The Algorithmic Beauty of Plants

Abstract

With 150 Illustrations, 48 in ColorThis edition of The Alogirthmic Beauty of Plants is the electronic version of the book that was published by Springer-Verlag, New York, in 1990 and reprinted in 1996. The electronic version has been produced using the original L A T E X files and digital illustrations. Front cover design: The roses in the foreground (Roses by D. R. Fowler, J. Hanan and P. ) were modeled using L-systems. Distributed ray-tracing with one extended light source was used to simulate depth of field. The roses were placed on a background image (photgraphy by G. Rossbach), which was scanned digitally and post-processed. PrefaceThe beauty of plants has attracted the attention of mathematicians for Mathematics and beauty centuries. Conspicuous geometric features such as the bilateral symmetry of leaves, the rotational symmetry of flowers, and the helical arrangements of scales in pine cones have been studied most extensively. This focus is reflected in a quotation from Weyl [159, page 3], "Beauty is bound up with symmetry."This book explores two other factors that organize plant structures and therefore contribute to their beauty. The first is the elegance and relative simplicity of developmental algorithms, that is, the rules which describe plant development in time. The second is self-similarity, characterized by Mandelbrot [95, page 34] as follows:When each piece of a shape is geometrically similar to the whole, both the shape and the cascade that generate it are called self-similar.This corresponds with the biological phenomenon described by Herman, Lindenmayer and Rozenberg [61]:In many growth processes of living organisms, especially of plants, regularly repeated appearances of certain multicellular structures are readily noticeable.... In the case of a compound leaf, for instance, some of the lobes (or leaflets), which are parts of a leaf at an advanced stage, have the same shape as the whole leaf has at an earlier stage.Thus, self-similarity in plants is a result of developmental processes.Growth and form By emphasizing the relationship between growth and form, this book follows a long tradition in biology. D'Arcy Thompson [143] traces its origins to the late seventeenth century, and comments:Organic form itself is found, mathematically speaking, to be a function of time.... We might call the form of an organism an event in space-time, and not merely a configuration in space. This concept is echoed by Hallé, Oldeman and Tomlinson [58]:The idea of the form implicitly contains also the history of such a form. vi PrefaceThe developmental processes are captured using the formalism of L-systems. They were introduced in 1968 by Lindenmayer [82] as a Modeling of plants theoretical framework for studying the development of simple multicellular organisms, and subsequently applied to investigate higher plants and plant organs. After the incorporation of geometric features, plant models expressed using L-systems became detailed enough to allow the use of computer graphics for realistic visualization of p...

Arterial Branching within the Confines of Fractal L-System Formalism

“…Although the formalism was originally developed for botanical structures, its language carries directly to arterial structures. Following that language and notation (Prusinkiewicz and Hanan 1989; Prusinkiewicz and Lindenmayer 1990), a basic L-system model for a tree structure consisting of repeated bifurcations, with axiom ω and production rule p , is given by: \documentclass[10pt]{article} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{pmc} \usepackage[Euler]{upgreek} \pagestyle{empty} \oddsidemargin -1.0in \begin{document} \begin{equation*}\begin{matrix}{\mathrm{{\omega}:}}\;X\\ p{\mathrm{:}}\;X{\rightarrow}F \left \left[-X\right] \right \left \left[\;+X\right] \right {\mathrm{,}}\end{matrix}\end{equation*}\end{document} where F represents a line of unit length in the horizontal direction and X is an auxiliary symbol that has no graphical representation but plays an essential role in the branching process. The square brackets represent the departure from ([) and return to (]) a branch point, whereas the plus and minus signs represent turns through a given angle δ in the clockwise and anticlockwise directions, respectively.…”

“…The variables of arterial branching λ, γ, θ 1 , and θ 2 , can be incorporated into the L-system formalism by using a so- called “parametric L-system” (Prusinkiewicz and Lindenmayer 1990) that would embody these variables. The first two variables are introduced by using a two-parameter step function F ( L , W ) in which L represents a step length and W represents a step width.…”

A Survey of Algorithmic Shapes

“…Early systems based on grammars were Lindenmayer systems, short L-systems, named after ARISTID LINDENMAYER [10]. They were successfully used for modeling plants [11] or fractal structures [12].…”

Developmental Computing

“…These problems are eliminated in parametric L-systems [14,21]. For instance, the relations expressed by the system of equations (3) are summarily expressed as a production…”