Visual mo dels of plant development Przemyslaw Prusinkiewiczy, Mark Hammely, Jim Hananz, and Radomr Mechy yDepartment of Computer Science University of Calgary Calgary, Alb erta, Canada T2N 1N4 e-mail: [email protected] z CSIRO - Co op erative Research Centre for Tropical Pest Management Brisbane, Australia e-mail: [email protected] From G. Rozenb erg and A. Salomaa, editors, Handbook of formal languages Springer-Verlag, 1996. To app ear. Visual mo dels of plant development 1 1 Przemyslaw Prusinkiewicz , Mark Hammel , 2 1 Jim Hanan , and Radomr Mech 1 Department of Computer Science University of Calgary Calgary, Alb erta, Canada T2N 1N4 e-mail: [email protected] 2 CSIRO - Co op erative ResearchCentre for Tropical Pest Management Brisbane, Australia e-mail: [email protected] Summary. In these notes wesurvey applications of L-systems to the mo deling of plants, with an emphasis on the results obtained since the comprehensive presenta- tion of this area in The Algorithmic Beauty of Plants [99 ]. The new developments include: { extensions to the L-system formalism that increase its expressivepower as needed for practical biological applications, { intro duction of programming constructs that enhance the use of L-systems as a language for describing developmental algorithms and as input for simulation programs, and { new biological applications of L-systems. Keywords: L-system, fractal, plant, mo deling, simulation, realistic image synthe- sis, emergence, arti cial life. There is nothing so practical as a goodtheory. Immanuel Kant (1724{1804) 1. Intro duction In 1968, Aristid Lindenmayer intro duced a formalism for mo deling and sim- ulating the developmentofmulticellular organisms [67], subsequently named L-systems. This formalism was closely related to the theory of automata and formal languages, and immediately attracted the interest of computer scien- tists [114]. The vigorous development of the theory of L-systems [46,113,116] was followed by its application to the mo deling of plants (for example, [28, 29, 30, 55, 74]). A series of p osition and survey pap ers by Linden- mayer addressed metho dological asp ects of mo deling using L-systems and their role in biology [66,70,71,72,73,75]. Concurrentadvances in computer graphics made it p ossible to visualize mo deled structures in forms ranging from schematic diagrams [30, 49] to realistic three-dimensional renderings of abstract branching structures [91, 120, 121] and real plants [101]. Visu- alizations have also revealed intriguing relationships b etween L-systems and 2 Prusinkiewicz et al. fractals [19, 20, 90, 100, 122]. Graphical applications of L-systems devised until 1990 were comprehensively presented in b o oks [94] and [99]. In the presentchapter wefocusonrecent results p ertinenttothemod- eling, simulation, and visualization of plant development using L-systems. These include, in particular: { extensions to the L-system formalism that increase its expressivepower as needed for practical biological applications, { intro duction of programming constructs that enhance the use of L-systems as a language for describing developmental algorithms and as input for simulation programs, and { new biological applications of L-systems, The organization of this chapter mimics the general pattern of theory construction in the natural sciences, where observed facts are distilled into a mathematical abstraction, and the resulting predictions are compared with realitytovalidate the theory [58]. First, we present mo dular plant architec- ture as our domain of interest and show that the essential asp ects of develop- ment at the mo dular level can b e viewed as rewriting pro cesses (Section 2.). In order to formally describ e the architecture of plants, we intro duce, af- ter Lindenmayer, the bracketed string notation to express the top ology of branching structures, and we extend this notation with symbols and con- structs based on turtle geometry to capture the shap e (Section 3.). Wethen present L-systems as a rewriting mechanism that simulates developmental pro cesses by op erating on strings, and use biologically motivated examples to illustrate the basic de nitions (Section 4.). More involved constructs and extensions of L-systems are intro duced to simulate fragmentation and the loss of mo dules (Section 6.), di erent asp ects of information ow within the mo deled structure (Section 7.), and interaction b etween plants and their en- vironment (Section 8.). Finally,wecharacterize the role of L-systems in the current practice of biological mo deling, and p oint out selected op en problems (Section 9.). Since the mo deling and visualization of plantdevelopmentisaninterdis- ciplinary area of research, wehave attempted to make the results legible to readers in various disciplines. Consequently,wehave emphasized the motiva- tions b ehind the theory,andavoided sp ecialized notions of formal languages, computer graphics, and biology. 