Curt McNamara, P.E.
Systems have a spatial basis. As such, they consist of a minimum of six forces, interconnected by four events. The forces are coupled to each other at the event nodes, and this coupling balances system forces internally, instant by instant. Forces may also be interchanged with other systems. This interaction or coupling between systems can be modeled as a shared flow of energy, information, or material. The energetic coupling is either an energy source – more than is required to maintain internal structure; or an energy sink – energy required to maintain internal structure. Such a flow may occur along a force member of the structure (i.e. a relation or connection), or it may occur at an event node (i.e. balance point) of the system. This paper will explore these two approaches to modeling systems coupling, and relate this coupling to the systemic effect of precession.
Keywords: System, tetrahedron, precession, flow, event
LOCATING A SYSTEM
In order to investigate system interaction, systems must be found and their components described. There are many approaches ranging from mathematical description of energy exchange (e.g. a control system) to prose explanations of human interaction. In A Fuller Explanation, Amy Edmondson explains Buckminster Fuller’s approach to thinking, and thereby detecting systems:
“Thinking, he explains, starts with ‘spontaneous preoccupation’; the process is never deliberate initially. We then choose to ‘accommodate the trend,’ through conscious dismissal of ‘irrelevancies’ which are temporarily held off to the side, as they do not seem to belong in the current thought. Fuller places ‘irrelevancies’ in two categories: experiences too large or too infrequent to influence the tuned-in thought, and those too small and too frequent to play a part. The process he describes is similar to tuning a radio, with its progressive dismissal of irrelevant (other frequency) events, ultimately leaving only the few experiences which are ‘lucidly relevant,’ and thus interconnected by their relationships.
Once an experience is isolated, the internal structure of that systemic thought can be established. Edmondson continues:
“Thinking isolates events; ‘understanding’ then interconnects them. ‘Understanding is structure,’ Fuller declares, for it means establishing the relationships between events.
“A ‘thought’ is then a ‘relevant set,’ or a ‘considerable set’: experiences related to each other in some way. All the rest of experience is outside the set-not tuned in. A thought therefore defines an insideness and an outsideness; it is a ‘conceptual subdivision of Universe.’ ‘I’ll call it a system,’ declares Bucky; ‘I now have a geometric description of a thought.’
“This is the conclusion that initially led Fuller to wonder how many ‘events’ were necessary to create insideness and outsideness. Realizing that a thought required at least enough ‘somethings’ to define an isolated system, it seemed vitally important to know the minimum number—the terminal condition. He thereby arrived at the tetrahedron. ‘This gave me great power of definition,’ he recalls, both in terms of understanding more about ‘thinking’ and by isolating the theoretical minimum case, with its four events and six relationships.” (Edmondson, 1987)
Recall that the tetrahedron is the simplest structure in Universe, because four points are required to separate systems from their surroundings in three-dimensional space. Two points define a line, three a plane, but four are required to separate space in two.
A system may be large and outside of us (e.g. a house), small and inside of us (e.g. a thought), or a web shared between systems (e.g. a meme (Dawkins, 1990)).
SYSTEM = STRUCTURE = TRIANGULATION
The tetrahedron consists of four event nodes joined by six relations. Each node of a tetrahedron is connected to every other node through a single relation. In more complex systems, nodes are interconnected to every other node as well, but there may be intermediate nodes between any two. All systems are triangulated – i.e. every node is interconnected to every other node via triangular relations. The tetrahedron is the minimal system.
Nodes are balance points between the internal system relations, and any energy flowing in or out of the system. The relations are the pathways for force within the system, and therefore distribute internal and external forces across the structure.
There are at least three distinct perspectives that apply to this type of systems modeling. In a system dynamics view the relations are flows (or rates of change), and the nodes are the stocks (or accumulations), representing the difference between the flows being balanced. For example, the nodes may represent populations of predator and prey, while the relations represent birth and death rates.
From an algebraic point of view the relations are variables, which are summed at the nodes. Using this perspective, the balancing points do not represent physical accumulations, but are algebraic summing junctions for the behavior of the forces represented by the relations.
