." The
illusion of simultaneity is one of the most important
illusions for us to consider. Musicians
may be able to comprehend nonsimultaneity better than
do others. Einstein emphasized the
importance of attempted spontaneous comprehension of
the nonsimultaneity of all the
events of Universe__a concept akin to our discovery that
in our Universe, none of the
lines can ever go simultaneously through the same points
(See Sec. 517 et seq.). What
Einstein is telling us is that there is no conceptual
validity to the notion that everything in
Universe is actually in simultaneous static array.
All words in the dictionary do not make one sentence;
all the words cannot
be simultaneously considered, yet each of the words
is valid as a tool of communication;
and some words combine in a structure of meaning. All
the words are memoranda of all of
humanity's attempts to communicate to self or to others
their understanding of the unique
evolvement of their separately viewed experiences. The
dictionary is the inventory of
unique aspects of the totally composited experiences
known as Universe.
A single event is integrally complex. As angles are
of size, events are conceptual, independent of frequency
of occurrence. An "original" or
"prime" energy event is conceptual. An energy event
is inherently complex. It is a nuclear
component, but it is not the nucleus. Nuclei__complexedly
composed of prime or original
energy events__are themselves "prime" and "original,"
originality being inherently
complex integrals. Energy transactions occur between
nuclei as an extramural complex of
events__as a "chemical compound."
All energy-event experimentation discloses omnioptimally
behavioral patterning of physical events. Every physical
event in nonsimultaneous scenario
Universe is characterized by three multidimensionally
interlinked vectors that interact
precessionally, i.e., at angles other than 180 degrees
to one another, as in the
multidimensional, helically zig-zagging pattern of lightning.
There are six positive and six negative degrees of
transformation freedoms, which provide 12 alternate
ways in which nature can behave
most economically upon each and every energy-event occurrence.
You have six vectors or
none for every energy event.
One set of three-vector groups corresponds to the
proton (with its electron
and anti-neutrino), and the other set of three-vector
groups corresponds to the neutron
(with its positron and neutrino). Each of these three
vector teams is identified by nuclear
When we bring together these two sets of three vectors
each, they integrate as six vectors
and coincidentally also make one tetrahedron (of six
vector edges). The tetrahedron is the
veritably conceptualizable unit of one energy quantum.
|one-half Planck's constant; or
|one-half spin; or
The open-ended tripartite spiral can be considered
as one energy event
consisting of an action, reaction, and resultant. Two
such tripartite-vectored "spirals," one
negative and one positive, combine to form the tetrahedron.
(See illustration 108.01.) The
tripartite vector set looks like an "erected cobra"
Z, that is, with two of its interlinked
vector lines on the ground and one erected. One erected
Z cobra erects its third vector
member clockwise, and the other Z cobra erects its third
vector in a counter-clockwise
direction in respect to its base.
We find that the triangular Z cobra, is not operating
in a plane because there
is no such thing as a plane. Therefore, one of the legs
sticks up a bit. We have a positive Z
cobra and a negative Z cobra, and one cannot nest in
the other. They will never be
congruent with one another, but they can complement
one another to become the
tetrahedron. An event is a triangle. A triangle is an
event. Two of them together make the
Each of the three-vector, action, reaction, and resultant,
minimum event Z
cobras has two open ends and two internal angles. The
two Z cobras have together four
ends and four internal angles. We will call the open
ends male and the internal angles
female. We can marry the two Z cobra, half quantum events
in an always consistent,
orderly manner, by always having a male end interconnected
with an internal female angle.
When all four such marriage ceremonies have been consummated,
we have produced one
tetrahedron, i.e., one quantum, i.e., one prime minimum
structural system of Universe.
When the end of one energy action comes over the middle
of another energy vector, there
is a precessional effect, a tensional effect. One energy
event gets angularly precessed, the
next energy event goes by the center of another mass,
and each one of them interaffects
the other. It is a basketry interweaving, where each
one precesses the other angularly so
that they hold together very much as a cotton ball.
The energy event of an action, a reaction, and a resultant
An energy event is illustrated by a diagram of a man
jumping from one boat
to another. At the top of the picture, a man standing
in one boat jumps. He does not glide
horizontally: he jumps.1 That is, he goes outwardly from
the center of Earth, and that is a
vector. That is an energy action in itself. He jumps.