2. Developmental mo dels of plant architecture 2.1 The mo dular structure of plants Mathematical mo dels in b otany corresp ond to various levels of plant organi- zation (Figure 2.1). In this pap er, we fo cus on the level of entire plants. We regard a plantas a spatial con guration of discrete constructional units or Visual mo dels of plant development 3 organelle organ crop biome molecule cell plant ecosystem Fig. 2.1. A hierarchy of levels of plant organization. One ob jectiveofmodelingis to predict and understand phenomena taking place at a given level on the basis of mo dels op erating at lower levels. Adapted from [124 , 131 ]. modules,whichdevelop over time. Typically, mo dules represent rep eating ba- sic structural comp onents of a plant, suchas owers, leaves, and interno des, or groupings of these comp onents, such as metamers (single interno des with an asso ciated leaf and lateral bud) and branches (Figure 2.2) [5, 43, 128]). (A di erent meaning of the term \mo dule" is also found in the litera- ture [4,38, 110].) The goal is to describ e the developmentofaplant, and in particular the emergence of plant shap e, as the integration of the development and functioning of individual mo dules. apical meristem apex apical segment internode metamer or bud (lateral) shoot unit branch inflorescence leaf flowers Fig. 2.2. Selected mo dules and groups of mo dules (encircled with dashed lines) used to describ e plant structure. 4 Prusinkiewicz et al. 2.2 Plant developmentas a rewriting pro cess The essence of plant development can be describ ed by a rewriting system that rep etitively replaces individual parent, mother,orancestor mo dules by con gurations of child, daughter,ordescendant mo dules. Assuming that all mo dules belong to a nite set of module types, the behavior of an arbitrarily large con guration of mo dules can be sp eci ed using a nite set of rewriting rules or productions. A pro duction sp eci es how to replace a single predecessor mo dule by a con guration of zero, one, or more successor mo dules. A simple example of this pro cess is shown in Figure 2.3. An occurrence map ' transforms the predecessor of the pro duction to the mother mo dules; the same map is then applied to the successor in order to determine the child mo dules [103]. predecessor successor p production occurrence ϕ ϕ−1 ϕ mapping production application parent children Fig. 2.3. Illustration of the concept of rewriting applied to mo dules with geometric interpretation. A parent mo dule is replaced by child mo dules in a sequence of 1 transformations ' p'. The replacement of mo dules in a structure may b ecome dicult when there are signi cant di erences in the geometry of a parent and its children. Several p ossibilities are illustrated in Figure 2.4. In case (a), mo dules lo cated at the extremities of a branching structure are replaced without a ecting the remainder of the structure. The pro duction applications may b e implemented using the mechanism just discussed (Figure 2.3). In case (b), the pro duction that replaces interno des divides the rewritten structure into a lower part (b elow the interno de) and an upp er part. The p osition of the upp er part is adjusted to accommo date the insertion of the child mo dules, but the shap e Visual mo dels of plant development 5 a) bud flower young fruit old fruit b) c) Fig. 2.4. Examples of pro duction sp eci cation and application: (a) development of a ower, (b) development of a branch, and (c) cell division. and size of b oth the lower and upp er part are not changed, and the children remain de ned entirely by the pro duction. Finally, in case (c), the rewritten structure is represented by a graph with cycles. The size and shap e of the pro duction successor do not exactly match the size and shap e of the prede- cessor, thus the geometry of the successor, the emb edding structure, or b oth must b e adjusted to accommo date the successor. The last case is the most complex, since the application of a lo cal rewriting rule may lead to a global change of the structure's geometry. Developmental mo dels of cellular layers op erating in this manner have b een presented in [16,17,26,99]. In this pa- per we will limit our interest to branching structures. This limitation o ers a useful compromise b etween breadth and depth in the resulting theory of development. The di erences between cases (a) and (b) are further discussed below. Case (a) is similar to the basic metho d of fractal generation using Ko ch construction, describ ed by Mandelbrot as follows [81, page 39]: One b egins with two shapes ,aninitiator and a generator.