Geometrical Systems Mapping

Curt McNamara, P.E.

Illustrated by

Gray Moon Gronberg


The geometric structure of systems has been established (Fuller, 1975). This paper will apply the methodology to real world systems. First, a simple control system will be mapped to the tetrahedron. Second, the vectorial nature of reality will be examined to determine the nature of connections in geometric models. Finally, basic systemic properties will be illustrated in this type of modeling. Examples of the properties to be mapped include: relation, node, boundary, structure, feedback, hierarchy, cycles and loops.

Keywords: Tetrahedron, modeling, system, minimal, geometric


    In Synergetics (Fuller, 1975) it was established that tetrahedrons are the minimal system representation. They are minimal since at least four points are required to describe a system in three-dimensional space, the reality that we all inhabit. Systems described in two-space — e.g. lines, planes, or fiction (Abbot, 1983) are theoretical constructs. All life (and therefore all systems), exist in three-dimensional space.

A tetrahedral system has six vector edges and four nodes. These components may be modeled as two events in relation to one another (Fig. 1). Each event consists of three parts: an action, a reaction, and a resultant. Every event in Universe acts, and is reacted against. An energy difference exists between the action and reaction, and this difference is the resultant. This energetic set can be viewed as an open triangle, balancing three components.

The vector edges represent a force in a direction (a magnitude plus an orientation). Vectors are used to model energetic relations. All real world activity is energetic – the only zero energy environment is found at the temperature of absolute zero. Forces have several interesting characteristics. Two forces can’t pass through the same point at the same time, so they can’t be coincident. Forces are composed of either particles or waves, and may represent flows, fields, gradients, coupling, or gradients. It is not possible for two forces to cancel each other precisely, due to the finite speed of energy in Universe (Fuller, 1997). Energy and force can be either physical or metaphysical (dreams and schemes can be systemic).

The difference between interacting forces is the resultant, and is termed precession. Observers often focus on an action or reaction. Surprisingly, it is the excess energy from the action/reaction (precession) that couples events and systems together. This energy can be shared at nodes or along relations (McNamara, 2001).

No event exists in isolation, but rather each event constantly couples to other events. When this happens, the vectorial components of the first interconnect with the vectorial components of the second. Once two events interconnect, a system is formed. For a system to persist, energy must be balanced at each corner (or node) of the system. The motion of an arm interacts with gravity and atmosphere, forming a system. Universe is an enormous set of related events. “Universe is the aggregate of all humanity’s consciously apprehended and communicated nonsimultaneous and only partially overlapping events” (Fuller, 1975).

Furnace System

Given that systems are at minimum two coupled events, what is an example of a minimal system? First we will examine the simple feedback system found in a furnace with thermostat. In McNamara (2001) this was described as a pair of events. A thermostat operates by comparing setpoint (action = desired temperature) with temperature (reaction = environment), and outputting a command (difference/resultant = output to the furnace). At the other end of the command link is the furnace. Its’ action is valve, its’ reaction fuel, and its’ resultant is flame. See Fig. 1. This model uses the two events that represent the fundamental interactions when the furnace is reduced to two parts.


System as Interaction of Two Events | Furnace as Two Interacting Events

Figure 1 (after Fuller)

It seems intuitively obvious that the paired events are thermostat and furnace, consisting of set-point/temperature/command and valve/fuel/flame. However, fitting the two events together in this fashion reveals that the nodes do not all balance correctly. Recall that at each node, three vectors come together. If the vectors don’t balance exactly, energy is exported or imported from Universe (at a higher, lower or equal level of structure). Examining the pairing of thermostat and furnace shows that energy is not balanced at every node. To examine the model from the perspective of nodal balance, a list of all twenty possible nodal interconnections can be made. This set is shown as Table 1.

