Synergetics (Fuller)
Synergetics is the name R. Buckminster Fuller (1895–1983) gave to a field of study and inventive language he pioneered, the empirical study of systems in transformation, with an emphasis on whole system behaviors unpredicted by the behavior of any components in isolation. His twovolume work Synergetics: Explorations in the Geometry of Thinking, in collaboration with E. J. Applewhite, distills of a lifetime of research into book form.^{ [1]}^{ [2]}^{ [3]}^{ [4]}
Since systems are identifiable at every scale, Synergetics is necessarily interdisciplinary, embracing a broad range of scientific and philosophical topics, especially in the area of geometry, wherein the tetrahedron features as Fuller's model of the simplest system.
Despite mainstream endorsements such as the prologue by Arthur Loeb, and positive dust cover blurbs by U Thant and Arthur C. Clarke, along with the posthumous naming of the carbon allotrope " buckminsterfullerene",^{ [5]} Synergetics remains an offbeat subject, ignored for decades by most traditional curricula and academic departments, a fact Fuller himself considered evidence of a dangerous level of overspecialization.
His oeuvre inspired many developers to further pioneer offshoots from Synergetics, especially geodesic dome and dwelling design. Among Fuller's contemporaries were Joe Clinton (NASA), Don Richter (Temcor), Kenneth Snelson (tensegrity), J. Baldwin (New Alchemy Institute), and Medard Gabel (World Game). His chief assistants Amy Edmondson and Ed Popko have published primers that help popularize Synergetics, Stafford Beer extended synergetics to applications in social dynamics, and J.F. Nystrom proposed a theory of computational cosmography.^{ [6]} Research continues.
Definition
"Synergetics" is defined by Bucky (as he preferred to be called) as:
A system of mensuration employing 60degree 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 multialternative, equieconomical action options ... Synergetics discloses the excruciating awkwardness characterizing presentday 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.^{ [7]}
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 (801.00842.07) 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.^{ [8]}
Amy Edmondson describes synergetics "in the broadest terms, as the study of spatial complexity, and as such is an inherently comprehensive discipline."^{ [9]} In her PhD study, Cheryl Clark synthesizes the scope of synergetics as "the study of how nature works, of the patterns inherent in nature, the geometry of environmental forces that impact on humanity."^{ [10]}
Here's an abridged list of some of the discoveries Fuller claims for Synergetics again quoting directly:
 The rational volumetric quantation or constant proportionality of the octahedron, the cube, the rhombic triacontahedron, and the rhombic dodecahedron when referenced to the tetrahedron as volumetric unity.
 The trigonometric identification of the greatcircle trajectories of the seven axes of symmetry with the 120 basic disequilibrium LCD triangles of the spherical icosahedron. (See Sec. 1043.00.)
 The A and B Quanta Modules.
 Omnirationality: the identification of triangling and tetrahedroning with second and thirdpowering factors.
 Omni60degree coordination versus 90degree coordination.
 The integration of geometry and philosophy in a single conceptual system providing a common language and accounting for both the physical and metaphysical.^{ [11]}
Significance
Several authors have tried to characterize the importance of synergetics. Amy Edmonson asserts that "Experience with synergetics encourages a new way of approaching and solving problems. Its emphasis on visual and spatial phenomena combined with Fuller's holistic approach fosters the kind of lateral thinking which so often leads to creative breakthroughs.".^{ [12]} Cheryl Clark points out that "In his thousands of lectures, Fuller urged his audiences to study synergetics, saying 'I am confident that humanity's survival depends on all of our willingness to comprehend feelingly the way nature works.'"^{ [13]}
Tetrahedral accounting
A chief hallmark of this system of mensuration is its unit of volume: a tetrahedron defined by four closestpacked unitradius spheres. This tetrahedron anchors a set of concentrically arranged polyhedra proportioned in a canonical manner and interconnected by a twistingcontracting, insideouting dynamic named the Jitterbug Transformation.^{ [14]}^{ [15]}


Corresponding to Fuller's use of a regular tetrahedron as his unit of volume is his replacing the cube as his model of 3rd powering.( Fig. 990.01) The relative size of a shape is indexed by its "frequency," a term he deliberately chose for its resonance with scientific meanings. "Size and time are synonymous. Frequency and size are the same phenomenon." ( 528.00) Shapes not having any size, because purely conceptual in the Platonic sense, are "prefrequency" or "subfrequency" in contrast.
