Hierarchic Structure

Abstract

We are concerned with structuration, the ways in which complex structures are built from simpler components, and especially with levels of organisation, primarily from a geometrical point of view, with the objective of developing a generalised crystallography extending from the atomic level, beyond regular crystals, towards biological structures which contain fibres, membranes and other sub-structures, some informational. Some levels may be random, some crystalline and some may be regular in different ways Materials characterised as textures must now be specified in more detail. The quasi-crystal affair has provided the stimulus for a fundamental re-estimation of orthodox crystallography.

“If this is the best of all possible worlds, then what the others may be like hardly bears thinking about” — Voltaire, (Candide), (1758).

Newton, having determined “the motions of the planets, the comets, the Moon and the sea”, was unfortunately unable to determine the remaining structure of the world from the same propositions because: “I suspect that they may all depend upon certain forces by which the particles of bodies, by some causes hitherto unknown, are either mutually impelled towards one another, and cohere in regular figures, or are repelled and receed from one another. Those forces being unknown, philosophers have hitherto attempted the search of Nature in vain; but I hope the principles here laid down will afford some light either to this or some truer method of philosophy”. Preface to the “Principia”, (1687). ¹

“It will be found that everything depends on the composition of the forces with which the particles of matter act upon one another; and from these forces, as a matter of fact, all phenomena of Nature take their origin”. R. J. Bosković. “Theoria Philosophiae Naturalis”, Venice, (1763). (sec. 1.5).

“A number of quite independent lines of argument converge toward the assertion that there is often a limit to the complexity of systems”. Richard Levins, ( “Towards a Theoretical Biology”, 3, 73–88, (1970)).

“Physics is simple only when analysed locally”. Misner, Thorpe and Wheeler, ( “Gravitation”)

Quasi-crystals are only part of the revolution in thinking about structure brought about by the computer and computer graphics² [8]. I want to direct your attention to the “Generalised Crystallography” first promulgated by J. D. Bernal [2] about 25 years ago. X-ray crystal structure analysis has been so successful that people have forgotten that most materials are not in fact crystalline. Discussions current in the 1930s can now be picked up again at a new level.

As you will see, I am trying to collect together many rather disparate parts of a general science of spatial structure and, since it is very difficult to organise them into a linear sequence, have begun to use the Hypertext system to handle them as a strongly connected network. This corresponds better, I believe, to the way ideas are organised in the brain. We can think only about a few items at a time, so that In order to organise our thinking about structures, we have to seize on the key feature of hierarchy.