Cooperative Trees by Adding Inosculated and Discrete Definitions to a DLA Design

Abstract

This paper presents a method to generate free-form branched structures from a small number of different constructive elements, based on the postulates of discrete or combinatorial design. The research is based on the study of fractal growth as a generator of complex tree-like structures, using references from other scientific approaches in which the possibilities of the DLA (diffusion-limited aggregation) model have been explored. The proposed method uses the Grasshopper visual programming language, and incorporates new topological strategies to improve the performance or robustness of the system through tree-tree (inosculation) and tree-soil (aerial roots) cooperations. The simulation demonstrates the effectiveness of the proposed method and its potential for the construction of structures with complex geometries from a discrete set of knots and bars and bioinspired strategies. The paper includes a review of the chosen design principles, the developed methodology and a recent physical test in Medellín (Colombia).

Full text

Cooperative Trees by Adding Inosculated and Discrete
Definitions to a DLA Design
Salvador Serrano Salazar1, José Carrasco Hortal2,
Francesc Morales Menárguez3, Juan Pablo Gutiérrez Salazar4
1,2,3Universidad de Alicante 4Universidad de Nacional de Colombia Sede Medellín
1,3{salvaserrano31|morales.francesc}@gmail.com
2jose.carrasco@ua.es 4fablab_med@unal.edu.co
This paper presents a method to generate free-form branched structures from a
small number of different constructive elements, based on the postulates of
discrete or combinatorial design. The research is based on the study of fractal
growth as a generator of complex tree-like structures, using references from other
scientific approaches in which the possibilities of the DLA (diffusion-limited
aggregation) model have been explored. The proposed method uses the
Grasshopper visual programming language, and incorporates new topological
strategies to improve the performance or robustness of the system through
tree-tree (inosculation) and tree-soil (aerial roots) cooperations. The simulation
demonstrates the effectiveness of the proposed method and its potential for the
construction of structures with complex geometries from a discrete set of knots
and bars and bioinspired strategies. The paper includes a review of the chosen
design principles, the developed methodology and a recent physical test in
Medellín (Colombia).
Keywords: DLA, discrete design, inosculation, branching structures, virtual-real
models
INTRODUCTION
This paper focuses on applying discrete or combina-
torial design to structures, based on the understand-
ing that this is desirable in constructive terms. The
objective of this work is to develop a method to de-
sign free-form branching structures based on a lim-
ited number of different knots and bars of the same
length.
FREE-FORM TO DISCRETE-FORM BRANCH-
ING DESIGN
Fractal growth by diffusion-limited aggre-
gation (DLA)
In recent years, the development of computer sim-
ulation tools has generated various lines of research
that explore how fractals can be used as design tools,
applying them to different scenarios. Fractals are
widely present in nature and we can learn from them
to reproduce growth patterns. Their complexity,
however, requires a bottom-up approach: the prop-
GENERATIVE DESIGN - Volume 2 - eCAADe 36 |103