Table 1. Possible nodal connections

Set-point/command/valve                                  Command/temperature/valve

Set-point/command/temperature                       Command/temperature/fuel

Set-point/command/fuel                                    Command/temperature/flame

Set-point/command/flame                                 Command/valve/flame

Set-point/temperature/valve                              Command/valve/fuel

Set-point/temperature/fuel                                 Command/flame/fuel

Set-point/temperature/flame                              Temperature/valve/flame

Set-point/valve/fuel                                           Temperature/fuel/flame

Set-point/valve/flame                                        Temperature/valve/fuel

Set-point/fuel/flame                                           Valve/fuel/flame



Examining this set, the following nodal combinations reveal themselves as potentially significant:

  • Thermostat – set-point/command/temperature
  • Heat – temperature/fuel/flame
  • Fire – valve/fuel/flame
  • Flow – command/valve/fuel

Note that none of these nodal sets showed up in the original events! The nodes termed fire (valve/fuel/flame) and thermostat (set-point/command/temperature) were originally modeled as events. It is clear that energy balancing can be done as events, as nodes, or as the combination of the two. For a system to exist, both nodal connections and events must be balanced. Note also that the node of fire/heat was not present in the event model.

If this set of connections is modeled, it will be obvious that a system cannot be formed. To make a tetrahedron, each vector must be a component of two nodes (only). Fuel exists as a component of three nodes, and as a vectorial edge it can only be a part of two nodes. Set-point is a component of only one node, and it needs to be a part of two.

How can these conflicts be resolved? Since set-point is a component of one node only, it must be a part of one node only. Tetrahedrons are not perfect containers of energy, rather the corners of the tetrahedron “leak” (Fuller, 1975). It is through these leaks that systems precessionally couple together. Therefore set-point can enter the system at a node (as an external input), and is thus outside the boundary of the system. Information (or energy) from set-point enters the system, but the system does not feed back energy or information to set-point. This defines a boundary of the system.

Fuel is more puzzling. With more careful analysis, fuel can be perceived as an input to the system as well (the storage tank or wood pile is outside the furnace). The furnace can’t operate without it, but it is not part of the coupled events of thermostat and fire. What does exist within the furnace is the flow of fuel to the node where fire exists. Flow of fuel is significant since it maintains the fire. It is also the logical output of command and valve, modulating energy flow into the system. Interestingly, if fuel enters at a node then four terms (three internal to the system plus fuel) lead to the possibility that two different equations can balance at that node.

What will replace these vector edges once fuel and set-point are removed? Since this modeling approach is based on energy, re-examine the events’ energetic relations. Recall that the inner workings of a thermostat contain a mechanical switch, which closes a circuit from a power supply, sending current to a valve. Where does the current come from? In a furnace system there is a power supply, presenting voltage at the thermostatic contacts. When the contacts close, a loop is created which allows current to flow to the valve. We can either revise the system to include the power supply, or bring it into the system as an external input. If the power supply voltage exists within the model then a loop is formed between voltage, current, and valve when the thermostat contacts close. The power supply is thus modeled as a parallel system, with one element of it (the voltage vector) coupling across a vector or relation (McNamara, 2001). The power supply might be modeled hierarchically (it is at a lower level of organization than the furnace) or heterarchically (it is at the same level of organization as the furnace).

The thermostatic node now includes voltage present on the contacts (via power supply), and current as command output when the contacts close. In addition, set-point and temperature remain as drivers of the contact closure. The thermostat operates as an informational comparator, switching energy (potential to kinetic) to control flow. Is this a system primitive?

Energetically, the contact on the thermostat converts potential energy (voltage) into kinetic energy (command = current), creating a loop for energy to travel. The set-point controls this conversion, and perhaps could be envisioned as setting the angle between voltage and current. In other words, does it take a little or a lot of energy to close the contacts? Obviously temperature can vary the angle as well.

