Tetrahedral Reasoning

(see other tetra-tools or return to the relational learning framework)
square-50cm-spacer.jpg tetrad3-colours.jpg
Fig. 1 - A 3D model of a tetrahedron

What is it?

  • A tetrahedron is a polyhedron with 4 triangular faces, three of which meet at each vertex.
  • It has 4 vertices and 6 edges. See Wikipedia or Wolfram for more information.

How does it help us?

  • It shows how the number 'four' is helpful to metadesigners.
  • This usually applies to the number of 'levels' we must orchestrate.
  • E.g. mapping society, technology, ecology and semantics within a whole design.
  • It therefore offers a framework of thinking that supports complex (i.e. manifold) innovations.
  • Four-fold innovations can become viral concepts, or memes that can propogate themselves.

What is the thinking behind it?

Things work better with the optimum number of ingredients. Juggling with too many will make us confused. But if we work with too few we may miss opportunities. Innovation in complex situations (e.g. teams, communities) can easily become difficult to comprehend. This may be because change defies the familiar boundaries of language. While thinking in 'fours' may not always show us how things 'really' are, it can inspire us to work beyond the 'thinkable'.

Four levels of usefulness

How does it work?

When trying to grasp a given system in a simple way, choose four interdependent (or co-creative) elements. Visualise, or represent them as four, colour-coded nodes (vertices) on a tetrahedron (e.g. see above). In a working situation it may be reasonable to consider including yourself in this 'world model' (see quadratic ethics).


  • 1 - Use the model to show how each node (e.g. one of the green letters) links directly to the other three letters (i.e. along the tetrahedron's edges).
  • 2 - Identify what each of these (six) numbered edges represent, within the logic and purpose of your system.
  • 3 - Remind yourself that each of the six 'links' are bi-directional - i.e. all four players may 'give' and 'receive'
  • 4 - Incorporate these (twelve) standpoints in you understanding of the whole system.
  • 5 - Remember that any shift in one of the twelve standpoints is likely to have an effect on the other eleven.

How have we applied it?


Moving from 2D to 3D

  • Many designers feel comfortable in playing with a 3D model.
  • The tetrahedron well illustrates the optimal values of a non-hierarchical team.
  • Its (4) nodes can represent interdependent agents, or players.
  • The tetrahedron works in the same way as the square whose corners are linked with diagonals (above right).

square-50cm-spacer.jpg Tetraeder-Animation.gif square-50cm-spacer.jpg euler-tetrad1.jpg square-50cm-spacer.jpg tetrad-balls-numbered1.jpg

  • Another way to visualise it is by thinking of adjacent atoms (e.g. 'collaborators')
  • Just imagine four equal spheres in the same working vicinity
  • When they are close-packed, each will touch all of the other three, simultaneously
  • This is special among spheres - i.e. if you add another sphere it will not touch all
  • This is an optimum, non-hierarchical representation using 3D forms
  • If you imagine lines connecting the centres of the spheres you have a tetrahedron.
  • Each of the vertices in a tetrahedron is a 'neighbour' of all the others

The Tool's Context

Human history has made us so accustomed to social/organizational hierarchies that we tend to assume they are 'natural'. When we speak of 'democracy' (e.g. ancient Greece or Iceland) we usually overlook the enormous scaling-up of national superpowers that are democracies. With the growth of hierarchies comes a reduction in what we call the consciousness of the network. (download article on Network Consciousness. Where some network theory explores what happens in social groups of over 100 (e.g. Dunbar's number) is approximately 150) this tool explores much smaller groups, or teams, of participants.


  • Plato wrote about the tetrahedron (one of his 'Platonic' solids).
  • It has 4 faces, 4 vertices (corners), and 6 edges
  • It is also non-hierarchical (Fairclough, 2005; van Nieuwenhuijze, 2005; Wood, 2005).
  • Buckminster Fuller was inspired by the '+2' in each case (see Amy C. Edmondson's interpretation of Euler+Fuller).
  • He called this constant relative abundance and used it in his idea of Synergetics (1975)
  • I am indebted to Paul Taylor, Otto van Nieuwenhuijze and Ken Fairclough, all of whom continued to develop and promote Buckminster Fuller's work
  • In 2005 ds21 researchers found that, by dividing design teams into 4 different groups we might produce interdependent sub-groups
  • In September 07 we realised that this might be a useful tool for achieving holarchic collaboration
  • The advantages of this approach resemble the computer system's peer-to-peer configuration
  • Heidegger (1964) spoke of a 'fourfold' state of being. 'These are characterised by
    • 1. being "on the earth" - but this also means:
    • 2. being "under the sky". Both of these also implicate:
    • 3. "remaining before the divinities" and:
    • 4. a "belonging to men's being with one another".
  • By a primal oneness the four-earth and sky, divinities and mortals-belong together in one...
    This simple oneness of the four we call the fourfold.' (Heidgger, 1964: p327)
  • Even when mortals turn "inward", taking stock of themselves, they do not leave behind their belonging to the fourfold.
  • When, as we say, we come to our senses and reflect on ourselves, we come back to ourselves from things without ever abandoning our stay among things. Indeed, the loss of rapport with things that occurs in states of depression would be wholly impossible if even such a state were not still what it is as a human state; that is, a staying with things. (Heidegger, 1964: p335)


  • Cowan, N., (2001), The magical number 4 in short-term memory: A reconsideration of mental storage capacity, in Behavioral and Brain Sciences (2001), 24: 87-114 Cambridge University Press Copyright ©2001 Cambridge University Press doi: 10.1017/S0140525X01003922. Published online by Cambridge University Press 30 Oct 2001
  • Fuller, Buckminster, (1949) "Total Thinking", reprinted in "Ideas and Integrities: A Spontaneous Autobiographical Disclosure" (1969), Ed. Robert W., Marks, Englewood Cliffs, NJ: Prentice-Hall.
  • Fuller, R. B., (1975), “Synergetics: Explorations In The Geometry Of Thinking”, in collaboration with E.J. Applewhite. Introduction and contribution by Arthur L. Loeb. Macmillan Publishing Company, Inc., New York.
  • Heidegger, M, (1964), 'Building, Dwelling, Thinking'. Basic Writings. London and Henley, Routledge & Kegan Paul.
  • Klingberg, Torkel (2009). The Overflowing Brain: Information Overload and the Limits of Working Memory. Oxford: Oxford UP. pp. 7,8. ISBN 0195372883
  • Miller, George A. (1956), ‘The Magical Number Seven, Plus or Minus Two: Some Limits on Our Capacity for Processing Information’, originally published in The Psychological Review, 1956, vol. 63, pp. 81-97

Relevant web resources

Relevant downloads

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