A tetrahedron is a geometrical figure with four triangular faces. Around 388 BC Plato included the regular (equal-sided) version as one of 5 unique polygons that we now call "Platonic solids". At Goldsmiths we have used this figure because of its unique properties. It is hard to explain why in a brief account...so please bear with this attempt to clarify all the relevant issues. First of all, we questioned the assumption that essays, dissertations, etc. should be imagined simply as narratives that link arguments in a serial 'string' with a beginning (introduction), middle (development) and an end (conclusion). This is a profoundly rhetorical model that emphasises the persuasive aspect of writing, rather than offering a better way to understand things in a comprehensive way. Many dyslexics seem to get frustrated or confused by this apparently simple model. (By definition), dyslexics are not stupid. Perhaps they resent the linearity because it implies that we should only include those relations that make sense for the narrative argumentation, rather than acknowledging a fuller set of relations. If 'real' situations are highly complex and interdependent, perhaps this calls for a process of 'mapping' that precedes the narrative stage. However, 'mapping' using, say, Tony Buzan's 'Mindmapping method may also be daunting for designer/artist writers, because it encourages the proliferation of elements without offering a relational (e.g. ethical) grammar. This raises questions about the optimum way to represent the relations between lots of interdependent things.image:clusters-assortment.gifFigure 1 - possible configurations for 'mapping' relations before writing them as a narrativeI have developed an elementary grammar of relations that can be exemplified in its most basic form as four elementary 'players' in a whole system:1) Author__ (the 'I' in verbs such as 'create', or 'design')
2) Reader (the 'you' that may benefit from what the writing's argument/proposition/theme has to offer)
3) Proposition (that which is to be written, and put before the reader)
4) Context (the full ecological, ethical background against which the proposition is made possible)Although these four elements have been listed in a series (1 to 4) they do not have a permanent hierarchy. Indeed, they more conveniently fit into the format of a tetrahedron. Whilst it always oversimplifies any given situation, it is a convenient template from which the reader can find his, or her own role. It enables her to 'write' her own predicament in a way that does not privilege any of the four elements.image:tetrahedron-drawing.jpgFigure 2 - a tetrahedronFigure 2 shows that the tetrahedron consists of four faces (triangles) that fit together as a solid. It also has four vertices ('corners') and six edges. I have tended to regard the vertices as 'players', and the edges as 'relations'. The 18th century mathematician Leonhard Euler (1707-83) explored the relations between faces, edges and vertices in all polygons, noting a constant relationship between them. This is usually referred to as 'Euler's Law'. It states that, in any polygon, the number of vertices plus the number of faces is equal to the number of edges plus two.i.e. V + F = E + 2In 1975, Buckminster Fuller described the 'additional 2' as the 'universal relative abundance'. He also claimed that the tetrahedron is so intrinsic to optimum relations in Nature that the human mind is, in effect, tetrahedral. Why is this? The tetrahedron is special in that it has 4 faces, 4 vertices (corners), and 6 edges. It is amusing to speculate how Plato could have used this figure to map out the (6) relations among the (4) characters in his play 'Gorgias'. This is a uniquely auspicious configuration because each of the vertices in a tetrahedron is an immediate 'neighbour' of all the others. When used to model relations, it exemplifies a peer-to-peer system. In other words, it illustrates the optimal values of a non-hierarchical team of 4 players.image:tetrad-balls-numbered.jpgFigure 3 - A tetrahdron implied by the close-packing of four spheresAnother way to visualise the tetrahedron is by thinking of adjacent atoms (e.g. 'players', or '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. We can remember 4 interdependent players quite easily, (i.e. mnemonically convenient) yet it is relationally 'rich'. It is optimal because of the sudden jump in relations between 3 and 4. (see below)2 players share 1 relationship
3 players share 3 relationships
4 players share 6 relationships
5 players share 10 relationship
6 players share 15 relationships__Further reading'''
- Baldwin, J., (1997), "Bucky Works", John Wiley and Sons, ISBN:0471198129
- Barr, S., (1989), "Experiments in Topology", Courier Dover Publications, ISBN:0486259331
- Wood, J., (2004), "The tetrahedron can encourage designers to formalize more responsible strategies", (Guest editors: Margo Blythman & Susan Orr), Journal of Art, Design, & Communication in Higher Education, Volume: 3 | Issue: 3, December 2004, Page(s): 175-192, ISSN: 1474-273X, (Editor, Linda Drew).
- Wikipedia entry for the tetrahedron
- See also AU website
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