Prime means sizeless, timeless, subfrequency. Prime is prehierarchical. Prime is prefrequency. Prime is generalized, a metaphysical conceptualization experience, not a special case.... ( 1071.10)
Generalized principles (scientific laws), although communicated energetically, do not inhere in the "special case" episodes, are considered "metaphysical" in that sense.
An energy event is always special case. Whenever we have experienced energy, we have special case. The physicist's first definition of physical is that it is an experience that is extracorporeally, remotely, instrumentally apprehensible. Metaphysical includes all the experiences that are excluded by the definition of physical. Metaphysical is always generalized principle.( 1075.11)
Tetrahedral mensuration also involves substituting what Fuller calls the "isotropic vector matrix" (IVM) for the standard XYZ coordinate system, as his principal conceptual backdrop for special case physicality:
The synergetics coordinate system  in contradistinction to the XYZ coordinate system  is linearly referenced to the unitvectorlength edges of the regular tetrahedron, each of whose six unit vector edges occur in the isotropic vector matrix as the diagonals of the cube's six faces. ( 986.203)
The IVM scaffolding or skeletal framework is defined by cubic closest packed spheres (CCP), alternatively known as the FCC or facecentered cubic lattice, or as the octet truss in architecture (on which Fuller held a patent). The spacefilling complementary tetrahedra and octahedra characterizing this matrix have prefrequency volumes 1 and 4 respectively (see above).
A third consequence of switching to tetrahedral mensuration is Fuller's review of the standard "dimension" concept. Whereas "height, width and depth" have been promulgated as three distinct dimensions within the Euclidean context, each with its own independence, Fuller considered the tetrahedron a minimal starting point for spatial cognition. His use of "4D" is in many passages close to synonymous with the ordinary meaning of "3D," with the dimensions of physicality (time, mass) considered additional dimensions.
Geometers and "schooled" people speak of length, breadth, and height as constituting a hierarchy of three independent dimensional states  "onedimensional," "twodimensional," and "threedimensional"  which can be conjoined like building blocks. But length, breadth, and height simply do not exist independently of one another nor independently of all the inherent characteristics of all systems and of all systems' inherent complex of interrelationships with Scenario Universe.... All conceptual consideration is inherently fourdimensional. Thus the primitive is a priori fourdimensional, always based on the four planes of reference of the tetrahedron. There can never be less than four primitive dimensions. Any one of the stars or pointtoable "points" is a systemultratunable, tunable, or infratunable but inherently fourdimensional. ( 527.702, 527.712)
Synergetics does not aim to replace or invalidate preexisting geometry or mathematics, it is designed to carve out a new foundation with a language that would serve to provide a new source of insights.^{[ citation needed]}
Starting with Universe
Fuller's geometric explorations provide an experiential basis for designing and refining a philosophical language. His overarching concern is the cooccurring relationship between tensile and compressive tendencies within an eternally regenerative Universe. "Universe" is a proper name he defines in terms of "partially overlapping scenarios" while avoiding any static picture or model of same. His Universe is "nonsimultaneously conceptual":
Because of the fundamental nonsimultaneity of universal structuring, a single, simultaneous, static model of Universe is inherently both nonexistent and conceptually impossible as well as unnecessary. Ergo, Universe does not have a shape. Do not waste your time, as man has been doing for ages, trying to think of a unit shape "outside of which there must be something," or "within which, at center, there must be a smaller something." ( 307.04)
U = MP describes a first division of Universe into metaphysical and physical aspects, the former associated with invisibly cohesive tension, the latter with energy events, both associative as matter and disassociative as radiation. (162.00)
Synergetics also distinguishes between gravitational and precessional relationships among moving bodies, the latter referring to the vast majority of cosmic relationships, which are non180degree and do not involve bodies "falling in" to one another (130.00 533.01, 1009.21). "Precession" is a nuanced term in the synergetics vocabulary, relating to the behavior of gyroscopes, but also to sideeffects. (326.13, 1009.92)
Intuitive geometry
Fuller took an intuitive approach to his studies, often going into exhaustive empirical detail while at the same time seeking to cast his findings in their most general philosophical context.
