LAWS OF GEOMETRIC SYSTEMS
Curt McNamara, P.E.
Summary. In Synergetics, Buckminster Fuller defined a consistent approach to modeling Universe as spatial systems composed of events and relations. This paper will summarize the methodology and present the system properties (or laws) that are an outgrowth of this approach.
Keywords: Synergy, system, tetrahedron, tensegrity, precession
For Fuller, Universe was all that humanity perceives. As Universe is perceived, distinctions appear. Each distinction defines a system, the first division of Universe. Since the system is distinct from Universe, it must divide space in two, whether the system is outside us (a physical experience) or inside us (a metaphysical thought). In order to divide Universe in two, a system must contain at least four events. This is because one event is a point, two a line, and three a plane. It requires four events to divide space into two.
If the four events are interconnected triangularly they form a tetrahedron, the simplest structure in universe. The most economical interconnection is geodesic (i.e. the shortest path). Systems with four or more events connected triangularly are stable because they can maintain their pattern of connections when the environment changes.
System as division
System is the first division of Universe, since Universe is the sum-total of our collective experiences. “Universe itself is simultaneously unthinkable. You cannot think about the Universe sum-totally except as a scenario. Therefore, for further examination and comprehension, you need a thinkable set, or first subdivision of Universe, into systems.” (Fuller, 1975, 361.00)
The system boundary may not be physical, but a product of our perception (or conception). “The definition of a system as the first subdivision of finite but nonunitary and nonsimultaneous conceptuality of the Universe into all the Universe outside the system, and all the Universe inside the system, with the remainder of the Universe constituting the system itself, which alone, for the conceptual moment, is conceptual.” (Fuller, 1975, 251.26)
If system divides Universe, what are the components of system? “After we subdivide Universe into systems, we will make further reductions into basic event experiences and to quantum units. We will then come to the realization that all structuring can be identified in terms of tetrahedra and of topology.” (Fuller, 1975, 362.00)
Traditional approaches in mathematics might use four points to define a tetrahedron. For Fuller, there was no such thing as a point. Something may appear to be a point because the level of resolution is inappropriate to see fundamental structure. But look closely and a grain of sand is composed of atoms, the atoms of particles, and the particles of quanta. Furthermore, the discoveries of physics indicate that each level of structure has an energy basis. For example, the energy of a particle may be constant, while only its’ mass or velocity can be determined (not both simultaneously). Given the reality of physical nature, a logical basis for system definition is to use energy events.
Thus, events are discrete and dynamic. Since the atomic or quantum nature of Universe composes all special-case (i.e. physical) experience, it is the basis for higher-level structures as well. Systems are therefore discrete and dynamic, as is Universe.
As stated previously, system perception defines a boundary, which divides Universe. Fuller tabulated the following ways this happens: “A system is the first subdivision of Universe. It divides all the Universe into six parts:
- First, all the universal events occurring geometrically outside the system;
- Second, all the universal events occurring geometrically inside the system;
- Third, all the universal events occurring nonsimultaneously, remotely, and unrelatedly prior to the system events;
- Fourth, the Universe events occurring nonsimultaneously, remotely, and unrelatedly subsequent to the system events;
- Fifth, all the geometrically arrayed set of events constituting the system itself; and
- Sixth, all the Universe events occurring synchronously and or coincidentally to and with the systematic set of events uniquely considered.” (Fuller, 1975, 400.011)
In other words, every system has an inside and an outside, a past and a possible future, and consists of its own events standing in relation to Universe.
If systems are composed of events, what is an event composed of? “A basic event consists of three vectorial lines: the action, the reaction, and the resultant. This is the fundamental tripartite component of Universe. One positive and one negative event together make one tetrahedron, or one quantum.” (Fuller, 1975, 537.15) An event is therefore dynamic and consists of forces (or energy). These forces are not in complete opposition (i.e. the action is not canceled by the reaction). Rather, there is a difference between the action and reaction, which is the resultant. The resultant has been termed precession, and occurs both internally to the system or event and also in the event or system coupled to the action. See Figure 1.
