Growth and Form

  • Alan L. Mackay
Conference paper


We examine general characteristics in the way things are built up and change. We follow Democritos is seeing the world of atoms as organising itself and not, as Plato and others wished us to believe, that the world was ordered by gods at a higher level. The most important ideas’ are atomicity and hierarchy but the key concept is that of description, where two structures may be related to each other as description and described, or descriptor and referent as in the genetic code. To restore a structure from its description further knowledge must be supplied. The joint system may evolve by this dialectical interaction to and fro from phenotype to genotype in the process of natural selection. Complex structures may result from local interactions as in cellular automata or from global minimising (or maximising) principles or by symbiosis. We show examples of a particular study of surfaces generated by simple descriptions, form being usually the interface between inside and outside. This duality can be carried over into language and art. Science, art, technology and society feed each other. The purpose of this meeting is for each to put various ingredients into the cooking pot, to stir them and to extract the useful new products which may emerge, repeating collectively what may happen in an individual mind.


Cellular Automaton Representational Space Periodic Minimal Surface Imaginary Dialogue Siamese Twin 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    Bernai, J. D. and Carlisle, C. H., “The Range of Generalized Crystallography”, Kristallografiya, 13, No. 5, 927–951, (Sept.-Oct. 1968). Soviet Physics — Cryst., 13, No.5, 811-831, (Mar.-Apr., 1969).Google Scholar
  2. [2]
    Brakke, K. A., “The surface evolver”, Experimental mathematics, 1, (2), 141–165, (1992).Google Scholar
  3. [3]
    Caspar, D.L.D. and Klug, A., “Physical Principles in the Construction of Regular Viruses”, Cold Spring Harbour Symp. Quant. Biol, 37, 1–24, (1962).CrossRefGoogle Scholar
  4. [4]
    Chattopadhyaya, D., Lokayata, a study in ancient Indian materialism, People’s Publishing House, Delhi, (1959).Google Scholar
  5. [5]
    Decety, J., Perani, D., Jeannerod, M., Bettinardi, V., Tadary, B., Woods, R., Mazziotta, J.C. and Fazio, F., “Mapping motor representations with positron emission tomography”, Nature, 371, 600–602, (13 Oct. 1994).Google Scholar
  6. [6]
    Diderot, D., Entretien entre D’Alembert et Diderot, 1769. Flammarion, Paris, (1965).Google Scholar
  7. [7]
    Diogenes Laertius, Lives of the Philosophers. IX, 44–45.Google Scholar
  8. [8]
    Ford, Brian J., Images of Science, British Library, (1992).Google Scholar
  9. [9]
    Haeckel, E. H., Generelle Morphologie der Organismen, I, 139, Georg Reimer, Berlin, (1866).Google Scholar
  10. [10]
    Haidane, J. B. S., [Question in a mock exam paper looking forward 25 years from 1931]. Brighter Biochemistry, Cambridge, (1931). Reprinted TIBS, (March 1981), p.91.Google Scholar
  11. [11]
    Kepler, J., The hexagonal snowflake, (1611), Oxford University Press, (1966).Google Scholar
  12. [12]
    [Titus Carus] Lucretius, The Nature of the Universe, trans. R. Latham, Penguin, Harmondsworth, (1951).Google Scholar
  13. [13]
    Mackay, A. L., “Generalised Crystallography”, Izv. Jugoslav. Centra za Krist., 10, 15–36, (1975).Google Scholar
  14. [14]
    Mackay, A. L., “De nive quinquangula — On the pentagonal snowflake”, Kristallografiya, 26, 910-919 (1981). Soviet Physics — Crystallography, 26, (5), 517–522, (1981).Google Scholar
  15. [15]
    Mackay, A. L., “Generalised crystallography”, Comp, and Maths, with Appl. 12B, Nos. 1/2, 21–37, (1986).Google Scholar
  16. [16]
    Mackay, A. L., “Post-classical Crystallography”, Colloques de Physique C3, Les Editions de Physique, 47, No.7, C3-153-C3-163, (July, 1986).Google Scholar
  17. [17]
    Mackay, A.L., “Lucretius: Atoms and Opinions”, Symmetry, 1, No.l, 3-17, (1990) and Interdisciplinary Scientific Reviews, 16, (2), 125–139, (June 1991).Google Scholar
  18. [18]
    Margulis, L. and Fester, R., (eds.), Evolution and Speciation: Symbiosis as a Source of Evolutionary Innovation, M.I.T. Press, (1991).Google Scholar
  19. [19]
    Needham, J., Order and Life, Cambridge University Press, (1936), (MIT Press, 1967).Google Scholar
  20. [20]
    Pinker, S., The Language Instinct, Allen Lane, London, (1994).Google Scholar
  21. [21]
    Saunders, P. T., (ed.), Collected Works of A. M. Turing, Morphogenesis, North-Holland, Amsterdam, (1992).Google Scholar
  22. [22]
    Simon, H. A., “The Architecture of Complexity” in The Science of the Artificial, MIT Press, (1969).Google Scholar
  23. [23]
    Thomson, W., “On the division of space with minimum partitional area”. Phil. Mag., 4, 503–574, (1877).Google Scholar
  24. [24]
    Thompson, D’A. W., On Growth and Form, Cambridge University Press (1916).Google Scholar
  25. [25]
    Whyte, L. L. (ed.), Aspects of Form, Indiana Univ. Press, Bloomington, (1971).Google Scholar
  26. [26]
    Whyte, L. L., Wilson, A. G. and Wilson, D., (eds.) Hierarchical structures, American Elsevier, New York, (1969).Google Scholar
  27. [27]
    Configurations and Interactions of Macromolecules and Liquid Crystals, Discuss, of the Faraday Society, No. 25, (1958).Google Scholar
  28. [28]
    Wolfram, S., Mathematica, Addison-Wesley, (1988).Google Scholar

Copyright information

© Springer-Verlag Tokyo 1996

Authors and Affiliations

  • Alan L. Mackay
    • 1
  1. 1.Department of CrystallographyBirkbeck College, (University of London)LondonUK

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