methods are applicable to a given rewriting system. The bi-directional model consists of 1) a forward process constructs the derivation tree with production rules, and 2) a backward process realizes the tree with shapes in a stepwise manner (from leaves to the root). Each inverse-derivation step involves essential geometric-topological reasoning. With this bi-directional framework, design constraints and objectives are encoded in the grammar-shape translation. In architectural design applications, the bi-directional grammar features consistent interior connectivity and a rich spatial hierarchy. Open problems There are many open problems involving studies of L-systems. For example: Characterisation of all the deterministic context-free L-systems which are locally catenative. (A complete solution is known only in the case where there are only two variables). Given a structure, find an L-system that can produce that structure. Types of L-systems L-systems on the real line R: Prouhet-Thue-Morse system Well-known L-systems on a plane R2 are: space-filling curves (Hilbert curve, Peano's curves, Dekking's church, kolams), median space-filling curves (Lévy C curve, Harter-Heighway dragon curve, Davis-Knuth terdragon), tilings (sphinx tiling, Penrose tiling) See also Digital morphogenesis Fractal Iterated function system Hilbert curve that provides diffusing-chemical-reagent simulations (including Life-like) Stochastic context-free grammar SpeedTree The Algorithmic Beauty of Plants Notes Books Przemysław Prusinkiewicz, Aristid Lindenmayer – The Algorithmic Beauty of Plants PDF version available here for free Grzegorz Rozenberg, Arto Salomaa – Lindenmayer Systems: Impacts on Theoretical Computer Science, Computer Graphics, and Developmental Biology D.S. Ebert, F.K. Musgrave, et al. – Texturing and Modeling: A Procedural Approach, Burry, Jane, Burry Mark, (2010). The New Mathematics of Architecture, New York: Thames and Hudson. Aristid Lindenmayer, "Mathematical models for cellular interaction in development." J. Theoret. Biology, 18:280—315, 1968. External links Algorithmic Botany at the University of Calgary L-Systems: A user friendly page to generate fractals and plants from L-Systems. Branching: L-system Tree A Java applet and its source code (open source) of the botanical tree growth simulation using the L-system. Fractint L-System True Fractals OpenAlea : an open-source software environment for plant modeling, which contains L-Py, an open-source python implementation of the Lindenmayer systems "powerPlant" an open-source landscape modelling software Fractint home page A simple L-systems generator (Windows) Lyndyhop: another simple L-systems generator (Windows & Mac) An evolutionary L-systems generator (anyos*) Page about using L-systems and genetic algorithms to generate music. eXtended L-Systems (XL), Relational Growth Grammars, and open-source software platform GroIMP. A JAVA applet with many fractal figures generated by L-systems. My