A third view is that system interconnection consists of events in relation to one another (Fig. 1). Each event consists of a vector set of Action, Reaction, and Resultant. For every Action there is a Reaction. However, the Reaction does not completely cancel the Action. The energy difference between the Action and the Reaction is the Resultant. Fig. 1 models Event+ and Event-, each consisting of these three forces. The relations in Fig. 1 are the vectors of Action, Reaction, and Resultant, and each node balances three vector forces to maintain system structure.
Figure 1 (after Fuller)
System: Interaction of Two Events Furnace: Two Interacting Events
Regardless of the model perspective, there must be sufficient variety of connections to internally balance the system. This requires that the forces, which are summed at the nodes or balance points, consist of both inward and outward components. If all the forces were outward, the structure would expand and fall apart or disintegrate. There would be nothing holding it together! Similarly if all the forces were inward the system would implode. In a tetrahedron, three relations or vectors are compressive (holding the structure together) while three are tensional (pulling the structure apart). It is the balance of both positive and negative forces that creates a whole structure. This interplay of forces is known as tensional integrity (tensegrity), and balances all the relational forces in parallel at all system event nodes.
If the forces exactly balance at the nodes, the system will be stable (but isolated). If forces aren’t completely balanced internally the system may be unstable, have energy needs that require connection to another system, or may separate from other systems.
Internal system balance results in a combination of relational edge lengths and connections, giving the system a preferred shape and structure. As a result of the internal system connections and the balancing properties of the nodes, energy applied to one node of a system is distributed across the entire structure. The simultaneous balancing at event nodes means that the system acts to maintain its’ shape when exposed to external forces, resulting in system “set-points” or goals. The interplay of internal and external forces gives rise to the property that systems “push back”. When one system interacts with another, each system attempts to maintain its own internal balance, structure, and connections.
Since no system is truly isolated and no energy exchange is perfectly balanced, systems must import or export energy. “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) Even when systems are perceived as distinct from Universe, connections are always present. These connections are due both to the laws of thermodynamics (energy flow is required for existence), and also 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)
An example of balancing system behaviors in human activity systems can be seen in the interplay of System 3 and System 4 in the Viable System Model, and also in the Team Syntegrity model (Beer, 1994). The system maintaining the present state of a firm must be balanced with the system creating the new future.
What is the basis of systems interconnection? As noted previously, systems require continuous interchange of energy with their surroundings. The rate of change of energy varies – some interactions are slow (the blue mountain and the white cloud) while some are fast (fruit flies and a banana). Regardless of the rate, energy exchange is required to maintain systems and events. This energy follows the standard laws of physics. For every action, there is a reaction. Since actions and reactions do not precisely cancel each other, there is excess energy, which is termed a resultant (see Figure 1).
At Node A of the illustration (the top one in this perspective), Action-, Reaction-, and Reaction+ balance. Simultaneously, Node B balances Action+, Reaction-, and Resultant+; Node C balances Action-, Resultant-, Resultant+; and Node D balances Action+, Reaction+, and Resultant-.
The two-event system illustrated is a furnace. Event+ is the control, or thermostat. Its’ parts are Set-point (Action+), Temperature (Reaction+), and Command (Resultant+, or the difference between the action and reaction). The coupled event (Event-) is a furnace, consisting of Valve as Action-, Fuel as Reaction-, and Flame as Resultant-.
The implication from the illustration is that all four nodes balance perfectly. In practice, no system is isolated from its’ surroundings, but is interconnected to surrounding systems. This interconnection is via energy exchange, either at the event nodes (balance points or accumulations) or along a relation (flows or rates of change). Therefore our equations of balance need additional terms to connect this system into the larger matrix of systems. In the furnace example fuel may come from another system.
These are three basic types of interconnection: “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)
Referring back to the model shown in Figure 1, each vertex (or event node) balances three forces from the set of possible actions, reactions, and resultants. This balance may leave enough energy to allow interconnection to another system. For example, the oxygen atom is balanced, yet has an energy sink that can be filled by the energy source of another atom or two. (Note 1)
This insight can be used to examine system models for connections to Universe. If the connection is at one node, a single flow will pass from one system to the other. Even if each system connects via a tetrahedral node, where each side has three vector relations that contribute to the new node (see Figure 2), there is a net flow of energy from one system to the other. It is possible to theorize systems where energy flows or is pumped first from one to the other and then back again through a single node. Such a set of interconnected systems would still have a net flow over time (either positive, negative, or balanced).
