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Symmetry, Branching, Modularity: Lego Tree

The tree experiment was designed to test out whether some characteristics of natural trees (branching, symmetry) could evolve as a consequence of the environment. The design of a tree in nature is a product of conflicting objectives: maximizing the exposure to light while keeping internal stability.

The experimental design for the tree has a narrow attachment base: Only three knobs. This provides very little sustentation for cantilevering, so the structure will have to be balanced to reach out. A ``light'' resource, coming from directions up, left and right, has one value per column or row. Light is ``absorbed'' by the first brick it touches -- and the fitness points given are equal to the distance from the absorption point to the x or y axis. The highest fitness would be a structure reaching out to completely cover the left, right and top borders (see fig. 2.29 and table 2.12).

**Figure 2.29:** Tree experiment
$\resizebox*{0.7\textwidth}{!}{\includegraphics{lego/treescheme.eps}}$

There were no symmetry-oriented operators in our experiments, as could be, for example a ``reverse'' recombination operator that switched the orientation of a subpart. This means that symmetry is not encouraged by representational biases. Instead, the problem setup requires balancing the total weight of both sides. The tree did evolve, however, with a central symmetry with branches reaching out, by evolving the same type of solution separately on both sides.

The general layout of the evolved tree has several similarities with that of a real tree: there is a (somewhat twisted) trunk, with branches that become thinner as they reach out, and ``leaves'', bulky formations that maximize the surface at the end of the branch.

Table 2.12: Setup of the tree experiment.

Bricks	{1,2,4,6,8,10,12,16}
Max Bricks	127
Base	(0,-1)-(2,-1)
x Range	(-50,52)
y Range	(0,45)
Fitness	f_L+f_R+f_T
	where
	$f_{L}=\sum ^{45}_{j=0}\max \left( 0,-\min \{x:(x,j)\in S\}\right)$
	$f_{R}=\sum ^{45}_{j=0}\max \left( 0,\max \{x:(x,j)\in S\}\right)$
	$f_{T}=\sum ^{52}_{i=-50}\max \left( 0,\max \{y:(i,y)\in S\}\right)$
	S = structure

The tree is, among all our experiments, the one that most clearly illustrates the emergence of nested levels of organization, key indicator of what we call complex organization

Level zero: individual bricks.
Level 1: diagonal stacks and horizontal stacks of long bricks (on the branches), stacks of small bricks (at the tips),
Level 2: trunk, U shaped branches, T structure at the top.
Level 3: two tridents on top of each other, and a roof.
Level 4: the entire structure.

Figure 2.30: Evolved Tree (internal model and built structure)

$\resizebox*{0.75\textwidth}{!}{\includegraphics{lego/tree6a.eps}}$

**Figure 2.30:** Evolved Tree (internal model and built structure)
$\resizebox*{0.75\textwidth}{!}{\includegraphics{lego/tree6a.eps}}$

Recombination Example

Next: Recombination Example Up: Evolution of Adaptive Morphology Previous: Optimization

Pablo Funes
2001-05-08