Events are structured thus as elementary particles are. The physicist sees trajectories of particles as traces of energy in the cloud chamber. Such tracks don’t go to infinity, but rather interact with a second particle (the reaction) and go off in another direction (the resultant). The resultant of the action and reaction of the first particle is termed precession (Resultant + in Figure 1). The reaction of the two particles also results in a precessional effect to the second particle (Resultant – in Figure 1). Two events interacting in this fashion have three force vectors each. The vectors are coupled together to produce a minimal tetrahedron from a different perspective, one that is structured by forces or relations along the six edges. There are now two ways to form the minimal system. The first technique is to find four events as the nodes of a tetrahedron. An alternate way is define six relations as the edges of a tetrahedron as shown in Figure 1. Note that these relations are shown as forces, and these forces must balance at each node. The event node then becomes a point of inter-relationship between forces.
“Two Triangular Energy Events Make Tetrahedron: The open-ended triangular spiral can be considered one ‘energy event’ consisting of an action, reaction and resultant. Two such events (one positive and one negative) combine to form the tetrahedron.” Fuller, R. Buckminster, (1975) Synergetics. New York, USA: MacMillan. Figure 511.10. ©, and Courtesy of, The Estate of Buckminster Fuller, Sebastopol, CA.
How do these forces get balanced? In the tetrahedron model illustrated in Figure 1: “Triangles are inherently open. As one positive event and one negative event, the two triangles arrange themselves together as an interference of the two events. The actions and the resultants of each run into the actions and the resultants of the other. They always impinge at the ends of the action as two interfering events. As a tetrahedron, they are fundamental: a structural system. It is a tetrahedron. It is structural because it is omnitriangulated. It is a system because it divides Universe into an outsideness and an insideness__into a macrocosm and a microcosm.” (Fuller, 1975)
The events that systems consist of are inter-related, and the relations may be represented as vectors. “All the interrelationships of system foci are conceptually representable by vectors. A system is a closed configuration of vectors. It is a pattern of forces constituting a geometrical integrity that returns upon itself in a plurality of directions.” (Fuller, 1975, 400.09) Each event node of a stable system eventually connects back to itself via connections with other nodes. These connections constitute systemic structure, and define an internal system space. This space exists in physical systems (e.g. real world) as well as metaphysical systems (e.g. thoughts and generalized principles). “Effective thinking is systematic because intellectual comprehension occurs only when the interpatternings of experience events’ star foci interrelationships return upon themselves. Then the case history becomes ‘closed.’ A system is a patterning of enclosure consisting of a conceptual aggregate of recalled experience items, or events, having inherent insideness, outsideness, and omniaroundness.” (Fuller, 1975, 400.25)
A system may be outside us (a house) or in our thought. “Thought is systemic. Cerebration and intellection are initiated by differential discernment of relevance from nonrelevance in respect to an intuitively focused-upon complex of events which also intuitively suggests inherent and potentially significant system interrelatedness.” (Fuller, 1975, 400.06) This form of perception or thinking isn’t programmed. It happens when focus is used to dismiss irrelevant information. Thought forms by following relevant connections. “You cannot program the unknowns you are looking for because they are the relationship connections and not the things. The only thing you can program is the dismissal of irrelevancies.” (Fuller, 1975, 509.30) This process happens by tuning in the set of connections we are searching for.
“We may now say that what we do in thinking, after deliberately excluding the irrelevancies and thereby inadvertently isolating the considered set, is to further subdivide Universe into four parts:
- All of the parts of Universe that are externally irrelevant because too large and too infrequent;
- All the events of Universe that are internally irrelevant because too small and too frequent to be resolvable and discretely differentiated out for inclusion in our interrelationship considerations;
- All of the lucidly relevant remainder of Universe, which constitutes the considered and reconsidered set of experiences as viewed from outside the set; and
- The lucidly relevant set as viewed from inside the set.” (Fuller, 1975, 509.07)
When a system is modeled as a set of relations, events are the balancing points of forces at the nodes of the system. This seems intuitively clear in the case of an individual balancing work, family, and community; or balancing the need for sleep, food, and intellectual stimulation.