Nodal coupling may be either a low-energy association like gas molecules to one another, or energy transfer between a source or sink. Gas molecules couple together temporarily, as the countervailing force to keep the connection is weak. Examples of energy transfer at a node with a one-way flow would be a mosquito or fly biting an animal, or oil being extracted from a well. The length of time the energy transfer occurs is a function of the energy the extracting system uses to maintain the connection. For example, the mosquito uses energy to remain attached to the skin, and an oil well uses energy to drill, to pump, and to guide the flow. In many cases the energy applied to the resource-rich system does not benefit that system but merely allows the extraction to take place. In these cases the coupling node can be modeled as a stock or accumulation of resources, with one system acting to extract resources from another. The connection is maintained by the extracting system, as there may be no benefit to the system that energy is being extracted from.
It can be seen that this type of interconnection does not result in a higher level of organization between the two interconnecting systems. Research on embodied energy (or emergy, Odum, 1983) and on dissipative structures (Prigigone, 1980) indicates that complex systems use local storage of energy to guide or amplify energy transfer from other systems into their own. It is clear that this can take place within one of the systems coupled via a single node. For example, human activity systems use locally stored emergy (knowledge of oil deposits) to guide further transfer of energy from these deposits. This allows the human activity system to increase in size and complexity as the other system decreases in size and complexity. Is it possible for systems to interact in a way that increases variety or complexity in each one? (Note 2)
Note that a nodal connection between systems requires at least three force components from each system (three is the minimum number of connections per node in geometric systems). These six forces balance between the two systems.
To create higher complexity, and sustainable connections between systems, energy transfer between two nodes is required. One connection transfers energy from System A to System B, while the other transfers energy from System B to System A (see Figure 3). This allows each system to use energy from the other to increase its’ own storage, which can then increase autocatalytic structures in each system. As these structures develop, there is an increase in emergy in both systems.
This two-way flow of energy can be seen in the bee seeking nectar as it pollinates flowers. It appears that the bee does not intend to pollinate, and does so simply as a by-product of visiting flowers (although flowers have evolved to place pollen in locations where it will attach to the hairs of the bee!). However, if the bee did not pollinate, the flowers would not thrive. If the flowers did not thrive there would be less nectar for the bees. Since the act of pollinating benefits both flowers and bees, this is a sustainable two-way flow, persisting over generations of flowers and bees. In the bee system an increase in pollen increases local storage of honey, aiding in the growth of the colony. Reproductive success of the flower system is increased, and the flower system thereby grows in size and interconnects genetically via pollination.
To reiterate: if the modeling viewpoint is nodes as stocks or accumulations, and the connection is at one node, then one system is transferring energy or materials from the other system. There may be energy flow required to maintain the transfer, but this may not benefit the resource-rich system. For each system to grow either autocatalytically or sustainably, a second node must be involved where energy or material from each system is passed to the other.
Relational or Flow Interconnection
Alternatively, the modeling viewpoint can be that systems couple via the interconnecting relations. If the connection is across a relation, the flow represented by that relation is being shared with the other system. An example is a waterwheel in a stream. The flow must still connect at nodes between systems, with one node supplying the flow to the other system, while the second node returns it (it appears uncommon for one system to share the entire flow of another). In this case the relation that represents the flow may either have parallel paths between nodes, or it may have intermediate nodes between its primary ones.
See Figure 4 for an example of the waterwheel and the stream. The coupling relation of the stream is the entire flow of water. This flow may be divided into many parallel paths, with a waterwheel in one segment of a parallel path. This illustration gives a sense that flow coupling can easily scale up and down in size and pathways. One system may couple to many others either via parallel relations, or through many small segments. An example of parallel flow coupling is when rainfall that would normally flow to a river is captured for human use. Part of the rainfall pathway is captured, while part flows to the river as normal. Flow coupling can be seen as the dual of stock or accumulation coupling.