erties of simple objects need to be defined so they
turn into complex objects when assembled (Kautz
and Shutters 2011). For example, Greenberg (2008)
describes two main optimization strategies in natural
branched structures: on the one hand, the “logic of
the bifurcation angle” tries to consider the constant
angular range in which transfer of fluids and loads are
effective to avoid turbulences; on the other hand, the
“phyllotactic logic”, a distribution of branches follow-
ing a spiral pattern to maximize solar exposure avoid-
ing self-shadows.
One of the most studied models for generat-
ing fractal structures is diffusion-limited aggregation
(DLA), proposed by Witten and Sander (1983). This
model describes the formation of clusters when par-
ticles in motion collide with other particles that re-
main fixed (see Figure 1). In architecture, DLA has
been mainly applied to the study of urban growth
(Batty et al. 1989) (Chan and Chiu, 2000). Other
works have addressed smaller scale problems, using
the DLA to design spaces according to environmental
considerations such as sunlight (Kim and Lee 2015) or
the interaction between people and their roles within
a company (Sun et al. 2015).
Diffusion-limited aggregation has also been
used to design structures. Previous work has ex-
plored its potential as a tool to model the morphol-
ogy of plants and algae (Serrano et al. 2017). Other
studies, such as Busch et al. (2011), propose a modu-
lar material system in which the aggregate particles
are assimilated to discrete elements that come to-
gether to form an emergent structure. In the same
line, Kachri and Hanna (2014) propose a system in
which the structure is formed by the merger of oc-
tahedrons truncated on their faces, which is then
analysed and structurally optimised. Narahara (2009)
also incorporates structural evaluation into the DLA
model, through physical simulations of the equilib-
rium of the generated clusters.
Combinatorial Design
Systems that correspond to the definition of combi-
natorial design are described in the works of both
Busch (2011) and Kachri and Hanna (2014). Busch
et al. (2011) clearly describes the stages the algo-
rithm must solve: attraction, particle fixation, align-
ment and random rotation, all of which are necessary
to generate an unpredictable configuration (see Fig-
ure 2).
For Sánchez (2016), combinatorial design in ar-
chitecture embraces strategies based on the permu-
tation and repetition of discrete units of material.
In a similar way, Borhani and Kalantar (2017) high-
light the relevance of a new constructive language
in which the geometry of discrete structural ele-
ments allows to assemble them through interlocking.
Retsin (2016) points out the advantages of discrete
assembled systems, highlighting the automation ca-
pacity of these constructive processes through the
serialization of actions. Dierichs and Menges (2013)
propose a set of aggregation rules that, as in the DLA,
are based on kinetics and collision to build a structure
from discrete elements. This paper focuses on ap-
plying combinatorial design to structures, based on
the understanding that this is desirable in construc-
tive terms. The objective of this work is to develop
a method to design free-form branching structures
based on a limited number of different nodes and
bars of the same length.
Figure 1
The NetLogo “DLA
Alternate Linear”
model: a variation
of the original DLA
model [1].
Figure 2
Aggregation of
discrete units
proposed by Busch
(2011).
104 |eCAADe 36 - GENERATIVE DESIGN - Volume 2

New topology strategies: inosculation and
aerial roots
Literature on the relationship between growth,
shape and stability in plants is extensive. Mattheck
(1998) described how the strategies developed by
trees to avoid collapsing can inspire the design of
structures. Some tree species such as redwoods are
capable of self-grafting, that is, their branches can be
reattached to the trunk creating a buttress that helps
the tree to resist wind loads more efficiently, as sug-
gested by Preston [2]. This phenomenon is called in-
osculation and can also occur between different indi-
vidual trees, thus leading to structural collaboration
(see Figure 3). On another scale, the reconnection of
diverging branches is called anastomosis. An exam-
ple is the reticulate venation observed in the leaves
of some tree species. Laguna (2008) suggested that
the veins that separate and rejoin are more abundant
in the areas of the leaf that are subjected to greater
structural stress due to the leaf’s weight. Klemmt
(2014) compared the advantage of reticular venation
patterns (with anastomosis) versus dendritic vena-
tion patterns (without anastomosis) such as DLA clus-
ters, in terms of mechanical stability.
Figure 3
Examples of trees
with inosculated
branches [3,4,5].
Figure 4
Tree of the Ficus
genus in Alicante
(Spain).
The development of aerial roots is another strategy
used by some tree species to increase their stability
(see Figure 4). When the aerial roots of Banyan trees
and other types of ficus trees touch the ground, they
can merge and form pseudo woody trunks that al-
low horizontal branches to reach great lengths and
a large overall tree size (Jackson 1986).
Inosculation and aerial roots can be interpreted
as topological strategies that increase the stability of
branch structures, in which constantly diverging el-
ements result in the appearance of significant defor-
mations when submitted to loads. The method pre-
sented in this study puts forward some geometric
procedures to incorporate these strategies into the
DLA model.
MODEL ASSEMBLY
Particles
The Grasshopper visual programming tool, part of
the Rhinoceros software, was used to implement a re-
cursive routine. The proposed model was based on
the NetLogo “DLA Alternate Linear” model [1], trans-
ferring it into a three-dimensional version. The sys-
tem’s rules are defined below:
• Moving particles arose in random positions in
the upper face of the prism in which the sim-
ulation space was enclosed. They followed
a vertical downward trajectory covering the
same distance at each iteration, so they main-
tained a constant falling speed.
• The particles in motion could be added to the
first capturing particles that remained fixed
on the floor of the simulation space. If they
did not succeed, they passed by and were dis-
carded.
• The particles that were captured became part
of the cluster and became capturing particles.
Knots
As a criterion to homogenise the structural elements,
the number of joints between bars was proposed to
be reduced to 3 (see Figure 5). The first of them (knot
GENERATIVE DESIGN - Volume 2 - eCAADe 36 |105