System relations are both internal and external, as events do not exist in isolation. For an event to act, react, and have a resultant, another event must be involved. When an organism moves a limb it does so in relation to gravity and atmosphere! In Figure 1, each event interacts with the other, forming a system. Such a system is balancing forces internally, and also maintains connections to Universe.
The minimal system model of Figure 1 involves only two events. Systems composed of four events may be tetrahedral as well. In this case, each node is an event consisting of a minimum of three components — action, relation, and resultant. This is a hierarchical model. “Systems are aggregates of four or more critically contiguous relevant events having neither solidity nor surface or linear continuity. Events are systemic.” (Fuller, 1975, 400.26) More complex systems are built from the same set of tools – events in triangular relations.
As every event interacts with Universe, it is logical to examine the ways that systems connect to each other. “Interconnection of Systems: If two adjacent systems become joined by one vertex, they still constitute two systems, but universally interjointed. If two adjacent systems are interconnected by two vertexes, they remain two systems, interlocked by a hinge. If two adjacent systems become adjoined by three vertexes, they become one complex system because they have acquired unit insideness and outsideness.” (Fuller, 1975, 400.53) This insight can be used to examine system models for connections to Universe. If the connection is only at one or two nodes, it is distinct from the other system. As a thought experiment, consider gases as interjointed systems, liquids as interlocked systems, and crystals as systems bonded to one another. (Bono)
Systems are distinct from Universe, yet connect to it for energy exchange. “There may be no absolute division of energetic Universe into isolated or noncommunicable parts. There is no absolutely enclosed surface, and there is no absolutely enclosed volume. Universe means ‘toward oneness’ and implies a minimum of twoness.” (Fuller, 1975, 307.03) Systems are perceived as distinct from Universe, yet connections are present. These connections exist because systems co-evolved with each other and did not arise spontaneously from the void. In fact, Universe requires all pieces for continued existence. “Universe is the minimum of intertransformings necessary for total self-regeneration.” (Fuller, 1975, 309.00)
This transforming or flow is a constant flux that pervades existence. “Every system, as a subdivision of the total experience of Universe, must accommodate traffic of inbound and outbound events and inward-outward relationships with other systems’ aspects of Universe.” (Fuller, 1975, 400.25) This flux consists of events, relationships, and the fundamental force of energy. “All systems are continually importing as well as exporting energy.” (Fuller, 1975, 400.11) Geometric system models can therefore be analyzed for energy flow. System structure may also evolve to store energy locally, if access to local energy is a more economical pathway. When energy is stored in a concentrated form it is called embodied energy, or emergy. (Odum, 1983) In addition, highly structured systems require the internal exchange of energy for continued existence, embodying energy in another fashion. In the language of Fuller, metaphysical knowledge (know-how) applied to Universe is wealth, the equivalent of emergy.
There is a multitude of ways that each system connects to Universe. Each of these connections enables a system to move or evolve via that connection. For example, a connection/relation of water or energy allows varying amounts of water or energy to flow, and a connection/relation of information allows importing and exporting that as well. Sustainable system design can use these connections to evolve closer to ecosystem standards. Each connection to a higher, lower, or equal level of system is a degree of freedom for system evolution.
For example, ecosystems have evolved to utilize many flows. Organism waste products and decay are used as compost for growth of other organisms. These flows are at “right angles” to apparent system purpose: they are not intentional from the perspective of the organism. This system property is known as precession and is discussed later in this paper. Human production systems can evolve along similar lines by utilizing “waste” products and interconnecting with other systems.
Balance and tensegrity
Systems thus consist of events in relation. If the events are coupled to each other triangularly the system is stable and is termed structure. Structures persist because they balance forces. The minimal structure (tetrahedron) in Figure 1 indicates that triangular relations are basic to structure.