If the connection is across a triangular face, forces are being simultaneously balanced at three nodes. There will be energy exchange at the nodes, and there must be a combination of positive and negative forces on the interconnecting relations. Each interconnecting node now has an additional connection, and the forces and flows of the connections are shared between the interconnecting nodes. Since the combination now has acquired insideness and outsideness, there is only one system present. Note that this example does not scale up and down as the previous flow example using two nodes.
It has been suggested that gases are systems interjointed at a node, liquids are systems interlocked across a relation, and crystals are systems bonded across a face to one another. (Bono)
Fuller has stated that precession is a primary means of system interconnection. “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) Refer back to Figure 1 for an illustration.
Recall that a resultant (and therefore precession) exists because an action is not completely cancelled by a reaction. An event (or system) in relation to Universe generates an internal resultant, and an external resultant is also generated from the interaction. In a general sense each node in Figure 1 has a resultant, even though it may not labeled as such. This is because at each node of a tetrahedron two vectors come together but do not cancel each other out, leaving the excess energy as a resultant.
Nodes or relations are shared when excess energy is shared between systems. The system energy equations then increase to include the relations or vectors from both systems. A shared system node may include several actions, reactions, and resultants in this case.
Precessional coupling occurs when excess energy (more than is required to maintain connectivity at a nodal set of actions, reactions, and resultants) is used by another system. The excess energy is a difference. Therefore precession is a difference, and precessional coupling happens when that difference is shared between systems. As noted before, this sharing may happen at a node or along a relation. Interestingly, information is also modeled as a difference (Bateson, 1988).
How is precessional coupling related on systems interconnection? Each case of interconnection is an example of energy sharing between systems, and is therefore precessional.
In nodal interconnection one system has stored energy (a stock or accumulation) that another system couples to. The system obtaining energy acts precessionally to the storage of excess energy in the other system. The coupling is at “right angles” to the intent of the energy-rich system, which had been acting to accumulate energy. The flow is from one system to another. It is possible to calculate the nodal impact that humanity is having on natural systems (Wackernagel, 1995).
In the case of bi-nodal interconnection, there are two energy flows between the systems during the time of contact. Each flow is precessional. The bee obtains nectar and collects pollen precessionally. The flower distributes pollen and gives nectar precessionally. Each system benefits from the interconnection. Bono has suggested that bi-nodal interconnection can be found in liquids, where atoms share bonds. In this case the positive energy from the proton-rich system is being balanced with the negative energy with the electron-rich system.
Flow interconnection transforms energy precessionally in the act of coupling. The linear flow of the stream is changed into the rotary motion of the waterwheel and its’ shaft. In a similar fashion the rotary motion of wheels can be transformed into the linear motion of a surface (for example a moving vehicle or a belt-type sanding machine). As noted previously, flow coupling is the dual of stock or accumulation coupling. There is “excess energy” in the system that has the flow. A second system couples to the flow precessionally.
Triangular coupling consists of energy being shared between nodes and across a system boundary. The result is that a new system is formed. At the instant of contact there is precessional coupling, and then forces equalize, forming a new system. This type of interconnection can also happen in bi-nodal interconnection, if the systems remain in contact because the new energy relations balance at the shared nodes.
It is clear from this discussion that all systems interconnection is precessional, yet each case of coupling differs. Nodal coupling is unidirectional material/energy transfer, bi-nodal coupling is bi-directional material/energy transfer, flow coupling is unidirectional energy/material transfer, and triangular coupling is bi-directional energy balance.
What are the systemic effects from each type of coupling? When systems interconnect at one node, the net energy flow benefits the resource-poor system. There is no countervailing force to maintain the resource in the resource-rich system. This architecture could be sustainable, if the rate of flow to the resource-poor system can be maintained by the resource-rich system. Wells work marvelously, as long as the flow taken out of the well can be replaced by the surrounding aquifer. When the rate increases beyond this the flow is unsustainable. Similarly, human activity systems can be created by a small set of people and used by others, as long as the rate of use can be sustained by the system. A local park may be used by those outside the city sustainably, until the rate of usage exceeds the natural and human energy put into maintaining it. There is a variety of work occurring in the area of environmental restoration and remediation (Whole Earth Review, 2001).