A) corresponded to the first capturing particles lo-
cated on the ground and was the starting piece of
the structure. The second (knot B) corresponded to
the bifurcations resulting from the aggregation of
particles, and was conditioned to continually form
the same angle. The last connecting piece (knot C)
connected the branches with the bridges (inoscula-
tion) and with the pseudo trunks (aerial roots). The
knot shapes approximated the smooth transitions
between tree branches as described by Mattheck
(1989). The structure’s bars had a circular section.
The length of bars arising from the aggregation of
particles had a single fixed size. The bars appearing
as bridges that added stability to the structure could
be of different lengths, but their diameters would re-
main the same as previous ones.
Branches
So that the linear elements arising from particle ag-
gregation had the same length and could be assem-
bled with the suggested knots, a series of geometric
restrictions was defined as follows:
• Each capturing particle had an “aggrega-
tion ring” corresponding to the circumference
drawn by the generators of a straight cone
whose vertex was the particle and whose gen-
erator angle with respect to its axis was the
same as the angle of the knot B.
• When a particle moving in a given iteration
reached below the collision distance (e) of
the “aggregation circumference”, the particle
became aggregated, creating a new bar that
joined the capturing particle at the point of
the circumference closest to the captured par-
ticle.
• When aggregated, the bar was created as well
as its twin symmetric pair with respect to the
cone’s axis, so that a complete bifurcation was
produced from the capturing particle.
• In the case where two particles were at the
necessary distance of aggregation, the one
closest to the circumference would be the
only one to be added.
Figure 5
Proposed discrete
set of transitions:
knots A, B and C.
Figure 6
Generating
branches:
aggregation of
particles in motion
• Knot A was located in the first capturing parti-
cles. The aggregation rules were the same ex-
cept that the twin of the aggregated bar was
not generated.
When this process is repeated over successive itera-
tions, a branched structure gradually builds itself, in
which the same restricted angles between the bars
are continuously maintained (see Figure 6).
Bridges
The knot C was used to create bridges between differ-
ent bars of the structure generated by the aggrega-
tion of particles, thus introducing the phenomenon
of inosculation into a DLA pattern.
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Figure 7
Segment leaning
on two straight
lines forming the
same angle with
respect to both.
Figure 8
Example of
application of the
bridge creation
restrictions. The
bridges that did not
meet the conditions
were discarded
(red), the others
were generated
(blue).
Figure 9
Aerial roots
generation:
intersection
between the two
cones and arbitrary
selection of one of
the resulted lines.
Figure 10
Example of how the
auto-intersection
discarding
algorithm worked.
A geometric problem arose: that of finding a seg-
ment that would lean on two straight lines forming
the same angle (α) with respect to both. To solve this
problem, a graph was made of a typical case (see Fig-
ure 7). It was necessary to define the plane whose in-
tersection with the straight lines produced the points
enabling to draw the segment we wanted to create.
The expression (1) indicates the distance between
the intersection planes and the segment with the
minimum distance between the straight lines. For
each pair of lines, there were 2 or 4 possible solutions,
depending on angle α and the angle formed by the
lines between them.
d=±√−h2·cos2(a)
4·tan2(θ)·(cos2(θ)−sin2(θ)) (1)
At each iteration, the routine calculated the hypo-
thetical bridges between all the bars (see Figure 8).
Of all those possible solutions, those that presented
the following conditions were discarded:
• One or two ends of the bridge were supported
outside the bars.
• One or two ends of the bridge were located
very close to a bar knot.
• The bridge length was less than LBmin or
greater than LBmax.
Aerial roots
While bridges created tree-tree or branch-branch col-
laboration situations, aerial roots increased tree-soil
collaboration. This connection was addressed by
adding type A knots where the root started from the
ground and type C knots when meeting the sup-
ported branch. The routine calculated all possible
aerial roots at each iteration. To do this, an intersec-
tion was built between conical surfaces with the ver-
tex at the midpoint of each of the structure’s bars. The
first cone defined the geometric location of the lines
that formed the angle α with these bars, while the
second cone defined the geometric location of the
lines that formed the angle β with the ground. The
resulting intersection segments (if any) were two and
GENERATIVE DESIGN - Volume 2 - eCAADe 36 |107