For example, his sphere packing studies led him to generalize a formula for polyhedral numbers: 2 P F^{2} + 2, where F stands for "frequency" (the number of intervals between balls along an edge) and P for a product of low order primes (some integer). He then related the "multiplicative 2" and "additive 2" in this formula to the convex versus concave aspects of shapes, and to their polar spinnability respectively.
These same polyhedra, developed through sphere packing and related by tetrahedral mensuration, he then spun around their various poles to form great circle networks and corresponding triangular tiles on the surface of a sphere. He exhaustively catalogues the central and surface angles of these spherical triangles and their related chord factors.
Fuller was continually on the lookout for ways to connect the dots, often purely speculatively. As an example of "dot connecting" he sought to relate the 120 basic disequilibrium LCD triangles of the spherical icosahedron to the plane net of his A module.(915.11Fig. 913.01, Table 905.65)
The Jitterbug Transformation^{ [15]} provides a unifying dynamic in this work, with much significance attached to the doubling and quadrupling of edges that occur, when a cuboctahedron is collapsed through icosahedral, octahedral and tetrahedral stages, then insideouted and reexpanded in a complementary fashion. The JT forms a bridge between 3,4fold rotationally symmetric shapes, and the 5fold family, such as a rhombic triacontahedron, which later he analyzes in terms of the T module, another tetrahedral wedge with the same volume as his A and B modules.
He models energy transfer between systems by means of the doubleedged octahedron and its ability to turn into a spiral (tetrahelix). Energy lost to one system always reappeared somewhere else in his Universe. He modeled a threshold between associative and disassociative energy patterns with his TtoE module transformation ("E" for "Einstein").(Fig 986.411A)
"Synergetics" is in some ways a library of potential "science cartoons" (scenarios) described in prose and not heavily dependent upon mathematical notations. His demystification of a gyroscope's behavior in terms of a hammer thrower, pea shooter, and garden hose, is a good example of his commitment to using accessible metaphors. (Fig. 826.02A)
His modular dissection of a spacefilling tetrahedron or MITE (minimum tetrahedron) into 2 A and 1 B module serves as a basis for more speculations about energy, the former being more energy conservative, the latter more dissipative in his analysis.(986.422921.20, 921.30). His focus is reminiscent of later cellular automaton studies in that tessellating modules would affect their neighbors over successive time intervals.
Social commentary
Synergetics informed Fuller's social analysis of the human condition. He identified "ephemeralization" as the trend towards accomplishing more with less physical resources, as a result of increasing comprehension of such "generalized principles" as E = Mc^{2}.
He remained concerned that humanity's conditioned reflexes were not keeping pace with its engineering potential, emphasizing the "touch and go" nature of our current predicament.
Fuller hoped the streamlining effects of a more 60degreebased approach within natural philosophy would help bridge the gap between C.P. Snow's "two cultures" and result in a greater level of scientific literacy in the general population. (935.24)
Academic acceptance
Fuller hoped to gain traction for his nomenclature in part by dedicating Synergetics to H.S.M. Coxeter (with permission) and by citing page 71 of the latter's Regular Polytopes in order to suggest where his A & B modules (depicted above), and by extension, many of his other concepts, might enter the mathematical literature (see Fig. 950.12).