In the pollination example, the bee did not intend to pollinate, yet it does so and benefits both systems. The bee’s presence on a flower with the appropriate structure transfers pollen to the bee. It is seeking energy (nectar) and finds it in a flower. The energy required for the pollen contact is not part of the intentional action and reaction, which is the bee gathering nectar. It is excess energy from the bee’s physical contact that provides the possible pathway for the pollen to be gathered and shared. The flower has excess energy in the form of pollen. Instead of waiting for wind to distribute it to another flower, the bee is an agent that can couple to this energy. Is it possible to evolve nodal interconnections so that sustainable systems are formed? As an example, Fuller based his life on the effect of the bee. He calculated that if he dedicated himself to pollinating (i.e. planting ideas in heads) that Universe would provide for him and his family. If Universe did not provide, then he would re-examine his strategies and re-adjust them.
The pollination example is not an isolated phenomenon! At a physical level precession may occur via interactions (generalized principles) between one system and another, yet systems also interact through intermediaries. “Because … vegetation is rooted, it cannot reach the other vegetation to procreate. To solve this regenerative problem Universe inventively designed a vast variety of mobile creatures__such as birds, butterflies, worms and ants__to intertraffic and cross-pollinate the vast variety of vegetation involved in the biochemical refertilization complexities of ecology, as for instance does the honeybee buzz-enter the flowers to reach its honey while inadvertently cross-fertilizing the plants.” (Fuller, 1979, 326.12) The capture of solar radiation by vegetation is also a precessional effect. “Precessional 180-degree efforts beget 90-degree effects such as the Sun’s radiation impoundment on Earth by the photosynthesis of agriculture (around the land) and photosynthesis of algae (around the waters of Earth), which regeneration occurs as precessionally impounded life-sustaining foods. The 180-degree Sun radiation effect precesses Earth’s atmosphere in 90-degree circumferential direction as wind power, which wind power in turn precesses the windmills into 90-degree rotating.” (Fuller, 1979, 533.10)
Flow coupling can be sustained as long as the flow being shared is within the carrying capacity of the system. The nodal example of the well can also be modeled as flow coupling. Obviously, waterwheels in streams are sustainable! Partial use of river flow for industry can be sustainable. The condition of the water returned to the river is the primary indicator of the sustainability. Conditions for system sustainability have been laid out by The Natural Step organization (The Natural Step). Interestingly, they appear to have a tetrahedral basis.
Commentary about precession often focuses on the fact that primary contact such as action and reaction is often perceived to be the most important part of an interaction, while the most interesting part of the interaction is actually that which flows along the difference relation (i.e. the resultant or precession). In that sense, precession is the unseen effect of systems upon one another, most particularly when that effect appears at “right angles” to the apparent intent of the system.
How does this view of Universe connect with the Newtonian view that “for every action there is an equal and opposite reaction”? In other words, what keeps 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.
In the case illustrated in Figure 1 the forces that initiate the interaction are Action+ and Reaction-. As a result of this interchange, each event has its’ own internal vector response of action/reaction/resultant. The obvious interaction is the Action/Reaction pair, which has corollaries in arguments and political systems. The law of precession informs us that even when parties disagree or fight to a win/lose endgame, there are precessional effects within each system. If two parties argue, the loser will often change behavior in ways that are unintended by the winner!
Clearly humanity uses precession in the design of technological artifacts. This type of precessional effect is found everywhere: “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)
It is also clear that precession begets precession. Humanity’s ability to precess (move) across the face of the Earth changed society at right angles to this technology. The technology itself (i.e. cars) has effects (exhaust, urban sprawl, social change, freedom of association, movement of materials beyond ecosystem boundaries, and possibly global warming) that are at right angles to its’ intended purpose.