Working through this process has revealed that two of the variables – set-point and fuel – come from outside the system. It may seem as if temperature is the same! However, temperature needs to be part of the modeled system for two reasons. First, it is part of the instantaneous working of the thermostatic balance. Second, temperature is the controlled variable, and is altered by the combustion node where heat is created from the burning fuel. The temperature vector is not contained within the system, but rather the furnace system lives as an element on this larger vector. Heat is added at one point on the vector, and temperature is sampled at another. This is another example of relational coupling.

This mapping represents energy (either kinetic or potential) at every edge. Fuel as potential energy is an input, along with set-point as an informational input which alters the operating point. The system nodes are now:

  • Temp/fuel-flow/flame (heat is produced from burning flow)
  • Temp/voltage/command (temperature balances with command and set-point)
  • Voltage/flame/field (furnace ignition)
  • Flow/command/field (command to magnetic field in valve controls flow)

Note that the relation valve has been replaced by magnetic field. This is a more descriptive term for the energetic operation of the valve. To complete the loop, current is transformed into field, opening the valve. Voltage>current>field then defines a loop for energy to travel back to the power supply.

This mapping is shown as Figure 2. Temperature is a “large” vector that is both influenced by, and influences the system. It couples into the thermostatic switching node – where voltage from the power supply links (via flow of current) to the valve (or does not). Set-point enters the system here as well, by determining the angle of separateness of the voltage and current (the closeness of the thermostatic contact). The above set of nodes is balanced!


Figure 2

Interestingly, the node of voltage/flame/field must represent two circuits. One circuit is the valve loop. However, this node is also the origin of the flame vector to the combustion node. What is the nature of the flame vector itself? In modern furnaces, the flame comes from an igniter, which generates an arc across contacts. This is powered from the voltage source, and could be modeled as two vectors, one going to the combustion node and one returning from it. This loop is independent of the loop of current to the valve. The igniter is another location where hierarchical structure may be required in the model.

Note that in older furnaces, a pilot light was used. This was not based on power supply, but existed as a parallel path alongside the fuel flow vector (a small amount of fuel was constantly burned to create a small flame at the combustion node).

In what areas might the model need additional detail? If the actual operation and energy basis of the valve is analyzed, another perspective emerges. The valve receives a command as current. The current is transformed to a magnetic field within the valve, which acts against the force holding the valve closed. This force typically comes from a spring (force represented by spring constant k), to keep the valve closed. So the valve operation is the balancing of magnetic field with spring constant. When the magnetic force exceeds the spring force, the valve opens, allowing fuel to flow into the combustion node. Since the spring force is a constant entering the system, it can be shown as an additional input at the valve node.


The loop of electricity flows from voltage source, to contacts, to command, and then to magnetic field, where it completes a circuit back to the source. To make a similar pathway for fuel requires that fuel entering the system be metered by valve, be contacted with fire, and be transformed into heat. This heat then couples to the temperature of the larger system. To close the loop back to fuel requires that we model the other vectorial components of the combustion node. Combustion is disassociation of organic compounds via oxidation. Combustion requires oxygen as an input, and its’ by-products are soot, water vapor, and CO2 (among other gases). These gasses and vapors return to the earth system, where they will eventually be recycled as fuel.


What is the nature of the energetic interactions at the nodes? From elementary physics it is clear that energy can neither be created nor destroyed. It can be transformed, stored, or released. Energy is coupled between at least three vectors at each node of a system. These vectors cannot pass through the same point at the same time, nor can they cancel each other out. Fuller termed these interactions interferences:

“There are six fundamentally unique patterns of the resultants of interferences. The first is a tangential avoidance, like knitting needles slipping by one another. The second is modulated noninterference, as in frequency modulation. The third is reflection, which results from a relatively direct impact and a rebound at an acute angle. The fourth, which is refraction, results from a glancing impact and an obtuse angle of deflection. The fifth is a smash-up, which results in several parts of one or the other interfering bodies going away from one another in a plurality of angular directions (as in an explosion). The sixth is a going-the-same-way, “critical-proximity,” attraction link-up such as that established between the coordinated orbiting of Earth and Moon around the Sun.” (Fuller 517.101, 1975)