“Triangulation is fundamental to structure, but it takes a plurality of positive and negative behaviors to make a structure. For example:
- always and only coexisting push and pull (compression and tension);
- always and only coexisting concave and convex;
- always and only coexisting angles and edges;
- always and only coexisting torque and countertorque;
- always and only coexisting insideness and outsideness;
- always and only coexisting axial rotation poles;
- always and only coexisting conceptuality and nonconceptuality;
- always and only coexisting temporal experience and eternal conceptuality.” (Fuller, 1975, 610.11)
The listing of paired forces illustrates a variety of ways in which structures persist. The observation was also made that forces balance at the event nodes. “We find that the structural system—is a complex of energy events which interacted with one another (to) produce a stable pattern; but some of them were trying to explode and some of them were trying to come together—(some wanted to) escape the system, and others were containing the system.” (Fuller, 1997) If this balance is complete the system is stable (but may be isolated!). If forces aren’t balanced internally the system may be unstable, with excess energy available to connect to another system, or separate from a system.
Interestingly, paired forces must exist in equal numbers. For example, there are three compression members and three tensional members in a tetrahedron (compression “holds the events together”, while tension “pulls them apart”).
The above list of system properties can be used to determine the structural forces holding a system together while simultaneously pushing it apart. The list can also be used to locate systems, by using inverse pairings of the fundamental system property being studied. For example, if one is investigating systems with the property of x, look for non-x.
The paired forces of tension and compression are known as tensional integrity, which has been shortened to tensegrity. Geodesic domes are tensegrity structures that can carry large loads by balancing force across the structure. Snelson has created sculptures based on tension and compression, and his early designs sparked Fuller’s adoption of the principle. (Snelson)
What brings events and forces into relation? Surprisingly, it is due to the finite nature of Universe! “Next thing, engineers say that the civilian doesn’t realize it, but every action has a reaction. … Since we have the speed of light, we now know we don’t have instantaneous Universe. Therefore, there is the action and there is another vector, or a resultant. Every action has its’ reaction and its’ resultant, and they are not the same.” (Fuller, 1997) The action of an event results in a reaction. The reaction is from a second event, coupling the actions and reactions of the events together. Resultants then balance the actions and reactions, forming a system. The time duration of the system may vary, and it may or may not be stable.
Resultants exist because actions are not completely cancelled by reactions. An event (or system) in relation to Universe has an internal resultant, and provokes an external resultant from the interaction. This effect has been termed precession. “Precession is the effect of bodies in motion on other bodies in motion. And all Universe is a complex of bodies in motion, so all the inter-effects are precessional. The effect of the sun on the earth, the gravitational pull, is to make the earth go into orbit around the sun, at 90 degrees not at 180 degrees. So the pull is 180 degrees, the resultant is 90 degrees, and this is precession. I find this is one of the motions that man is not really used to, he really thinks about his 180 degrees, and he expects 180 degrees all the time, not realizing this other angle, the resultant, precession. I find, then, that there is an action and its’ reaction is at some angle other than 180 due to a complex of other forces operating, and there is a resultant, so it is some kind of a ‘Z’ form … Not in a plane necessarily. This is a typical energy event, of a three-vector affair.” (Fuller, 1997) See Figure 1 for an illustration.
A fundamental property of systems is that they “push back”. This is due to tensional integrity (tensegrity) balancing forces between system components. Every system has a boundary as well as internal set-points (or goals). When one system interacts with another, each system attempts to maintain its own structure and direction. Action/reaction is the focus, yet the resultants (precession) are often the most important results of the interaction. Even when one system attempts to cancel the action of another, there are always two resultants. One is internal to the actor, one to the reactor.
Precession is a fundamental system property. In addition to the orbit of satellites, precession is the driving force of energy creation. Wood is burned to give off heat. Gasoline is combusted to force pistons to move, and this energy is precessionally transferred to a vehicle. Movement of the vehicle transfers heat to the roadway precessionally. The ability to move in this fashion has precessional effects on society and on the atmosphere of Earth.