Modeling precession with graphs
The examples are shown as directed graphs (or digraphs). A more intuitive approach is to model them as three-dimensional structures, but this type of model is easily done in a computer (Koutsofios). As such, these examples are known as a structural model. Structural models that illustrate hierarchy are known as “interpretive structural models”, and have been extensively developed by Warfield (1990).
Structural models have varying numbers of connections at each node. In contrast, geometric models require a minimum of three connections per node (as the tetrahedron), and typically require that all nodes connect to each other. In the case of the octahedron the “end nodes” are not required to connect to each other to maintain structure (see Figure 7).
These figures show precession as a shared relation, which results in a transfer of energy at “right angles” to the relation. Other precessional examples may be present when systems contact at one node (for example as a gas would do), or when they share a face (the triangular area of a tetrahedron). This type of sharing has been mentioned in Atkin (1974) as “Everyman is a Polyhedron”.
The following figures illustrate a variety of geometric models shown as graphs. In the case of Figure 6, a stream is shown as a combination of earth, sky, and a difference in potential (head to end). Arrows are used to show flows of water, and labels are used to name the flows. The right side of Figure 6 is a structural model of a water-wheel, showing the pieces that comprise it and the forces that characterize the relations between the pieces. The two models are combined when the flow (relation) of the stream is in contact with the rotation of the wheel and shaft. At this point there is a precessional effect in the water wheel as the rotation of the shaft is used as energy.
Stream and Waterwheel
Figure 7 is more abstract yet, and shows an octahedral model of system components with names for each relation. On the right side of Figure 7 is a tetrahedron illustrating the paired forces and properties that often characterize systems interconnection. Consider chemical bonding for one example. The two system models can interconnect across the appropriate relations.
System Properties as Octahedron, System Molecule as Tetrahedron
1) Each atom was systemic by itself, yet the combination is a new system, which shows properties quite unexpected by the individual atoms. There is nothing in the properties of hydrogen or oxygen individually that predicts the beauty of the snowflake or the ways that living systems utilize water.
2)This situation is echoed in the difference between complex ecosystems where many individual entities have evolved together to a high level of organization, and highly extractive systems like present technological economies where variety in the natural world is decreased as the complexity of the manufactured world increases. It appears that the manufactured world is not sustainable at this time (Brown, 2001). There are also many examples from human activity systems in which nodal or one-way transfer has proven to be inadequate to maintain a connection between systems. It appears that perceiving systems monitor their energy relations and seek connections that aid their own persistence.
Atkin, Ronald. (1974) Mathematical Structure in Human Affairs. New York: Crane, Russak and Company, Inc.
Bateson, Gregory. (1988) Mind and Nature. New York; Bantam.
Beer, Stafford. (1994) Team Syntegrity. New York: John Wiley.
Bono, Rick. Applied Synergetics. A comprehensive site devoted to the work of Fuller. In particular, this page shows how to make geometric models using simple materials. See the tetrahedron model for an example of modeling interconnected systems. http://www.cris.com/~rjbono/html/model.html
Brown, Lester. (2001) State of the World 2001. New York: Norton and Company.
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
Dawkins, Richard. (1990) The Selfish Gene. New York: Oxford University Press.
Edmondson, Amy. (1987). A Fuller Explanation. Birkhauser: Boston, USA. The best introduction of the work of Buckminster Fuller. 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. MacMillan: New York, USA.
Fuller, Buckminster 1979. Synergetics 2. MacMillan: New York, USA. Fuller’s magnum opus, encapsulating his thought on geometric systems. 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. Buckminster Fuller Institute: Sebastopol, CA, USA. An on-line version is available at the Buckminster Fuller Institute:
Harary, Norman, and Cartwright. (1965) Structural Models: An Introduction to the Theory of Directed Graphs. New York, Wiley.
Ikoso Kits. Offers simple geometric modeling kits. 85210 Willamette St., Eugene. OR, 97405 USA. 1-800-231-0104
Koutsofios, Eleftherios. Drawing Graphs with Dot. AT&T Bell Laboratories.
Krausse and Lichtenstein. (1999). Your Private Sky, 524 pp. Germany: Lars Muller. An excellent compilation of materials on Bucky’s life and work. Consists of a biology/chronology of Fuller, and numerous examples of his designed artifacts are included.
