4. Simulating Growth and Form
I
n this section we will discuss a number of methods which have applied
in modeling growth and form of marine sessile organisms. The models
have been, more or less, been arranged according to level of abstraction with
respect to the actual growth process, starting at the highest level of abstraction . The first section on L-system models, focuses on a method capturing
the iterative structure of a modular organism into an algorithmic form, using
formal languages . In the second section an alternative method for modeling
the iterative structure in seaweeds is presented, based on iterative geometric constructions. In the examples shown of the L-system models and the
example using iterative geometric constructions, the level of abstraction is
relatively high and no model of the influence of the physical environment
is present. In the third section a method, stemming from computational
physics, is presented which is very suitable for modeling hydrodynamics
in three-dimensional irregular geometries, as frequently found in marine
sessile organisms. In the fourth section on Laplacian growth models a twodimensional model is discussed which is in some ways intermediate between
approaches such as L-systems and models which have a representation of the
fluid environment. In the fifth section a simple three-dimensional model of
the growth process, the aggregation model, is discussed, while the model of
the fluid environment is extended to a full three-dimensional model of hydro -
dynamics, using the results from the second section on the lattice Boltzmann
method. In the sixth section, the three-dimensional growth model of marine sessile organisms is further extended to a model of accretive growth,
including the influence of light, hydrodynamics, and biological regulation
mechanisms. In the last section a model with the relatively lowest level of
abstraction is presented, which simulates one specific internal component
of the growth process, the fluid transport in the gastrovascular system of
a hydrozoan.
4.1 L-systems
4 .1.1 Introduction to Modeling Using L-systems
L-systems were introduced by A. Lindenmayer as a mathematical model of
multicellular organisms that form linear or branching filaments (Linden -
mayer 1968, 1971). The models are inherently dynamic, which means that the
form of an organism is considered "an event in space-time, and not merely
a configuration in space" (D'Arcy Thompson 1917). The whole organism is
treated as an assembly of discrete units, called modules. Although the nature
J. A. Kaandorp et al., The Algorithmic Beauty of Seaweeds, Sponges and Corals
© Springer-Verlag Berlin Heidelberg 2001
I
n this section we will discuss a number of methods which have applied
in modeling growth and form of marine sessile organisms. The models
have been, more or less, been arranged according to level of abstraction with
respect to the actual growth process, starting at the highest level of abstraction . The first section on L-system models, focuses on a method capturing
the iterative structure of a modular organism into an algorithmic form, using
formal languages . In the second section an alternative method for modeling
the iterative structure in seaweeds is presented, based on iterative geometric constructions. In the examples shown of the L-system models and the
example using iterative geometric constructions, the level of abstraction is
relatively high and no model of the influence of the physical environment
is present. In the third section a method, stemming from computational
physics, is presented which is very suitable for modeling hydrodynamics
in three-dimensional irregular geometries, as frequently found in marine
sessile organisms. In the fourth section on Laplacian growth models a twodimensional model is discussed which is in some ways intermediate between
approaches such as L-systems and models which have a representation of the
fluid environment. In the fifth section a simple three-dimensional model of
the growth process, the aggregation model, is discussed, while the model of
the fluid environment is extended to a full three-dimensional model of hydro -
dynamics, using the results from the second section on the lattice Boltzmann
method. In the sixth section, the three-dimensional growth model of marine sessile organisms is further extended to a model of accretive growth,
including the influence of light, hydrodynamics, and biological regulation
mechanisms. In the last section a model with the relatively lowest level of
abstraction is presented, which simulates one specific internal component
of the growth process, the fluid transport in the gastrovascular system of
a hydrozoan.
4.1 L-systems
4 .1.1 Introduction to Modeling Using L-systems
L-systems were introduced by A. Lindenmayer as a mathematical model of
multicellular organisms that form linear or branching filaments (Linden -
mayer 1968, 1971). The models are inherently dynamic, which means that the
form of an organism is considered "an event in space-time, and not merely
a configuration in space" (D'Arcy Thompson 1917). The whole organism is
treated as an assembly of discrete units, called modules. Although the nature
J. A. Kaandorp et al., The Algorithmic Beauty of Seaweeds, Sponges and Corals
© Springer-Verlag Berlin Heidelberg 2001