Copyright 1999, Estate of Buckminster Fuller, all rights reserved.
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The Estate of Buckminster Fuller|
P.O. Box 3248
Santa Barbara, CA 93130
Dear Robert Burkhardt:
In reply to your letter I will first review a personal experience which is relevant to my answering your question.
[Editor's note: The following story appears on pp. 65-66 of Cosmography where it is more specific and slightly elaborated. There Professor X is identified as Professor Weiss, and the year is identified as 1968 when Bucky had been appointed to a Hoyt fellowship at Yale.]
Every once in a while in the late 1950s and in the 1960s philosophy scholars would say to me, if you ever are confronted by Professor X at Yale, some of your basic theories are liable to be cancelled out. On my appointment in the '60s to a short-term fellowship at Yale, the students immediately asked Professor X and myself to appear together on the Yale University TV program. Though we had as yet not met we both accepted, so I met Professor X for the first time in the TV broadcasting studio. The studio director seated Professor X and myself opposite one another at a stout table. On the director's signal that our "recorded in advance" program was starting, Professor X thumped the table resoundingly with his fist saying, "Don't tell me that this table is not solid!" I replied, "How can you see me over here to so defy me while looking at me through what you call solid spectacles?" I then explained that glass is an aggregate of very high-frequency atomic events and that a good analogy would be an alignment between Professor X and myself of a number of revolving airplane propellers one behind the other, rotating so fast we cannot see any of the blades of any of the propellers. If Professor X very, very slowly reaches his fist toward me an invisible "solid," like his eyeglasses, would thump his fist or cut it off.
Because the speed of the propellers are directly coupled to their controlled speed motors, it is possible through gears to time and aim a machinegun to shoot its bullets through, between the blades. I explained to Professor X that the speed of light is so swift that it can readily pass through the glass's atomic electron's circular motion patterns. By making the glass lenses a little thicker, the distance the photons of light must travel (at 186,000 miles per second) is increased enough to permit mild interference with the gyrations of the electrons of the atomic components.of the glass. As does a baseball get angularly redirected by a batter's tick-foul, this refractive direction changing of the light-photon passage makes possible lens correction of our vision.
Professor X asked the studio to cancel the program and walked out. Thus we discover how seemingly "hard realities" may be only mathematical differentiations of frequency and angle operative in pure principle.
You are wrong, Robert, in speaking about tensegrity as being one of my innovations. It is one of my discoveries that, since there are no "things," no "solids," only "events operating in pure principle," and since no events touch other events in Universe, Universe is coordinatingly coherred, formed, and transformed only tensionally, repulsively, electromagnetically, gravitationally, and even the event electron is as remote from its nucleus as is the Earth from the Moon as compared in terms of these regenerative system's respective diameters.
I simply found that the Universe is compressionally discontinuous and only tensionally continuous. The structural integrity of Universe is tensional as Kepler discovered. I gave this phenomena the name "tensegrity."
Academic and professional civil (structural) engineering assumes only solid, rigid compressional continuums to be the heart of structural designing effectiveness and safety as checkable by other engineers. The engineers work with physical-tests-proven data regarding the various solids' unique compressive and tensile strengths per given square areas with, here and there, a "solid" high-tensile strength metallic alloy sinew to hang, tie, or hold together their compressional continuum bundle.
There exists only one exception to the foregoing and that is in the world of mechanical engineering. There we find the wire wheel whose structural integrity between the rim and the hub is accomplished exclusively by tensed wire spokes. I had to seek an experientially demonstrable, lucidly cogent means of translating the exclusively tensional continuum of the wire wheel's special structural integrity case into a scientific generalization that could revolutionize engineering.
In my synergetics, always-and-only experientially-based systemic geometry, I discover first that all experiencible somethings -- be they apples, cows, thoughts, clouds, somethings, anythings or rocks -- are systems which always divide all Universe into all the Universe outside the system (the macrocosm) and all the Universe within the system (the microcosm) and the remainder of the Universe which constitutes the system itself that divides the macrocosm from the microcosm.