McNamara, Curt. firstname.lastname@example.org Additional information on geometric systems and procedures for modeling can be found at the author’s web-site:
Odum, Howard 1983. Systems Ecology. John Wiley: New York, USA. A brief exposition on emergy can be found at:
Prigogine, Ilya. (1980) From Being to Becoming: Time and Complexity in the Physical Sciences. San Francisco: W. H. Freeman & Co.
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.
The Natural Step. World-wide organization devoted to advancing sustainability.
Urner, Kirby. Synergetics on the Web. A website devoted to Fuller’s work, particularly Synergetics. Site has numerous computer-based representations of geometric systems.
Wackernagel, Mathis. (1995) Our Ecological Footprint. Gabriola Island, British Columbia: New Society Publishers.
Warfield, John. A Science of Generic Design: Managing Complexity Through Systems Design. Intersystems Publications. Salinas, California. (1990). 610 p. [IASIS 90/004]
Whole Earth Review, Spring 2001 (2001). Whole Earth, 1408 Mission Ave., San Rafael, CA. 94901 USA
Zometools. A source for geometric modeling tools (struts and hubs to make polyhedra).
1526 S. Pearl St. Denver, CO, 80210 USA 1-888-966-3386, 1-303-733-2880
Modeling a System Geometrically
To make a geometric model a system must first be separated from Universe. While some systems are clearly distinct from Universe (for example a tree), others require a tool to identify their events and relations. One technique is to take a piece of paper and write down the central idea in the center of the page. Then brainstorm to generate system components (boundary, feedback, goals, pattern). Write down each name or idea and connect it to the central one. Proceed in this fashion, creating a network linked back to the center. Now look one level out and add terms or ideas related to that level. Take the time to re-write the map, using an organizing idea to re-structure the central connections.
A second approach is to use a technique like soft systems methodology. On a piece of paper, list the CATWOE elements of the system under consideration. C stands for Customer (everyone who stands to gain), A for Actor (performs the process), T for the transformation process, W for weltanschauung (world-view), O for Owner (can start or stop the system), and E for environmental constraints (outside the system). Develop these into a “big picture” of the situation. Once this part is done, identify potential processes (relations) and nodes (events) in the model.
The next step is combining the parts into a structure. Obtain one of the following: colored paper, a geometric modeling set (e.g. Zometool, Roger’s Connections, or Ikoso Kits), or sticks and modeling clay.
If colored paper is used for the modeling, cut out pieces and label each one with the name of an event or relation in the system. Each one can be a “cloud” floating around. Place the pieces on a large sheet of paper, and draw in connections between them. Take adequate time and move pieces around and re-draw the connections until it feels right. As the parts are re-arranged, think of the simplest geometric structures — the tetrahedron and the octahedron. Since these are the simplest models that have structural integrity, try and fit the map to one of them. If there are more event nodes than 4 (for the tetrahedron) or 6 (for the octahedron), try and combine or limit them. Similarly, if there are less than 4 or 6 event nodes there may be some parts of the system missing. Look back at the system of interest, paying particular attention to the boundaries. There may be some elements outside the boundary that should be put back in, or some inside that should be omitted. Consider modeling the system as interacting geometric structures if there are more event nodes than four or six. An example would be two octahedrons connected together or a tetrahedron with an internal node as Figure 8.
If geometric modeling tools are used, pick an event node and place that name on a vertex (use a piece of masking tape). Then pick the event node that seems most closely connected to the first and connect them. Find a third and connect that in as well. Continue in this fashion, re-arranging the vertices and connections as needed. Once again, try and fit the model to the tetrahedron or octahedron. Note that in the tetrahedral model, each node has connections to three others, while in the octahedral model each node connects to four.
Modeling clay and sticks can be used in the same fashion as the geometric modeling tools. If the modeling clay is colored, different colors can be used to represent different nodes or types of material. As above, try to fit the model to the tetrahedron or octahedron.
With all of the above, it is possible that the model does not fit neat geometric structures! That is fine, but take the time to think of what links or nodes are missing. Using this approach will identify additional connections using the property of synergy present in the system. Recall that if connections or relations are not present between event nodes, the system will collapse down to one that does have relations between all nodes. What is keeping the system from collapsing?
