Read here about Synergetics, the name Fuller gave to his energetic/synergetic geometrical explorations. It was those explorations that led Snelson and Fuller to discover the structures and coordination that Fuller named tensegrity. Read here about Synergetics in general, how tensegrity forms a subset of Synergetics' investigations, and how to dismbiguate Fuller's use of the term from other later uses. To read about the two volume work names Synergetics, see Fuller's Two Volume Synergetics.
Synergetics is the empirical study of systems in transformation, with an emphasis on total system behavior unpredicted by the behavior of any isolated components, including humanity’s role as both participant and observer. Since systems are identifiable at every scale from the quantum level to the cosmic, and humanity both articulates the behavior of these systems and is composed of these systems, synergetics is a very broad discipline, and embraces a broad range of scientific and philosophical studies including tetrahedral and close-packed-sphere geometries, thermodynamics, chemistry, psychology, biochemistry, economics, philosophy and theology. Despite a few mainstream endorsements such as articles by Arthur Loeb and the naming of a molecule “buckminsterfullerene,” synergetics remains an iconoclastic subject ignored by most traditional curricula and academic departments.
Buckminster Fuller (1895-1983) coined the term and attempted to define its scope in his two volume work Synergetics. His oeuvre inspired many researchers to tackle branches of synergetics. Three examples: Haken explored self-organizing structures of open systems far from thermodynamic equilibrium, Amy Edmondson explored tetrahedral and icosahedral geometry, and Stafford Beer tackled geodesics in the context of social dynamics. Many other researchers toil today on aspects of Synergetics, though many deliberately distance themselves from Fuller’s broad all-encompassing definition, given its problematic attempt to differentiate and relate all aspects of reality including the ideal and the physically realized, the container and the contained, the one and the many, the observer and the observed, the human microcosm and the universal macrocosm.
For newcomers to tensegrity and synergetics, it is helpful to recall Fuller's earlier name for the endeavor: synergetic/energetic geometry. This conveys his unique focus on force and structure and how they deploy across the ancient, well understood menagerie of polyhedral geometrical forms.
Definition in Synergetics, the Book
Synergetics is defined by R. Buckminster Fuller (1895-1983) in his two books Synergetics: Explorations in the Geometry of Thinking and Synergetics 2: Explorations in the Geometry of Thinking as: "A system of mensuration employing 60-degree vectorial coordination comprehensive to both physics and chemistry, and to both arithmetic and geometry, in rational whole numbers... Synergetics explains much that has not been previously illuminated... Synergetics follows the cosmic logic of the structural mathematics strategies of nature, which employ the paired sets of the six angular degrees of freedom, frequencies, and vectorially economical actions and their multi-alternative, equi-economical action options... Synergetics discloses the excruciating awkwardness characterizing present-day mathematical treatment of the interrelationships of the independent scientific disciplines as originally occasioned by their mutual and separate lacks of awareness of the existence of a comprehensive, rational, coordinating system inherent in nature." (Synergetics, Sec. 200.01-203.07).
Other passages in Synergetics that outline the subject are its introduction (The Wellspring of Reality) and the section on Nature's Coordination (410.01). The chapter on Operational Mathematics (801.00-842.07) provides an easy to follow, easy to build introduction to some of Fuller's geometrical modeling techniques. So this chapter can help a new reader become familiar with Fuller's approach, style and geometry. One of Fuller's clearest expositions on "the geometry of thinking" occurs in the two part essay "Omnidirectional Halo" which appears in his book No More Secondhand God.
Fuller's geometric system is at heart a direct extension of all previous three dimensional, real world geometric investigations, though expressed in Fuller's own rhetoric and expositional style called "thinking out loud."
Geometers found most of Fuller's insights "trivial." Each discovery by Fuller, including tensegrity and a tetrahedral space-filler that Coxeter deemed to be a significant discovery, can be seen as manipulations of known polyhedra.
Math in general was dismissive of practical geometry during Fuller's lifetime. Part of Fuller's main agenda was to comprehend the world as we encounter it without resorting to infinite series, limits, and and irrational numbers--in short, without calculus. While such geometrical explorations have a long and venerable history--indeed, it can be argued they founded classical mathematics--in recent decades geometry has been sidelined by abstract, purely algebraic methods that plotted, for example, the planetary exploration routes travelled by all of humanity's spacecraft. Jean Dieudonne, a proponent of this approach, said, "Down with Euclid! Death to Triangles!"  Such attitudes are not receptive to Fuller's geometrical innovations.