The listing of precessional effects useful to humanity is surprisingly long. “Leverage, Sun power, wind power, tidal power, paddles, oars, windlasses, fire, metallurgy, cooking, slings, gears, electromagnetic generators, and metabolics are all 180-degree efforts that result in 90-degree precessional intereffects. … Fish fan their tails sideways to produce forward motion. Snakes wriggle sideways to travel ahead. Iceboats attain speeds of 60 miles per hour in a direction at right angles to wind blowing at half that speed. These results are all precessional.” (Fuller, 1975, 533.12)
Because systems (and thought) are structured as space is structured, a set of spatial system properties can be found. “In addition to possessing inherent insideness and outsideness, a system is inherently concave and convex, complex, and finite. A system may be either symmetrical or asymmetrical. A system may consist of a plurality of subsystems. Oneness, twoness, and threeness cannot constitute a system, as they inherently lack insideness and outsideness.” (Fuller, 1975, 400.05)
The fact that systems are spatial structures means properties of shape apply to systems. “All systems are polyhedra. Systems having insideness and outsideness must return upon themselves in a plurality of directions and are therefore interiorally concave and exteriorally convex. Because concaveness reflectively concentrates radiation impinging upon it and convexity diffuses radiation impinging upon it, concavity and convexity are fundamentally different, and therefore every system has an always and only coexisting inward and outward functionally differentialed complementarity. Any one system has only one insideness and only one outsideness.” (Fuller, 1975, 400.04) This property means that system boundary defines the region at which energy is internally reflected or externally diffused. In the example of an organism, internal energy powers growth and movement yet is only indirectly transferred to the environment, while external energy, (e.g. sunlight and wind) is diffused by the body.
Movement and Change
In the previous discussion of relations, the concept of systemic connections to Universe (environment) was discussed. Every system has a boundary where strong internal connections and weaker external connections meet. For living systems, the boundary is where water, air, food, information, and energy is exchanged. This exchange occurs at an event node, where a system balances forces. This balancing may result in movements and changes either internally (system structure) or externally (in relation to Universe and environment). For example, balancing human production systems for sustainability may require closing loops between human systems and the environment, or coupling human activity systems to each other as nature does. These changes occur as events, and as connections or relations.
Given the set of possible relations between every system and Universe, it is not surprising that everything is connected. “Nothing stands in a vacuum of Universe. Nothing can change locally without changing everything else.” (Fuller, 1975, 537.01) The basis for possible changes is the structure of space, and therefore system. Changes do not occur in a plane or on a line, as these are imaginary concepts. Instead, everything changes along one of the twelve possible directions (positive and negative movements of six vectors) indicated by the relations defined by the edges of a tetrahedron.
There is a plurality of ways that systems can move and evolve. “Systems can spin. There is always at least one axis of rotation of any system. Systems can orbit. Systems can contract and expand. They can torque; they can turn inside-out; and they can interprecess their parts.” (Fuller, 1975, 400.60-61) From this set of six possible motions, twelve degrees of freedom are possible (a system can spin or orbit in either direction).
This gives every system a tremendous set of possibilities for motion and evolution. “We start with Universe as a closed system of complementary patterns—i.e., regenerative, i.e., adequate to itself—that has at any one moment for any one of its subpatterns 12 degrees of freedom. There is an enormous complexity of choice. We start playing the game, the most complicated game of chess that has ever been played. We start to play the game Universe, which requires absolute integrity. You start with 12 alternate directions and multibillions of frequency options for your first move and from that move you have again the same multioptions at each of your successive moves. The number of moves that can be made is unlimited, but the moves must always be made in absolute respect for all the other moves and developments of evoluting Universe.” (Fuller, 1975, 537.02)
So at each moment every system can change direction in six ways. For example, it can decide to spin clockwise while orbiting counter-clockwise, or contract while interprecessing. How are these decisions made? As usual, they happen when they are the most practical. “Nature always employs only the most economical intertransformative and omnicosmic interrelatedness behavioral stratagems. With each and every event in Universe – no matter how frequently recurrent – there are always 12 unique, equieconomical, omnidirectionally operative, alternate-action options, which 12 occur as four sets of three always interdependent and concurrent actions, reactions, and resultants. This is to say that with each high frequency of recurring turns to play of each and all systems there are six moves that can be made in 12 optimal directions.” (Fuller, 1979, 537.06)
In the case of a system orbiting, there may be a decision of what to orbit. For example, in a work place some employees orbit around others, some around managers, and some around projects. This orbiting can be clockwise (positive?) or counter-clockwise (negative?). Since employees are complex systems, their sub-systems may be orbiting around different objects (i.e. work effort around project team, while social life circles around another individual). Political systems composed of individuals have many orbits and coordinating focus on a topic can be a daunting task! In addition, each system will have their own perception of the situation, leading to different system models. A key benefit to the type of modeling practice suggested in this paper is that a shared model can be built with differing viewpoints included or aligned.