Defining the connections
Once a preliminary structure has been determined, set it aside and think back to the connections between the nodes. What do these connections represent?
In many system models the event nodes can be quantified or counted. If this is true, that quantity must be capable of changing over time. In this type of model the connections carry material — for example if the system were a farm and the node were corn, the connections could be soil, sun, air and water. The quantity of corn will cycle as each of these is varied over time. The geographic limits of the system will drive the quantities of the constituents (air, soil, sun, water), thereby determining the quantity of corn. Figure 8 represents such a model.
Corn as the Center of a Tetrahedral System
Note that in the above example, the four nodes of air, earth, sun and water are modeled as a tetrahedron. This is a generic representation of a biological system, which provides an environment for many plants. The corn seed (which later becomes a plant) is a fifth node, modeled as an internal node to the tetrahedron. Note that the corn node itself represents a complex system, connecting to the environmental tetrahedron at the level above it. This model is hierarchic.
Each connection can therefore support a flow, as well as maintain the structure. What forces or flows hold the nodes together? What forces push them apart? In a mechanical system, each edge would either be in compression (pulling together), or tension (pushing apart). Think about the example of molecules, where the attraction of electrons to protons holds them in a shell, while the repulsion between electrons distributes them across the shell. Differing atoms also form molecules when electrons are shared between shells of different atoms. Refer to Figure 7 and visualize which edges of the model are actions, which are reactions, and which are resultants of the action/reaction.
While the forces that hold a system together may seem static — attraction and repulsion with constant values — the loops that flow between nodes are clearly dynamic. What insights does the static structure give to the dynamic loops that the system supports?
Inputs and outputs
In the corn example, it is easy to see that the end product comes from the constituents (corn comes from soil, sun, air and water). It is also clear where soil comes from — over time, the cycles of sun, corn, air and water have resulted in soil (although this is a larger scale model). But where does the sun come from? Since it is clear that the sun is an input to this system and not dependent on it, the sun is an input node to the model.
Similarly, the corn node may need to be connected as an output. If corn is grown as an intentional practice, typically it is stored to be sold or eaten at a later time.
Note that there are several loops in the corn model. Water flows from the soil into the corn, then into the air, and then back to the plant. Fertilizer (if modeled) flows from the soil into the corn and then back into the soil as the corn plant decays.
Flows and storage
In the corn example above, energy flows into the model from the sun. Energy is stored as corn at that node (embodied energy or emergy). Does energy flow from the soil (nutrients and fertilizer) to the corn plant? “Walk around” the model and define what energy flows in the connections between the parts. Find at least two loops in the model. Arrows can be used to show attraction, repulsion, or flow.
Since this model can be used to represent information or knowledge, the relations can also carry information. Previously, information has been talked about as if it were a flow (like a river, carrying water from one place to another). Of course, that water had to come from somewhere — in fact, it came from rain and snow that came from clouds. So, the visible river is one aspect of a water loop.
In the same way, the visible flow of information is one part of a larger loop. This larger loop creates the environment (source and potential) that then creates the visible flow. In the case of the river, it empties into the sea or lake and thence evaporates, once again driving the loop. So it is with information. The sea of written material — the stuff of culture — pervades human consciousness and provides fertile grounds for the creation of new materials. Each flow is part of a loop that supports it. When the information flows are mapped, the loops and their support in the system can be mapped as well.
Examine the model and define what information flows in the connections between the nodes, and which nodes and connections define a loop where information is used as feedback. Once again, use arrows to show attraction, repulsion, or flow.
Now that there is a preliminary model, think back to the connections between the system being modeled and the outside world. These connections define the boundary in one of the ways discussed above (space, energy, information, or time). What differences are defined by the boundary/connections between the problem/opportunity and the system it is embedded in?
List the spatial boundary (edge of a pond), the informational boundary (how the system looks from the outside as compared to the inside) the energetic boundary (what flows in and out), and the temporal boundary (over what range of time is the model valid, what is the period of the behavior modeled).