We are living in a period of such absurdly blind acceptance of the Cartesian system of co-ordinates, based on the cube structure with three axes in 90 degree co-ordination developed more than four centuries ago. Instead, more recently, Buckminster Fuller, came out with a much more natural way of co-ordinate system based on his invention of the geodesic dome, a structure of triangularly-interconnected elements that has the best ratio of weight to enclosed space of any artificial construction so far developed. More recently, Fuller has received much public acclaim for having predicted, with his geometry, the existence of spherical molecules. The experimental discovery of the Buckminster fullerene, a spherical and extraordinarily stable large molecule of carbon, is only a few years old.
|Departing from convention, this geometry replaces the cube with the regular tetrahedron as its principal unit of volume. The four-sided tetrahedron is the simplest possible enclosure, which is why mathematicians call it a "simplex". Drawn as a cage, or wire frame, it has four windows, four corners and six edges. No space-enclosing network has fewer windows (facets) than four. The cube (or hexahedron), by contrast, has six facets, eight corners, and twelve edges.Given the status of the simplex as "simplest space-enclosing network", the decision to use its regular form as a unit of volume makes some sense. As a consequence of this decision, we obtain whole number volumes for other familiar shapes (including for the cube).|
|Fuller's geometry goes by the name of Synergetics and has been developed by experimentally observing the behaviour of spheres of equal diameter, when packed as close as possible to form regular geometric figures. The basic and most simple stable geometric configuration of synergetic geometry is the tetrahedron, formed by four spheres laying next to each other, in perfect triangular configuration forming four angles of 60 degrees. Other important elements are the octahedron (formed by six closest-packed spheres) and the vector equilibrium, which is the result of twelve spheres nested around a thirteenth, central sphere, in omnidirectional closest-packing, 60 degree co-ordinated configuration. The cube, which is at the basis of our present-day construction methods and of the x-y-z Cartesian co-ordinate system, is not in and by itself a stable configuration. Eight spheres forming a cube are inherently unstable. To gain stability, they must be artificially stabilised by interconnecting them in the way the tetrahedron is connected. In this way, two tetrahedra of four spheres each, joined at their respective centers, form one cube of eight spheres. The cube and dodecahedron are both space-fillers, meaning they fill space without gaps. The tetrahedron and octahedron fill space in complements with twice as many tetrahedra as octahedra.|
|It happens that this geometry, as developed by Fuller, is in perfect accord with how crystals grow in their various forms, and that its application in engineering reveals to us the possibility of very efficient structures in terms of economy of raw materials and strength of the resulting construction.|
Now how could the discoveries of Fuller be utilised to form a co-ordinate system and why should we venture to do such a task, seeing that the Cartesian x-y-z co-ordinates have done perfect (or almost perfect) service for such a long time?
For one, Cartesian co-ordinates may be a convenient mathematical construct, but they do not accord with nature's ways any more than modern chemistry will ever be able to duplicate the conditions of living organisms.