4.6. ACCRETIVE GROWTH
to generate a gradient in morphologies similar to that observed in Haliclona
oculata. The growth axes mechanism applied in the f(a, {3) component,
where the growth direction of previous growth layers partly determines the
growth direction of a new growth layer, may be a realistic mechanism. In for
example the sponge Axinella polypoides (see Fig. 2.19),the dense central axis
of skeleton material may be the biological representation of such a growth
axis. In organisms with an accretive architecture (Figs. 1.4 and 2.17), where
usually the direction of growth of new layers is again determined by the
previous directions, the biological representation of growth axes might be
the location of cells which are secreting skeleton material.
The sphere- and column-shaped forms generated with the L(8) model
(4.28) resemble the growth forms of the stony coral Montastrea annularis
shown in Fig. 2.31a and b. In this species photosynthesis represents the main
energy source (see Sect. 2.2.3). The growth forms in the range shown in
Fig. 2.31C and d, where the tapered hemisphere gradually transforms into
a substrate covering sheet cannot be captured with the current version of
the accretive growth model. In this version it is assumed that the object is
a manifold, where each triangle in the mesh is connected to a neighbor.
A simulation model of light-driven, accretive growth, in which substrate
covering sheets can be simulated, requires a modified version of the model
allowing for triangular meshes with free boundaries. In such a modified
version growth may occur by the addition of new polygons at the free edges
of the sheet. The branching form generated with the L(8) · h 2 ( •• ) model
(4.29) resembles growth forms of some types of stony corals, for example
the Caribbean species Acropora palmata. In this model branching, umbrellashaped forms emerge , where branches do not self-intersect, since the lightdriven growth process stops as soon as neighboring branches are at the point
of self-intersection.
When comparing the objects generated with the k(c) model in Fig. 4.39
and with the k(c) . h 2 ( • • ) model in Fig. 4.41, it can be observed that in the
simulations where growth is exclusively driven by local nutrient gradients
(using growth function (4.25» , highly complex surfaces develop with a ~ox
of around 2.35. In contrast the surfaces which develop in the experiment (see
Fig. 4.41) where growth is driven by local nutrient gradients and the local
amount of contact with the environment (using growth function (4.26» are
relatively smooth and vary between 2.00-2.31. In Fig. 4.39lobed objects without discernible branches are formed. A comparison between Figs. 4.39 and
4.41 demonstrates the effect of the amount of contact with the environment
h 2 ( •• ), causing the formation of branches in Fig. 4.41. Both the lobed and the
branching morphologies can be found in actual sponges and stony corals.
This result suggests that sensitivity to the amount of contact with the environment, for example by a relatively low contribution of translocation of
nutrients from the place of absorption to more remote sites, could playa role
in the formation of branches.
In a comparison between the ranges shown in Figs. 4.39 and 4-41 using
the ratio R of the total sink nodes to the total number of object nodes , listed
in Tables 4-4and 4.5,for both ranges R becomes lower for increasing Penumbers . The decreasing R ratios indicate an increasing degree in compactness.
It is quite remarkable that for the range shown in Fig. 4.41the values for ~ox
increase with increasing degree of compactness, while based on the observations done on the R ratios alone the opposite would be expected. Probably
141
to generate a gradient in morphologies similar to that observed in Haliclona
oculata. The growth axes mechanism applied in the f(a, {3) component,
where the growth direction of previous growth layers partly determines the
growth direction of a new growth layer, may be a realistic mechanism. In for
example the sponge Axinella polypoides (see Fig. 2.19),the dense central axis
of skeleton material may be the biological representation of such a growth
axis. In organisms with an accretive architecture (Figs. 1.4 and 2.17), where
usually the direction of growth of new layers is again determined by the
previous directions, the biological representation of growth axes might be
the location of cells which are secreting skeleton material.
The sphere- and column-shaped forms generated with the L(8) model
(4.28) resemble the growth forms of the stony coral Montastrea annularis
shown in Fig. 2.31a and b. In this species photosynthesis represents the main
energy source (see Sect. 2.2.3). The growth forms in the range shown in
Fig. 2.31C and d, where the tapered hemisphere gradually transforms into
a substrate covering sheet cannot be captured with the current version of
the accretive growth model. In this version it is assumed that the object is
a manifold, where each triangle in the mesh is connected to a neighbor.
A simulation model of light-driven, accretive growth, in which substrate
covering sheets can be simulated, requires a modified version of the model
allowing for triangular meshes with free boundaries. In such a modified
version growth may occur by the addition of new polygons at the free edges
of the sheet. The branching form generated with the L(8) · h 2 ( •• ) model
(4.29) resembles growth forms of some types of stony corals, for example
the Caribbean species Acropora palmata. In this model branching, umbrellashaped forms emerge , where branches do not self-intersect, since the lightdriven growth process stops as soon as neighboring branches are at the point
of self-intersection.
When comparing the objects generated with the k(c) model in Fig. 4.39
and with the k(c) . h 2 ( • • ) model in Fig. 4.41, it can be observed that in the
simulations where growth is exclusively driven by local nutrient gradients
(using growth function (4.25» , highly complex surfaces develop with a ~ox
of around 2.35. In contrast the surfaces which develop in the experiment (see
Fig. 4.41) where growth is driven by local nutrient gradients and the local
amount of contact with the environment (using growth function (4.26» are
relatively smooth and vary between 2.00-2.31. In Fig. 4.39lobed objects without discernible branches are formed. A comparison between Figs. 4.39 and
4.41 demonstrates the effect of the amount of contact with the environment
h 2 ( •• ), causing the formation of branches in Fig. 4.41. Both the lobed and the
branching morphologies can be found in actual sponges and stony corals.
This result suggests that sensitivity to the amount of contact with the environment, for example by a relatively low contribution of translocation of
nutrients from the place of absorption to more remote sites, could playa role
in the formation of branches.
In a comparison between the ranges shown in Figs. 4.39 and 4-41 using
the ratio R of the total sink nodes to the total number of object nodes , listed
in Tables 4-4and 4.5,for both ranges R becomes lower for increasing Penumbers . The decreasing R ratios indicate an increasing degree in compactness.
It is quite remarkable that for the range shown in Fig. 4.41the values for ~ox
increase with increasing degree of compactness, while based on the observations done on the R ratios alone the opposite would be expected. Probably
141