Fuller presented in a rhetoric not valued by any standard mathematical program. Fuller does not lay out his system in the currented accepted mathematical style, by way of hypotheses, axioms, and proofs. The Euler-type drive to derive the "essence" of a mathematical system in a few axioms is exactly the kind of math thinking that Fuller is avoiding. Applewhite's codification of Fuller's thought as embodied in Synergetics, while useful as a strategy to capture Fuller's ideas in such a way as to enable future random access, is counter-productive when people think Fuller did systematize his thought. Fuller never did that; he insisted on a certain fuzziness that would enable him to continually tackle the issues as they presented themselves.
Furthermore, mathematics as taught today is not interested in telling its stories in different ways. While other disciplines, such as literature, cultivate and value alternative symbolic renditions, math does not. For example, Shakespeare's sonnets are studied along with Auden's, although both speak of love; yet math courses teach one method of three digit multiplication alone. Some argue that mathematic education has discarded alternate rhetorical and theoretical frameworks in its rush to speed young learners towards mastery of calculus; and this is precisely a goal Fuller wished to avoid completely.
Fuller also fought to find centered models of reality. This helps explain his profound excitement when exploring nucleated cuboctahedra. Alexander's recent work shows that the drab continuity of numbers can also be seen as a poor reflection of the "centeredness" of reality. Reality, and particular living reality, presents itself in dynamic vivid centers surrounded by non-centers. Arithmetic is the decided opposite message--that any number is the same as any other. He also sought to find cetneredness in numbers, one reason he returned to his integers repeatedly, and found satisfaction in the 1-2-3-4-3-2-1 nature of tossing out nines. Tensegrity elucidates its "centers" as floating disconnected islands in a non-centered, omnipresent tensional web, an interesting new perspective on the issue of centeredness.
Space and the Number Line
Fuller rejects that space is Cartesian. He accepts that Descartes' mapping of orthogonal axes, as expressed in number tuples (pairs or triplets) does provide a useful mapping of distinct sightable objects in space. Fuller was thrilled with Nasa's space programs and never suggested that they drop their linear equations. Fuller did urge NASA and anyone that would listen to reject the notion that space is ordered in number tuples. In fact, Fuller suggests that thinking this way misleads researchers.
Fuller rejects that the continuity of space *is* the continuity of the number line. Fuller uses the number line of course, though he--like Plato--shows an abiding suspicion of irrational and transcendental numbers. But numbers to Fuller are always, well, just that: numbers. They are not reality, they emerge from reality, from human brains. Fuller holds that the continuity of space is based on an alternation of co-dependent co-originating co-arising oppositional pairs such as concave/convex, event/novent, space/matter and tension/compression. Numbers cannot fully express this alternation, since they irrevocably uniform--they are all NUMBERS. This is one reason that tensegrity compression/tension alternation--and tetrahedron/octahedron alternation as an all-space filling pattern--so deeply excited Fuller--it seemed a better model for the alternating nature of reality.
Fuller rejects the abstraction of the mathematical point as it is taught in algenraic-based mathematics. He finds it an in adequate model of a point in real continuous space. Note that this idea of a point is NOT ancient. The discrete point of Euclid's point/line/plane as we know it today was really elaborated in the arithmetized calculus of Weierstrass, just a century ago.
Other Critics, Like Fuller, Of the Arithmetization of Geometry
Fuller's reaction to arithmetization is typical of Weierstrass's oponents. James Pierpont of the American Mathematical Society, an ultimate insider, was also uneasy at Weierstrass' break with the past. Pierpont spoke of these extreme arithmetizations at Yale University in February 1899 when Bucky was four years old,
"The mathematician of to-day, trained in the school of Weierstrass, is fond of speaking of his science as "i c die absolut klare Wissenschaft. ' ' Any attempts to drag in metaphysical speculations are resented with indignant energy. With almost painful emotions he looks back at the sorry mixture of metaphysics and mathematics which was so common in the last century and at the beginning of this. The analysis of to-day is indeed a transparent science. Built up on the simple notion of number, its truths are the most solidly established in the whole range of human knowledge. It is, however, not to be overlooked that the price paid for this clearness is appalling, it is total separation from the world of our senses."
Fuller often calls of this separation from sensed experience "flying blind," just as Pierpont does.
Fuller's innovation is to take the critique of Weierstrass a controversial step further. Fuller says, the extreme arithmetization of our concepts of the world, and in particular the action of matter in space, leads to a dangerous seperation from life itself.
The arithmetized thinker relies too heavily on infinity, Fuller contends, for the most simple calculations, and the implied utterly uniform dependence on number--uniform, meaning a singular constituent unit as opposed to the clear non-consistent dualities that do compose the world--is a debilitating habit of mind that leads the typically arithmetized university graduate to overlook many highly nurturing, compatible and fruitful design solutions to contemporary problems.
Links and references
See the life of Fuller for more information on geometry and influences on his work.