He is the action. The action was not
just horizontal, it was also vertical. It was mildly
vertical in that he went outwardly. As he
jumps, the boat goes into reaction and shoots off the
other way. A moment later, he lands,
and the second boat moves in a complex that is both
horizontal and vertical. There is a
reaction and a result, so there really is a four-foldedness
going on. It may appear as
threefolded because the man does not jump very high.
We should consider it as a
tetrahedron of very low altitude.
(Footnote 1: In this way, we begin to discover that force
diagrams in engineering result from over-simplification.
At the outset, the boats are more or less parallel
to one another. As the man
jumps from the stern of the boat, it turns and whirls
around, so that the reaction is
following the resultant. They are not going in opposite
directions. The reaction and
resultant run into each other. Notice that it begins
to look like a triangle, but with a
vertical component, so it ends up as our friend, tetrahedron.
Engineers have been proud of pointing out that the
engineers and lay society is that engineers know that
every action has its reaction and that
lay society thinks only of the actions. Before the speed
of light was measured, light
seemed, to all humanity, to be instantaneous. Since
we now know experientially that
neither light nor any other phenomenon is instantaneous,
we may conclude that an action
and the vectors that it creates are neither simultaneously
occurring nor instantaneous.
Because vectors have discrete length, whose dimension
represents the energy mass
multiplied by its velocity, every action vector has
two terminals__a "beginning" and an
"ending" at the end of its noninstantaneous action.
The beginnings and the endings are
nonsimultaneously occurrent. Therefore, the "ending"
terminal of an action's vector
occurs later than its "beginning." Therefore, every
action must have a reaction vector at its
"beginning" terminal and a resultant vector at its "ending"
terminal. The reaction vectors
and the resultant vectors are never angled at 180 degrees
to the action vectors. They are
always angled precessionally at other than 180 degrees.
Mechanically and chemically, a steerable rocket embraces
a complex of
internal and external events. Both airplanes and steerable
rockets are complexes of internal
and external energy-event transactions and omni-interacting,
resultant "motions" in
Universe transcendental to Earth motions, where the
extraterrestrially positioned. Since the Earth is moving
as a dependent motion-complex in
respect to the Sun's and other planets' motions, and
since the Sun is engaged in a plurality
of internal and external motions in respect to the galactic
system, and since the galactic
system is a complex of motions in respect to other galaxies
and supergalaxies, and so on,
and since the whole set of motion events are nonsimultaneous
and of uniquely variant
durations, and since the intereffects of the events
vary vastly in respect to eons of time, it
is obvious that any thinkably meaningful conceptual
coordination of event
interrelationships in the meager lifetime limits of
humans is inherently limited to a
relatively local set within Universe and within a time
sense, and the relationships may be
measured only in respect to the angle and frequency
magnitude characteristics of any one
subsystem of the totality.
Vectorial Orientation and Observation
The angles of orientation and the dimensional fixes
of vectorial energy-event
manifestations are always conceptually oriented and
positioned in respect to the optionally
selected axis of conceptual observation.
Fixes consist of both angular and dimensional observations.
To be experiential, we must have an observer and the
To be experimental, we must have the articulator,
the articulated, and the
The vectorial angulation of both the experientially
observed and the
experimentally articulated is always referential to
the axis of conceptual observation of the
observer or the articulator, respectively. These always
and only coexisting functions of
experience and experiments embrace the fundamental parameters
of operational science.
"My life" is the progressive harvestings of the information
accruing in the attempt to be both adequate and accurate.
The harvest is stored in the
brain bank. Life consists of alternate observing and
articulating interspersed with variable-
recall rates of "retrieved observations" and variable
rates of their reconsideration to the
degrees of "understandability."
Resonantly propagated evolution oscillatingly induces
quanta__both metaphysical and physical__formulated vectorially
between four "star-
- understanding, and
Articulation: The articulations are ever reenacted,
each time hoping to
reduce the tolerance magnitude of residual inaccuracy
of either observation or articulation.
Axis of Reference
The axis of reference is the axis of conceptual observation.
The axis of
observer reference frequently occurs spontaneously:
as the line between the nose and the
The direction of a vector is an angular one in respect
to an omnidirectional
coordinate system having a specific central point and
a specific set of external points at
specific angles and distances from one another and from
the central point.
Our definition of an opening is that it is framed
by trajectories. Every
trajectory in a system has at least two crossings, but
these crossings are as viewed,
because the lines could be at different levels from
other points of observation.
Copyright © 1997 Estate of R. Buckminster Fuller