Dr. Arthur Loeb provided a prologue and an appendix to Synergetics discussing its overlap with crystallography, chemistry and virology.
Fuller originally achieved more acceptance in the humanities as a poetphilosopher and architect. For example, he features in The Pound Era by Hugh Kenner published in 1971, prior to the publication of Synergetics.^{ [16]} The journal Nature circled Operating Manual for Spaceship Earth as one of the five most formative books on sustainability.^{ [17]}
Errata
A major error, caught by Fuller himself, involved a misapplication of his Synergetics Constant in Synergetics 1, which led to the mistaken belief he had discovered a radius 1 sphere of 5 tetravolumes. He provided a correction in Synergetics 2 in the form of his T&E module thread. (986.206  986.212)
About synergy
Synergetics refers to synergy: either the concept of whole system behaviors not predicted by the behaviors of its parts, or as another term for negative entropy — negentropy.
See also
 Cloud Nine
 Dymaxion House
 Geodesic dome
 Quadray coordinates
 Octet Truss
 Synergetics coordinates
 Tensegrity
Notes
 ^ Synergetics, http://www.rwgrayprojects.com/synergetics/synergetics.html
 ^ Fuller, R. Buckminster (1963). No More Secondhand God. Carbondale and Edwardsville. pp. 118–163. ISBN 0809302470.
 ^ CJ Fearnley, Presentation to the American Mathematical Society (AMS) 2008 Spring Eastern Meeting, p. 6. Retrieved on 20100126.
 ^ E.J. Applewhite, Cosmic Fishing: An Account of Writing Synergetics With Buckminster Fuller. Macmillan, 1977.
 ^ E.J. Applewhite. The Naming of Buckminsterfullerene. The Chemical Intelligencer, July, 1995 (Vol. 1, No. 3), edited by Istvan Hargittai (Institute of General and Analytical Chemistry, Budapest Technical University)
 ^ Nystrom, J. F. (October 1999). "Tensional computation: Further musings on the computational cosmograph". Department of Electrical and Computer Engineering, University of Idaho.
 ^ Synergetics, Sec. 200.01203.07
 ^ Fuller, R. Buckminster (1963). No More Secondhand God. Carbondale and Edwardsville. pp. 118–163. ISBN 0809302470.
 ^ Edmondson, Amy C. (1987). A Fuller Explanation: The Synergetic Geometry of R. Buckminster Fuller. Boston: Birkhauser. pp. ix. ISBN 0817633383.
 ^ Cheryl Clark, 12 degrees of Freedom, Ph.D. Thesis, p. xiv
 ^ Synergetics, Sec. 251.50
 ^ Edmondson 1987, pp. ixx
 ^ Clark, p. xiv
 ^ Fuller, Buckminster (October 20, 1975). "Vector equilibrium jitterbug".
 ^ ^{a} ^{b} Verheyen, H.F. (1989). "The complete set of Jitterbug transformers and the analysis of their motion". Computers & Mathematics with Applications. 17, 1–3 (1–3): 203–250. doi: 10.1016/08981221(89)901600.
 ^ Hugh Kenner. The Pound Era. ISBN 0520024273
 ^ Adam Rome. Spaceship Earth: Five Books that Pinpointed Our Planetary Limits. Nature Magazine. November 26, 2015. pg. 443.
References
 R. Buckminster Fuller (in collaboration with E.J. Applewhite, Synergetics: Explorations in the Geometry of Thinking [1], online edition hosted by R. W. Gray with permission [2], originally published by Macmillan [3], Vol. 1 in 1975 (with a preface and contribution by Arthur L. Loeb; ISBN 002541870X), and Vol. 2 in 1979 ( ISBN 0025418807), as two hardbound volumes, reeditions in paperback.
 Amy Edmondson, A Fuller Explanation, EmergentWorld LLC, 2007.