Given the above, what types of interactions are defined by this minimal system? At the thermostatic node the switching might be modeled as critical proximity when it closes, and reflection when it opens. Note that critical proximity is similar to the orbit of satellites around a planet. “Though lines (subvisibly spiraling and quantitatively pulsative) cannot go through the same point at the same time, they can sometimes get nearer or farther from one another. They can get into what we call “critical proximity.” Critical proximity is the distance between interattracted masses__when one body starts or stops “falling into” the other and instead goes into orbit around its greater neighbor, i.e., where it stops yielding at 180 degrees and starts yielding to the other at 90 degrees.” (Fuller 518.101, 1975) The loop of voltage/current/magnetic field is an orbit when the thermostat is closed. When the thermostat opens, the loop is broken and the vectors decoupled. The voltage vector is then reflected back to the power supply.

The valve node is modulated interference, as varying command modulates the flow of fuel to the combustion node. When command is not present the spring pulls the valve closed, switching the node to reflection. With high temperatures or low set-point, the valve is closed. With low temperatures or high set-point the valve is continuously open. As the set-point nears the temperature there are a series of openings and closures as the connection is modulated by temperature. Valving is a fundamental systemic operation.”The frequency and magnitude of event occurrences of any system are comprehensively and discretely controllable by valving, that is, by angle and frequency modulation. Angle and frequency modulation exclusively define all experiences, which events altogether constitute Universe.” (Fuller 208.00, 1975)

The node where fuel and flame produce heat is a smash-up. Fire is disassociation of fuel into the vector components of heat, light. and waste.

As noted previously, the power supply node in Fig. 2 completes two independent loops. One is the circuit through the valve, the other the ignition current to the combustion node. These are critical proximity when combustion is happening, and tangential avoidance when it is not (there is no voltage or current present to reflect when the loops are open).

How does this nodal balancing relate to + and – events? In the Fig. 2 model, one event now consists of current/fuel flow/flame. The other is field/voltage/temperature. For the + event, voltage is the action, field the reaction, and temperature the resultant. The – event consists of current as the action, fuel flow as the reaction, and flame as the resultant. Note that this model was not reached directly from event modeling, but rather a three step process of event followed by node followed by energetic balancing was performed. Obviously these three types of modeling interact to produce a stable model. Event balancing and nodal balancing interact to form a stable system.

Modeling Lessons

What conclusions can be drawn about geometric systems mapping? First, that the obvious paired events may not be the fundamental interactions. Second, that elements assumed to be inside the system are sometimes outside the system. Third, that the nature of the interactions at the nodes is a)part of a small set of fundamentals; and b)may change from one type to another as the system operates. Fourth, that even “simple” systems may have substantial complexity. This complexity may come from hierarchy (i.e. a system component is actually a manifestation of an underlying system), or it may come from a coupled system (heterarchy, i.e. a system at the same level of complexity). Fifth, systems may not map neatly to the tetrahedron, but require additional interconnections. In other words, the tetrahedron is the minimum structure required for closure in three-dimensional space.

Can this control system be mapped onto the woodsman and his fire? In this case the power supply is the woodsman. The thermostat is internal to his mind, or he may receive input from other individuals. Current is getting up to fetch a log (action has been decided), and valving consists of throwing that new log onto the fire. The flame loop is started with a match, then maintained as he arranges logs near existing flame. A coupled system is the one where he gathers wood to maintain future fires.

A note on embodied energy: the wood is embodied sunlight, the woodpile is that sunlight arranged for easy use, and the fire releases the sunlight as heat, light, and waster. Entropy decreases as the sunlight is captured and stored as wood, then as woodpile. Entropy then increases as wood is burned and the byproducts returned to the earth (Odum, 1983).

