## TETRAHEDRON

A tetrahedron is a polyhedron with 4 triangular faces. If the triangles are equilateral, the structure is then the first of the five regular polyhedra, the others being the cube, octahedron, dodecahedron and icosahedron.

Having a triangulated structure, the tetrahedron has the mechanical stability of triangles.

The tetrahedron is a limit structure, in that it is the minimum polyhedron: it could not have fewer edges. It also encloses the least volume for the most surface area, i.e. it has the greatest area/volume ratio of polyhedra. As a multi-faceted polyhedron approximates to a sphere, it tends to the other limit condition of enclosing the most volume for the least surface area.

The volumes of all systems can be expressed in tetrahedra, because the sum of the face angles of any polyhedron is evenly divisible by 720 degrees, which is the sum of the angles of a tetrahedron.

The ubiquitous tetrahedron is thus the unit pervading all synergetic accounting of space and time, which is to say that events and experiences are accounted for by mapping them onto a tetrahedral co-ordinate system or grid.

A great deal more could be written about this unit, which is what Fuller did in the two volumes of Synergetics.

The relevance of this to art and design may perhaps be seen by pondering Banham's (1960) discussion of the spatial conceptions of the Futurists and Elementarists:

"... the concept of space is considerably more than of a void containing objects, and it seems to come nearer to a three-dimensional grid... in that it appears to contain a regular, measurable, imaginary structure... Space in Elementarist art is indeed, continuous and open, and the work of art is a structure that makes its rectangularity manifest by giving body to its grid-lines and the planes and volumes between them, and this is still true when the grid, as in some van Doesburg paintings of the mid-Twenties, has been skewed out of the vertical". (p.191)

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THE FULLER MAP

© Paul Taylor 2001