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All the content following this section is a snapshot of the Wikipedia entry after I got through editing it, before turning it over to other members of our community to continue improving. I don't pretend it's complete. I had some trepidations in writing it, per this blog post.

I gave the Wikipedia page a boost (it was "a stub" when I took it on). I helped get the ball rolling again.

Here's a followup entry where I mention creating this Wikieducator page.

I've done a lot more writing on Synergetics elsewhere, including right here on Wikieducator.

What is Synergetics?

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."[1]

Here's an abridged list of some of the discoveries Fuller claims for Synergetics (see Controversies below) 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 great-circle trajectories of the seven axes of symmetry with the 120 basic disequilibrium LCD triangles of the spherical icosahedron. (See Sec. 1043.00.)
  • The rational identification of number with the hierarchy of all the geometries.
  • The A and B Quanta Modules.
  • The volumetric hierarchy of Platonic and other symmetrical geometricals based on the tetrahedron and the A and B Quanta Modules as unity of coordinate mensuration.
  • The identification of the nucleus with the vector equilibrium.
  • Omnirationality: the identification of triangling and tetrahedroning with second- and third-powering factors.
  • Omni-60-degree coordination versus 90-degree coordination.
  • The integration of geometry and philosophy in a single conceptual system providing a common language and accounting for both the physical and metaphysical. [2]

Tetrahedral Accounting

A chief hallmark of this system of mensuration was its unit of volume: a tetrahedron defined by four closest-packed unit-radius spheres. This tetrahedron anchored a set of concentrically arranged polyhedra proportioned in a canonical manner and inter-connected by a twisting-contracting, inside-outing dynamic named the Jitterbug Transformation.

Shape Volume Properties
A,B,T modules 1/24 tetrahedral voxels
MITE 1/8 space-filler, 2As, 1B
Tetrahedron 1 self dual
Coupler 1 space filler
Cuboctahedron 2.5 cb.h = 1/2, cb.v = 1/8 of 20
Duo-Tet Cube 3 24 MITEs
Octahedron 4 dual of cube
Rhombic Triacontahedron 5 radius rt.h < 1, rt.v = 2/3 of 7.5
Rhombic Dodecahedron 6 space-filler, dual to cuboctahedron
Rhombic Triacontahedron 7.5 rt.h = phi/sqrt(2)
Icosahedron ~18.51 edges 1 = tetrahedron's edges
Cuboctahedron 20 edges 1, cb.h = 1
2F Cube 24 2-frequency, 8 x 3 volume
Shape Volume A B T
A module 1/24 1 0 0
B module 1/24 0 1 0
T module 1/24 0 0 1
MITE 1/8 2 1 0
Tetrahedron 1 24 0 0
Coupler 1 16 8 0
Duo-Tet Cube 3 48 24 0
Octahedron 4 48 48 0
Rhombic Triacontahedron 5 0 0 120
Rhombic Dodecahedron 6 96 48 0
Cuboctahedron 20 336 144 0
2F Cube 24 384 192 0

Synergetics vols.gif
Whole number volumes

Synergetics abmod.gif
A & B modules

Corresponding to Fuller's use of a regular tetrahedron as his unit of volume was his replacing the cube as his model of 3rd powering.(Fig. 990.01) The relative size of a shape was 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, were "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, did not inhere in the "special case" episodes, were 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 involved substituting what Fuller called 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 unit-vector-length 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 was defined by cubic closest packed spheres (CCP), alternatively known as the FCC or face-centered cubic lattice, or as the octet truss in architecture (on which Fuller held a patent). The space-filling complementary tetrahedra and octahedra characterizing this matrix had prefrequency volumes 1 and 4 respectively (see above).

A third consequence of switching to tetrahedral mensuration was 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" was 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 -- "one-dimensional," "two-dimensional," and "three-dimensional" -- 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 four-dimensional. Thus the primitive is a priori four-dimensional, being always comprised of the four planes of reference of the tetrahedron. There can never be any less than four primitive dimensions. Any one of the stars or point-to-able "points" is a system-ultratunable, tunable, or infratunable but inherently four-dimensional. (527.702, 527.712)

Synergetics did not aim to replace or invalidate pre-existing geometry or mathematics, was designed to carve out a namespace and serve as a glue language providing a new source of insights.

Starting with Universe

Fuller's geometric explorations provided an experiential basis for designing and refining a philosophical language. His overarching concern was the co-occurring relationship between tensile and compressive tendencies within an eternally regenerative Universe. "Universe" is a proper name he defined in terms of "partially overlapping scenarios" while avoiding any static picture or model of same. His Universe was "non-simultaneously 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 described 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 distinguished between gravitational and precessional relationships among moving bodies, the latter referring to the vast majority of cosmic relationships, which are non-180-degree 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 side-effects. (326.13, 1009.92)

An 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 F2 + 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 cataloged 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 provided a unifying dynamic in this work, with much significance attached to the doubling and quadrupling of edges that occurred, when a cuboctahedron is collapsed through icosahedral, octahedral and tetrahedral stages, then inside-outed and re-expanded in a complementary fashion. The JT formed a bridge between 3,4-fold rotationally symmetric shapes, and the 5-fold family, such as a rhombic triacontahedron, which latter he analyzed in terms of the T module, another tetrahedral wedge with the same volume as his A and B modules.

He modeled energy transfer between systems by means of the double-edged 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 T-to-E 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 space-filling tetrahedron or MITE (minimum tetrahedron) into 2 A and 1 B module served 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 was 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 = Mc2.

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 60-degree-based 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)


Fuller hoped to gain traction for his ideas and nomenclature by dedicating Synergetics to H.S.M. Coxeter (with permission) and by citing page 71 of the latter's Regular Polytopes to suggest where his A & B modules (depicted above) might enter the literature (see Fig. 950.12). Dr. Arthur Loeb provided a prologue and an appendix to Synergetics discussing its overlap with crystallography, chemistry and virology.

However few if any academic departments, outside of Literature, have much tolerance for such an intuitive and/or exploratory approach, even with a track record of inventions and successes attached. Synergetics is difficult to pigeon-hole and is not in the style of any currently practiced discipline. E.J. Applewhite, Fuller's chief collaborator on Synergetics, related it to Edgar Allen Poe's Eureka, in terms of its being a metaphysical work. Fuller might have had more of an audience back in some Renaissance period, when natural philosophers still had an appetite for Neo-Platonism.


A two volume work of this size is going to have some outright mistakes. A major bug Fuller himself catches involves a misapplication of his Synergetics Constant in Synergetics 1, leading him to delude himself into thinking he had discovered a radius 1 sphere of 5 tetravolumes. He provides a patch 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 the output of a system not foreseen by the simple sum of the output of each system part, or simply — less used — another term for negative entropy — negentropy.


  1. Synergetics, Sec. 200.01-203.07
  2. Synergetics, Sec. 251.50


  • 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 hard-bound volumes, re-editions in paperback.
  • Amy Edmondson, A Fuller Explanation, EmergentWorld LLC, 2007.

External links