System Properties

What does this example tell us about structure, boundary, and hierarchy? It was quickly discovered that the system does not exist in isolation. It was difficult to tease out a fundamental set of connections, while still retaining distinct boundaries between the system under study and the rest of Universe. For example, does fuel exist within the system, or does it enter at a node? There are also a multitude of interactions to model at the fire node.

Boundary has a two-part dynamic. First, finding the internal connections that define the system, while still retaining external connections that are essential for structure and definition. Second, revealing relations to the external world that weren’t obvious from initial models. In this exercise system inputs were either consumed in the system (as the fuel), or they controlled system properties without being subject to feedback from the system (set-point). Is boundary a one-way flow? As noted elsewhere, fuel loops back through the system when waste products are modeled (although this time period is long compared to that of furnace operation).

The power supply may be seen as hierarchy – i.e. complexity at a higher/lower level that couples into the furnace. It needs to be present to supply voltage (and current) for the valving operation, yet it may be desirable that the complexity of it remains outside the system model under consideration.

The power supply may also be seen as a system at the same level (heterarchy) as the furnace. In this case the voltage vector or relation is part of the power supply, and both systems are joined together along this edge. Voltage appears at the thermostat node and returns from the magnetic field vector.

In the bonfire model, the biological organization and energy of the human actions take the place of the power supply. When the decision is made, this energy is used to put wood on the fire. This complexity is orders of magnitude more than the power supply.

Loops are present in this model as voltage present at the contacts is transformed to current which then creates magnetic field within the valve, from whence the electrons return to the power supply. The system cycles (oscillates slowly) as the thermostat controls temperature. Relations are the vector edges. Structure is the set of vector edges and nodes that define the geometric model.

Note that this system uses precession at every node. Fuel is metered into the system precessionally – the valve balances command (magnetic field) vs. spring constant. The excess is proportional to the valve opening, which allows fuel to flow into the system at a “right angle” to the valve. At the combustion node heat is generated precessionally as a result of rapid oxidation of fuel (fire). “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.” (McNamara, 2001)

Modeling Insights

The structure of the model varied as the model grew more detailed. The simplest model seemed obvious structurally, yet it did not hang together at the nodes. The final model has one obvious loop (voltage, current, field), two precessional effects (fuel flow and heat from fire), and two sub-systems (igniter and power supply).

The implications for system design seem to be of four types :

  • The obvious connections are not always the real connections.
  • Complexity is often hidden by focus on the obvious connections.
  • Systems that valve energy may be a fundamental archetype.
  • Systems that transform energy have hidden effects.

This system model may be an archetype of control of flow of potential energy. What other common systems display this archetype? Possible examples might be material processing, beaver dams, river bends, wind-borne seeds that germinate plants which modify the edges of mountains, stock market trading controls, and interest rates.

Note also that even though the control of flow defines the fundamental system structure, the point of the system is to acquire heat, precessionally from the process of fire. This system does not exist simply to burn or to flow, but to heat as a result of the flowing burn.



Abbot, Edwin A.. (1983) Flatland HarperPerennial: New York, USA. A novel describing life in two dimensions.

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.

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. 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:

Gronberg, Gray. This site offers geometric modeling instructions and materials. Very clear, and appropriate for all ages.

Ikoso Kits. Offers simple geometric modeling kits. 85210 Willamette St., Eugene. OR, 97405 USA. 1-800-231-0104

Krausse and Lichtenstein. (1999). Your Private Sky, 524 pp., (2000). Your Private Sky: Discourse. Germany: Lars Muller. An excellent compilation of materials on Bucky’s life and work. Consists of a biology/chronology of Fuller, numerous examples of his designed artifacts, excerpts of his writing, and commentary.

McNamara, Curt. (2001) Systems Coupling and Precession, Proceedings, ISSS 2001 Conference. 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:

Urner, Kirby. Synergetics on the Web. A website devoted to Fuller’s work, particularly Synergetics. Site has numerous computer-based representations of geometric systems.

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