Describing a system with relations
The preceding paragraphs described the structure of systems as described by events and relations. This methodology can be summarized with the following process: (Refer to Figure 1)
- Find an action of interest (Action+).
- Search for the reaction (Reaction-) that occurs in response to the action.
- Locate the corresponding resultant (Resultant-, the energy that balances the action/reaction).
- Define the internal reaction (Reaction+), internal resultant (Resultant+), and the external action (Action-).
- Describe the forces that characterize the actions, reactions, and resultants
- Identify the forces as compressive or tensional and pair them.
- Determine the energy balance at each node, thereby determining the stability of the system.
- Determine whether there is excess energy for change either
- a) Within the system or;
- b) Between the system and Universe.
Another approach to this process is to find paired forces. For example, consider an action/force to find lowest cost production, and an opposing reaction/force to maintain standards. What is the resultant of these two? Recall that there is an internal resultant in the system/event seeking lowest cost, and another in the system/event seeking standards. Once these four vectors have been identified solve the system for the other two vectors as above.
This modeling methodology helps in understanding the basics of systems and relations. Every event node balances a minimum of three forces, and all events in a system are in relation to one another. This modeling is more complex than the simplistic pairing of forces commonly employed in discussing complex situations.
With geometric systems defined, the inverse of a system can be found. “A system is the antithesis of a nonsystem. A nonsystem lacks omnidirectional definition. Nonsystems such as theoretical planes or straight lines cannot be found experimentally. We are scientifically bound to experientially discovered and experimentally demonstrable systems thinking.” (Fuller, 1979, 400.21) In other words, systems require enough connections to divide space. The minimal structures of mathematics (lines and planes) do not constitute a system. Similarly, models and equations that are single or two-dimensional or lack connections between events are nonsystems.
System as structure
This paper has identified the following system properties:
- The correspondence of system with spatial structure has been established.
- A set of spatial system properties has been identified.
- The basic nature of interaction (action/reaction/resultant) has been explored.
- System movement and evolution has been modeled as changes in relations.
- A systematic way to make a system model of relations has been suggested.
With these results in hand, systems practitioners can model systems and theories geometrically. This is a significant addition to existing practice, as system models and theories are often text-based or based on equations, and may not explicitly define dimension, connectivity, or shape. The addition of this tool to modeling toolkits connects the structure of space (and thought) to systems. Using this methodology will result in models with enough complexity to model the real world.
Bono, Rick. Applied Synergetics. A web page showing how to make geometric models using simple materials. See the tetrahedron model for an example of modeling interconnected systems. This section is part of a comprehensive site on Synergetics.
Buckminster Fuller Institute. The best place to find copies of materials for the student of Fuller and geometric thinking. Buckminster Fuller Institute; 111 N. Main Street; Sebastopol, CA 95472 Phone: 707-824-2242 Fax: 707-824-2243
Edmondson, Amy. (1987). A Fuller Explanation. Perhaps the best introduction to Fuller’s work. Unfortunately out of print, but it can be ordered through the Buckminster Fuller Institute in photocopy or viewed on the web at:
Fuller, Buckminster. (1975) Synergetics, 876 pp. New York, USA: MacMillan.
Fuller, Buckminster. (1979) Synergetics 2, 592 pp. New York, USA: MacMillan. Unfortunately out of print, but they can be ordered through the Buckminster Fuller Institute in photocopy. Both volumes have been combined at:
Fuller, Buckminster. (1997) Everything I Know. Sebastopol, CA, USA: Buckminster Fuller Institute. An on-line version is available at the Buckminster Fuller Institute:
McNamara, Curt. firstname.lastname@example.org Additional information on geometric systems as and procedures for modeling can be found at the author’s web-site:
Odum, Howard. (1983) Systems Ecology. New York, John Wiley. A brief exposition on emergy can be found at:
Snelson. A summary of Snelson’s work and its’ relation to Synergetics can be found at the web site Synergetics on the Web, created by Kirby Urner.