In electromagnetics, for instance in radio systems, there are the for the moment tuned-in programs of unique wavelength and frequency. In addition to the tuned-in frequency there are all the other for the moment non-tuned-in longwave, low-frequency macro set programs of broadcast tunabilities; and all the other for the moment non-tuned-in high-frequency shortwave micro sets of broadcast programs.
Each for-the-moment thought has its for-the-moment-relevant, tuned-in thoughts and those tuned-in thoughts' macro-irrelevants which are too large and too infrequent to be considered and the micro-irrelevants which are too frequent and too short in wavelengths to be conceivably relevant to the thought system considered.
The Universe does not operate as a one-dimensional straight line phenomenon. One dimensionality with no insideness or outsideness cannot be experimentally evidenced. Unveering, linear straightness cannot be experientially demonstrated. The Universe does not operate as a two dimensional planar phenomenon having no insideness nor outsideness. No such phenomenon can be experimentally evidenced. The Universe does not operate as an exclusively three dimensional phenomenon having no insideness or outsideness. An exclusively XYZ--perpendicular and parallel, straight-lined -- "frame of reference" cannot be experientially demonstrated. Demonstrable local Universe always and only operates as a convergent-divergent, nucleated, or vacantly centered, insideness and outsideness system; a growable or shrinkable, spherically expandable or contractible, radiant wave propagatable system; a gravitational, spherically embracing, pulsatively expanding and contracting synergetic system. There are no non-systems. There are no parts independent of systems.
There is no physically demonstrable "up" or "down" in Universe. None of the perpendiculars to our planet's surface are parallel to one another. The sensations misinterpreted is "up" and "down" are experientially demonstrable exclusively as "out" and "in" respectively. In is always specific "into the Moon," "into the Earth," and out is any direction, including "go in to go out" on the other side.
All systems have six primitive motions potential -- "degrees of freedom." -- the first four of which are integral to the system: (1) axial rotation, (2) torque, (3) expansion-contraction, (4) inside-outing (involution-evolution), (5) orbital travel, (6) precession or the effect of systems in motion upon other systems in motion.
Life begins with awareness. No simultaneous otherness; no awareness. No simultaneous otherness; no life.
Life, one small something, too small to be described as being other than a point-to-able "something" over there -- seen by another "something." One something by itself has no external relationships. No externality, no life.
The only interrelatedness of two overlappingly occurent somethings is betweeness: AB or BA.
Three simultaneously occurent somethings have also only three betweenesses: AB, AC, BC.
Four simultaneously, overlappingly occurent somethings (A, B, C, D) have six betweennesses: AB, AC, AD, BC, BD, CD. They have an only mutually differentiated insideness and outsideness. Four somethings produce a system, a tetrahedron.
|(Resembles a composite of Figs. 6.8 and 6.9 from Cosmography)|
|Figure 2:||Four Spheres When Closest Interpacked Form a Tetrahedron|
|(Similar to Fig. 411.05 from Synergetics)|
|Figure 3:||Falling Sticks: Six Vectors Provide Minimum Stability|
|(Similar to Fig. 621.10 from Synergetics or Fig. 6.58 from Cosmography)|
|Figure 4:||Investigation of the Requirements for a Structurally Stable Minimum System|
|(Similar to Fig. 401.01 from Synergetics or Fig. 6.59 from Cosmography)|
[Editor's note: A slightly different version of the following paragraph appears on pp. 182-184 of Cosmography.]
