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Growth and Form and Infinity

Who knows, maybe all our 'abstract forms' are 'forms in nature'.

Kandinsky (1937)

The perception of a visual shape often involves an implicit conjecture about the process that generated it. A plausible theory of perception may thus crucially involve an account of shape-generation and shape-transformation processes. (Michael Leyton articulates this perspective in his book "Symmetry, Causality, Mind"; he discusses Euclidean transformations (translation, rotation), as well abstractions of physical processes (dent, stretch, squeeze). Cf. also: Benjamin Kimia: Shapes as "shock-graphs".)

Applying this approach to the description of organic forms, we try characterize them in terms of the growth processes that produced them. Art generation and Gestalt perception thus meet biological morphology.

"I have called this book a study of Growth and Form, because in the most familiar illustrations of organic form (. . .) these two factors are inseparably associated, and because we are here justified in thinking of form as the direct resultant and consequence of growth (. . .) whose varying rate in one direction or another has produced (. . .) the final configuration of the whole material structure."

D'Arcy Wentworth Thompson (1917)


Ernst Haeckel: Monograph of the Challenger Radiolaria. (1887)

Ernst Haeckel: Art Forms in Nature. (1904)

Wassily Kandinsky: "Assimilation of Art" (1937). Complete Writings (Vol. 2, p. 803).

Susanne Langer: Mind: An Essay on Human Feeling, Vol. I. Baltimore: Johns Hopkins University Press, 1967.

D'Arcy Wentworth Thompson(1917): On Growth and Form, 1917. Second edition: Cambridge, UK: Cambridge University Press, 1942 (Vol. 1, p. 57).

Lindenmayer-systems: Growth processes in line patterns

"(. . .) not only the leaves repeat each other, but the leaves repeat the flowers, and the very stems and branches are like un-unfolded leaves. (. . .) To the pattern of the flower there corresponds a further pattern developed in the placing and grouping of the flowers along the branches, and the branches themselves divide and stand out in balanced proportions, under the controlling vital impulse (. . .) Musical expression follows the same law."

Basil de Selincourt: "Music and Duration." Music and Letters, 1 (1920), p. 288.

A mathematical method to describe plant growth processes was developed by Aristid Lindenmayer in the period 1968-1990. The most basic version of Lindenmayer's "L-systems" resembles the Context-Free Grammars that are often used in Computational Linguistics to characterize the constituent structure of natural language sentences. The formalism works differently in two respects: (1) The difference between terminal symbols and non-terminal symbols is abolished: all symbols can be rewritten, and all symbols can occur in output-expressions. (2) Every rewrite step rewrites all symbols simultaneously.

Another important difference with language-grammars is that the strings that are generated have a visual interpretation: they are strings of a "turtle-graphics" coding language. In "Turtle Graphics", a line consisting of straight segments is described from the perspective of a turtle who draws it: the length of segment 1, the angle to turn after segment 1, the length of segment 2, the angle to turn after segment 2, and so on. Combined with the commands "pen up" and "pen down", this allows the description of all possible drawings consisting of straght line segments.

References Lindenmayer-Systems

Przemyslaw Prusinkiewicz & Aristid Lindenmayer: The Algorithmic Beauty of Plants. New York: Springer-Verlag, 1990.

Gary W. Flake: "L-Systems and Fractal Growth". In: The Computational Beauty of Nature. Cambridge, MA.: MIT Press, 1998. [Online applet: choose "L-Systems" in the menu on the Java-Applets page. ]

References Turtle Graphics

Harold Abelson and Andrea diSessa: Turtle Geometry. The Computer as a Medium for Exploring Mathematics. Cambridge, Mass.: MIT Press, 1980.

Seymour Papert: Mindstorms. Children, Computers, and Powerful Ideas. New York: Basic Books, 1980.


Przemyslaw Prusinkiewicz: Algorithmic Botany.

Plant-modelling-software for games, movies and architects:

Bit-101 Action-Script Laboratory: 2001 sep 05–08; 2001 nov 09; 2001 dec 10; 2002 oct 30.

Mondriaan's trees: David Sylvester: "Mondrian in London," Studio International, December 1966.

Assigments 2001 (results):

   Puite en Roelofs


Infinite recursion.

If we imagine that the rewrite process is continued forever, Lindenmayer-systems may represent shapes which are fractal and self-similar, such as Koch-islands and spacefilling curves.

The Droste-picture on a computer screen with a zoom option: the image really has an unlimited degree of detail. Cf. also: Driessens & Verstappen: Ima Traveller (discussed on our page about 2D cellular automata). The visual experience of the Droste-demo and "Ima Traveller" is: to fall. This correlates directly with the dizzying nature of any vivid thought which involves infinite recursion.

When infinite recursion is mapped onto infinite iteration it becomes a visual structure that can be contemplated: a circle. But to think infinite recursion is to fall into an epileptic mental process which is not contained anymore by the rationality that launched it. This occurs if one attempts to think seriously about the liar-paradox. It also occurs if one attempts to seriously think any all-embracing philosophical thought which concerns the whole world with oneself in it. (This is what Wittgenstein called "das Mystische": "die Erfahrung der Welt als begrenztes Ganzes". It correlates directly with Kant's notion of "das mathematische Erhabene".)

George Brecht: Vicious Circles and Paradoxes.


Applet voor Koch-krommen
Bit-101 Laboratory: 2002 feb 05/11

Fractales sur le Web

Fractint Fractal Geometry PC Freeware.

The Radical Art page on paradox.

The Radical Art page on tautology.


Beyond the countably infinite

Cantor proved that the real numbers constitute an uncountable set. (I.e., they can not be conceived as the outcome of an infinitely continued recursive process.)
Turing used the same proof technique (the "diagonal argument") to establish that the "halting problem" for Turing machines cannot be solved by a Turing machine.
Gödel used a similar proof structure to show that a consistent formal system that allows the formalization of arithmetic allows the formulation of propositions which are true but unprovable within the system.


Gödel's Onvolledigheidsstelling.

Gregory Chaitin: The Unknowable. (On Turing, Gödel and Randomness.)


In the "Monadologie" Leibniz describes organic systems as "machines" which (unlike "machines" in the narrower sense of that word) consist of infinitely small parts. Find his exact formulation of this idea, and discuss it in relation with the present-day notion of fractal structures. Consider the relation between the fractal and the biological which is suggested by the literature on Lindenmayer-systems.


Gottfried Leibniz: The Monadology, 1714.


Remko Scha, 2001/2006.
Some links suggested by Jürgen Sturm and Chuntug Taguba.