# The Search for an Algebra or Calculus1 for Synergetics: A Key Element in the Task of Completing Synergetics

## A Missing Piece in Synergetics

Synergetics is a collection of passages and scenarios which provide illuminating insights into the way Universe operates. The collection of these insights taken as a synergetic whole suggests an omni-rational, geometrical coordination operative in Universe which Fuller himself characterizes with a broad scope in the following quotes:

All structural accounting of nature is accomplished with rational quantities of tetrahedra. The XYZ coordinates may be employed to describe the arrangements, but only in awkward irrationality, because the edge of the cube is inherently irrational in respect to the cube's facial diagonal.
-- R. Buckminster Fuller, Synergetics, Sec 540.11.
All co-occurring vectors have unique angles of direction as angularly referenced multidimensionally to a given observer's system axis, spin orientation, and system-orbit direction at the time of observation. All angularly referenced relationships inherently involve fourth-dimensional accommodation (and fifth-power accommodation, when referenced to the cosmic scenario). These relationships can be conceptually comprehended in Synergetics but can be expressed only in complex formula terms in the XYZ-CGtS system.
-- R. Buckminster Fuller, Synergetics, Sec 540.41.

So, Synergetics is unique and incisive in its exploration of the problem of identifying Nature's Coordinate System. However, Fuller's assertion that a Synergetics treatment makes Nature's relationships "conceptually comprehensible" and, implicitly, simpler than XYZ-CGtS is incompletely developed. Today Synergeticists constantly need to resort to traditional mathematical methods (especially for their computer software) that are considered irrational in Fuller's Synergetics (in particular, the Cartesian XYZ coordinate system and infinity). I think there is a great need to develop an algebra or a calculus that accommodates Synergetics principles and provides mathematical methods for problem solving. This (ambitious) missing component in Synergetics is necessary to establish it as a complete theory.

## Sampling of Possible Properties Required of a Synergetics Mathematical System

I am not very satisfied with this list. The idea is to make a list of those most fundamental Synergetics ideas, then see what type of mathematics that would suggest. The challenge is to be holistic (which is necessary in order to be true to Synergetics) and yet see clearly through the complexity that these diverse topics involve. So far, I haven't found the right level on which to look at the question.

## Speculations on Providing a Synergetics Calculus

• Accept the full infrastructure of the cultural heritage that is modern mathematics. Perhaps Synergetics is "just" a fruitful perspective (or point-of-view) which makes some problems comprehensible, but it isn't all that Fuller claimed it to be.
• Interpret linear algebra from a Synergetics perspective. Since linear algebra is independent of coordinate system (any basis will do, for example), we may be able to overcome Fuller's objections to XYZ and utilize the full machinery of vector space algebra (perhaps using a finite (tetrahedral?) field into order to overcome the problem of irrationals).
• Develop a vector and tensor interpretation for quaternion algebra (which is a division ring and not sufficient for the full apparatus of a vector space). Quaternion arithmetic represents transformations in space. Perhaps a module over the quaternions would provide the necessary abstraction?
• Building on Klein's Erlangen Programme, identify the group of geometrical transformations that "meet some criteria" in Synergetics. This may characterize Synergetics Geometry.

## Conclusion

The search for an algebraic system in which one may calculate synergetically is on-going. My plan is to continue exploring our cultural heritage of higher mathematics, physics, and Synergetics itself in pursuit of this missing link. Synergetics tells me that along the way, exciting precessional discoveries await me.

### Notes

1
Calculus in the sense of "to calculate" and not in the sense of the branch of mathematics called analysis.

## Author Information

Contact the author at cjf@CJFearnley.com. View the author's home page at http://www.CJFearnley.com/ and blog at http://blog.cjfearnley.com