I formed a tetrahedron of six two-feet-long, thin-walled one inch outside diameter steel tubes welded to four three-inch-diameter steel balls at the tetrahedron's four corners. I drilled and tapped (threaded) four holes on the inside of the four corner balls. I then connected those four corner balls perpendicularly to a single three inch diameter steel ball located at the center of volume of the tetrahedron having four drilled and reverse tap-threaded holes. I made the connection of the center ball with the four corner balls by means of four one-eighth-inch steel rods, each threaded oppositely at their respective two ends then inserted the positively and negatively threaded rods between the corner balls and the center ball, then screw tightened them together as with turnbuckles. The center ball could not be dislocated from the tetrahedron's exact center of volume. I then took a stilson wrench and found that without displacing it from its exact center of the tetrahedron I could rotate the center ball mildly in six different positive and six different negative directions. To counteract these in-place rotatabilities required twelve rods, three different tangential rods from each of the four corner balls of the tetrahedron with their central ends also fastened tangentially to the center ball. This produced twelve prime restraints on the center ball which could no longer be either dislocated from the tetrahedron's center nor be locally twisted in place. These twelve restraints proved to be the always and only twelve restraints necessary to cope structurally with the twelve degrees of freedom of all independent systems in Universe in respect to all other independent individual systems in Universe. They were the same twelve restraints we soon found to be necessary to stabilize the wire wheel.
|Figure 5:||Ball Restrained in Welded Steel Tetrahedron|
|Figure 6:||A System Within a System|
|(Similar to Fig. 401.00 from "Drawings Section" of Synergetics and Fig. 6.46 from Cosmography)|
|Figure 7:||Stabilization of Tension: Minimum of 12 Spokes|
|(Similar to Fig. 640.41A from Synergetics)|
|Figure 8:||Minimum of Twelve Spokes Oppose Torque: Universal Joint|
|(Similar to Fig. 640.41B from Synergetics or Fig. 6.61 from Cosmography)|
[Editor's note: A slightly different version of the following paragraph appears on pp. 42-43 of Cosmography.]
In my search for the experientially formulated generalization of tensegrity, I then constructed a necklace consisting of many twelve inch long, one-half inch diameter aluminum tubes strung on a dacron cord. I found that the more tubes included, the more "fluidly" flexible was the necklace. I found that the flexing neither altered the length nor bent any of the tubes. Clearly, the flexibility was provided entirely by the tension joints. Because the tubes were not providing flexibility, I progressively eliminated them one by one and the necklace became progressively more prominently angular. Finally I had only three tubes left and, for the first time, the necklace would no longer flex. It was a triangle with a triangular hole in it -- the hole being larger than my neck. It clearly demonstrated that the triangle is the only polygon that holds its shape despite its three completely flexible corners.
|Figure 9:||Tube Necklace|
|(Similar to Fig. 2.3 from Cosmography)|
At this stage in my exploration I discovered that neither physics nor engineering had a description or definition of what they meant by the word "structure." Structure to them is axiomatic and obvious -- the "solid" block of marble, the "rigidity of stone." Since I found my triangular necklace stable and since the triangular necklace which held its shape consists of three separate push-pull, firmly shaped aluminum tubes and three flexible dacron cord corner-angle coherers, I formulated my definition of structure as "a complex of events that interact to produce a stable pattern." I proceeded to explain that interaction as "a system whose component events are persistently interpositioned by a balance of the forces of interrepulsion and interattraction." I found the necklace structure to be just such a complex of push-pull coherence. I thus concluded that triangulation is essential to structure and that no other polygon is stable.
Since the minimum system in Universe, the tetrahedron, is entirely embraced by exactly four triangles and since the triangle alone produces a stable pattern I concluded that the tetrahedron is the minimum and simplest structural system in Universe. No wonder it and its contained octahedron together with its external, all-space-filling companion octahedra are the structural components of diamonds; no wonder that the chemist Vant Hoff was the first chemist ever to receive the Nobel Prize for his optical proof of the tetrahedronal configuration of carbon. No wonder Paul MacCready's Gossamer Albatross was light enough to be human-muscle pedalled in its flight across the English Channel, its structural components being made of tetrahedronally stabilized carbon fibers.
