Sage Journals: Discover world-class research

Abstract

The second edition of D'Arcy Thompson's On Growth and Form was written only shortly before the advent of computers made it possible to develop more sophisticated mathematical models of processes of growth, morphogenesis and pattern formation in nature. It also predates the blossoming of several branches of science with the conceptual tools to investigate complex phenomena such as self-organization, nonlinear dynamics and chaos, fractal geometry and self-organization. As a result, Thompson's aspirations sometimes fall short of his means — and occasionally he sets out on the wrong path. This image essay explores some of the instances in which Thompson surpassed his limitations or, alternatively, was constrained by them. In either circumstance, it illustrates how his themes remain areas of significant scientific activity today.

Keywords

pattern formation D'Arcy Thompson fractal geometry morphogenesis Alan Turing

Get full access to this article

View all access options for this article.

References

D'A. W. Thompson. 1992. On Growth and Form, 2nd edn. Cambridge University Press, 1942. Reprinted by Dover, New York.

A. Turing. 1952. The chemical basis of morphogenesis. Phil. Trans. R. Soc. B 237: 37–72.

J. D. Murray. 1990. Mathematical Biology. Berlin: Springer.

H. Meinhardt. 1982. Models of Biological Pattern Formation. London: Academic Press.

H. Meinhardt, P. Prusinkiewicz, and D. R. Fowler. 2003. The Algorithmic Beauty of Sea Shells. Berlin: Springer.

H. Bénard. 1900. Les tourbillons cellulaires dans une nappe liquide. Rev. génér. Des Sci. 12: 1261–1271.

Lord Rayleigh. 1916. On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Phil. Mag. 32: 529–546.

W. Hofmeister. 1868. Allgemeine Morphologie der Gewächse. In Handbuch der Physiologischen Botanik, Band 1, Abteilung 2, 405–664. Engelmann, Leipzig.

A. H. Church. 1904. On the Relation of Phyllotaxis to Mechanical Laws. London: Williams & Norgate.

10.

H. Vogel. 1979. A better way to construct the sunflower head. Math. Biosci. 44: 179–189.

11.

S. Douady and Y. Couder. 1992. Phyllotaxis as a physical self-organized growth process. Phys. Rev. Lett. 68: 2098–2101.

12.

S. Lau, I. de Smet, M. Kolb, H. Meinhardt, and G. Jürgens. 2011. Auxin triggers a genetic switch. Nature Cell Biol. 13; 611–615.

13.

R. P. Bhalerao and M. J. Bennett. 2003. The case for morphogens in plants. Nature Cell Biol. 5: 939–943.

14.

A. M. Turing. 1992. Morphogen theory of phyllotaxis. In Morphogenesis: Collected Works of A. M. Turing, Vol. 3, ed. P. T. Saunders. Amsterdam: North-Holland.

15.

J. Banks, quoted in W. Guthrie and J. Ferguson (eds). 1801. A New Geographical, Historical, and Commercial Grammar; and Present State of the Several Kingdoms of the World, 155. London: G. G. & J. Robinson.

16.

E. A. Jagla and A. G. Rojo. 2002. Sequential fragmentation: the origin of columnar quasihexagonal patterns. Phys. Rev. E 65: 026203.

17.

L. Goehring, S. W. Morris, and Z. Lin. 2006. An experimental investigation of the scaling of columnar joints. Phys. Rev. E 74: 036115.

18.

A. M. Worthington. 1908. A Study of Splashes. London: Longmans, Green & Co.

19.

S. T. Thoroddsen, T. G. Etoh, and K. Takehara. 2008. High-speed imaging of drops and bubbles. Annu. Rev. Fluid Mech. 40: 257–285.

20.

M.-J. Thoraval et al. 2012. Van Kármán vortex street within an impacting drop. Phys. Rev. Lett. 108: 264506.

21.

A. T. Winfree. 1987. When Time Breaks Down. Princeton: Princeton University Press.

22.

S. Sadek and R. Sultan. 2011. Liesegang patterns in nature: a diverse scenery across the sciences. In Precipitation Patterns in Reaction-Diffusion Systems, ed. I. Lagzi. Trivandrum: Research Signpost Publications.

23.

W. A. Bentley and W. J. Humphreys 1931. Snow Crystals. New York: McGraw-Hill.

24.

W. W. Mullins and R. F. Sekerka. 1964. Stability of a planar interface during solidification of a dilute binary alloy. I. Appl. Phys. 35: 444–452.

25.

E. Ben-Jacob, N. Goldenfeld, J. S. Langer and G. Schön. 1983. Dynamics of interfacial pattern formation. Phys. Rev. Lett. 51: 1930–1932.

26.

P. Ball. 2004. The joy of six: the growth and form of snow crystals. Interdisc. Sci. Rev. 29: 353–365.

Hits,Misses and Close Calls: An Image Essay on Pattern Formation in On Growth and Form

Abstract

Keywords

Get full access to this article

References