Great artists of all ages, from Botticelli to Kit Williams and William Morris to Pip Hackett, have drawn inspiration from the natural world. We surround ourselves with flowers and plants and designs that represent them. It is hard to imagine a healthy environment without the presence of plants. Above is one of the beautiful creations of premier artist and milliner Pip Hackett. Her fantastic hats, resplendent with horns, wings and feathers, take their inspiration from a variety of organic sources: Dancing until dawn through fields of intoxicating flowers and the silver-blue flash of a kingfisher over a stream murmuring in the morning light.

We have an intuitive understanding of the structure of organic materials. It would be very unusual to mistake an inorganic structure or a human artefact for a living organism. But what is it that enables us to recognise the organic? Many plants look very similar on different length scales, we say they have a fractal structure. A small twig is strikingly reminiscent of a full-grown tree, a single cauliflower floret looks very like the full cauliflower.

In 1968 the Swedish Biologist Aristid Lindenmayer devised a powerful new recursive technique for modelling plants. These models are known as Lindenmayer systems, or more usually just L-systems. They are closely related to cellular automata and produce intrinsically fractal structures. Lindenmayer described his L-systems in the following terms: “The central concept of L-systems is that of rewriting. In general, rewriting is a technique for defining complex objects by successively replacing parts of a simple initial object using a set of rewriting rules.”

Each model consists of two parts. It has a collection of transformation rules and a seed, which is often referred to as the axiom, to which the rules are progressively applied. One simple example has two transformation rules: A goes to AB, B goes to A. If we start with the seed A in the first generation. In each subsequent generation wherever A occurs it will be replaced by AB and wherever B occurs it will be replaced by A.

Starting with the seed A, the transformation rules lead to the text string AB in the second generation, the third generation is ABA, fourth generation: ABAAB, fifth generation: ABAABABA, sixth generation: ABAABABAABAAB. We rapidly reach quite complicated expressions that would be difficult to predict from our starting point. The greater the number of generations that we use, the greater the recursion and the more complicated the expression. The fractal nature of the expressions is derived directly from the method of construction.

The above animation shows a slightly more complicated model that is already beginning to look organic. The animation starts from the seed and shows how the L-system develops upto the 15th generation. The LSystem5 software developed by Laurens Lapre was used to generate the L-system that was used to produce the frames of the animation.

The results that are possible with L-systems are so realistic that they are now commonly used to model plants and other organisms in the computer games and computer graphics industries. For example, the above illustration of a field of sunflowers was completely computer generated by Oliver Deussen using L-systems. The fact that L-systems can be used to generate such realistic models of plants could be giving us important clues about the actual developmental biology of plants.

The above animation shows an example of how an L-system changes as one of its parameters is varied. In this case an angular parameter changes from frame to frame.

Text and animations by Nick Mee.                    

For further information see 'The Algorithmic Beauty of Plants' by Przemyslaw Prusinkiewicz and Aristid Lindenmayer. (1990) LSystem5 software by Laurens Lapre.