The minimum structural system in universe -- the tetrahedron can be tensegrity structured and a linear growth of the tensegrity tetrahedron becomes a tensegrity column. The carbon fiber is most probably constructed in exactly this way as the tensegrity tetrahedron column is demonstrably the strongest and lightest column structurally producible. This column and its method of assembly is shown on the next page.
|Figure 10:||Functions of Positive and Negative Tetrahedra in Tensegrity Stacked Cubes|
|( Figs. 730.11 and 730.12 from Synergetics with a hand-drawn addition and 730.11 caption omitted; Figs. 6.47 and 6.48 of Cosmography with a hand-drawn addition, 6.47 caption omitted and slight changes to caption text)|
|Figure 11:||Tensegrity Masts as Struts: Miniaturization Approaches Atomic Structure|
|( Fig. 740.21 from Synergetics)|
[Editor's note: A slightly different version of the following paragraph appears on p. 184 of Cosmography.]
These illustrations show my omnitetrahedra-comprised tensegrity mast, each of whose struts is comprised of tetrahedronal tensegrity masts each of whose micro-mast struts in turn consist of tetra-tensegrity masts until we reach the minitude of the atoms whose internal structuring is discontinuous compression-continuous tension. The tensegrity mast explains why carbon fibers have twelve times the strength per pound of minor carbon content structural steel and four times the strength per pound of the strongest aluminum alloy. Great wonder however that tetrahedra and their significance are not included in college preparatory school system curricula.
[Editor's note: A slightly different version of the following paragraph appears on p. 46 of Cosmography.]
I next undertook to discover why the 3 independent aluminum tubed, dacron-cord-cohered triangle held its shape. We discover that a pair of scissors is a pair of edge-sharpened levers pin-fulcrummed on each other, in which the longer the lever arms, the more powerfully they can cut. We discover that any two sides of our necklace triangle which are tensilely cohered are thus fulcrumed to one another at their commonly situate corner ends. We then discover that on the third side of the triangle we have a push-pull tube, which is firmly seizing the outer ends of the two levered together other tubes of the triangle, thus the third tube stabilizes the angle opposite it with minimum leverage effort. We find this minimum effort characteristic to be consistent with all natural behaviors which always accomplish their patterning work with minimum effort. In the triangle we have the principle of the leverage advantage holding the complex of events motionless in contradistinction to levers being used to move objects with minimum possible effort.
|Figure 12:||Triangle Levers|
|(Similar to Fig. 2.4 from Cosmography)|
In synergetics, we recognize experientially that all lines are histories, trajectories or traces of actual events. Vapor trails of airplanes jetting at high altitude are time and place histories along which the event jet airplane has taken place. The planes is always at the "now-new" event end of the line. The line-creating event is the airplane's jet engine's steam. The airplane's jet engine team is the "now-new" twin events beginning of the two lines. A line is an historical sequence of events. A point is an event and is the "now-new" beginning of a line.
A point or event to be seen, must be a system reflecting light from its outside surface. Obverse surface must have an insideness and reverse side. The now point is the event system of the line. If two vapor-trail-creating airplanes (systems) try to go through the same point at the same time -- wham. They prove it can't be done.
Contrary to academic plane geometry, synergetics discovers that two or more lines cannot go through the same point at the same time. When they attempt to do so we have an interference (as visibly witnessable in the high energy physicists' cloud chamber) resulting in smash-crack-aparts, reflections or refractions. If lines could go through the same point at the same time there would be no human sight. It is the reflection resultants of interferences that bounce back through our optical system that produces the images in the brain.
|Figure 13:||Interference Phenomena: Lines Cannot Go Through the Same Point at the Same Time|
|(Fig. 6.1 from Cosmography which is an updated version of Fig. 517.10 from Synergetics)|
[Editor's note: A slightly different version of the following two paragraphs appears on pp. 171-173 of Cosmography.]
The trails of two lines, one pre- and one post-crossing a point, or one only visibly superimposed at a distance apart from one another, produce an angle (). , an angle, is a visual experience -- an awareness of two other event somethings, history lines, interrelating as an angular overlay interrelationship. An angle is a conceptually imaginable interrelationship quite independent of the relative length of the angle's lines.
It takes time to measure "how long." Time is measured by numbers of completed cycles (circles). The angle is a fraction of a circle (). Angles are subcyclic. Angles are pre-time and -size conceptual. Angles are imaginatively conceptual independently of size and time. A triangle is an imaginatively conceptual triangular pattern independent of size or time.