were symmetrical with regard to the vertical plane
containing the supported bar (see Figure 9).
As in the case of bridges, for the creation of aerial
roots to be considered, the length of the roots had
to be within the [LAR min, LAR max]range; this re-
striction was necessary to avoid adding very short or
very long bars. If the intersection segments passed
this filter, the programmed routine arbitrarily dis-
carded one of the two and defined the other as the
aerial root to be added. In this way,a k not C appeared
at the junction between the supported bar and the
aerial root, as well as an A knot at the junction of the
latter with the ground.
Self-collision avoidance
To ensure the viability of the structure’s construction,
there were some exceptional cases in which bar ag-
gregations did not occur:
• If an aggregated branch crossed the surface of
the ground, it would be discarded.
• If an aggregated aerial root, branch, or bridge
intersected with another existing element,
then it would be discarded. If it was a branch,
it would not generate its twin pair when dis-
carded.
• If an aggregated branch element did not in-
tersect with an existing element but its twin
pair did, then the former would be generated
but the latter would not, leaving the bifurca-
tion knot’s mouth unoccupied.
• If two or more of the same type of elements
aggregated in the same iteration intersected
with one another, a discarding algorithm was
developed that counted the incompatibilities
of each bar with respect to the others, and
eliminated the one that had the biggest num-
ber of incompatibilities (see Figure 10). The
process was repeated until there were no bars
with incompatibilities left.
MODEL TESTING
Running the virtual
A digital implementation of the method was carried
out. The simulation was carried out within a three-
dimensional domain of 20x20x20 meters. The col-
lision distance was set at 0.1 meters. The length of
the bars generated by aggregation was 3 meters. For
bridges, a range of validity between 2 to 7 meters was
established. Figure 11 shows the growth of the ag-
gregate structure over 800 iterations.
One can observe how little material was added
after 300 iterations, while in the last 200 iterations
of the simulation the structure practically doubled
its maximum height. This is because when one cap-
turing particle caught another, it completed its cap-
ture capacity, but created a bracket in which two cap-
turing particles appeared; i.e., as the structure grew,
its ability to add more particles increased. At itera-
tion 400, two of the three trees were united by sev-
eral bridges, and developed some aerial roots. At it-
eration 600, the whole structure was highly consoli-
dated.
Figure 12 shows the difference between
branches generated by aggregation and branches
added by the algorithm as a strategy of stability in-
spired by plants. As in inosculation events, some
bridges connected the branches of the same tree
and others connected the branches of different trees.
These elements are expected to be effective at re-
ducing any displacements of the structure that could
be produced by external loads such as wind. The
aerial roots appeared when a bar formed an angle
close enough to the ground, that is, when it grew
in a horizontally-oriented direction. These elements
increased the supporting points and avoided pro-
nounced bends generated by branches expanding
laterally. At each iteration, all previously created ele-
ments could support a new bridge.
108 |eCAADe 36 - GENERATIVE DESIGN - Volume 2

Figure 11
Growth of the
structure over 800
iterations.
Figure 12
Structure after 800
iterations: branches
(grey), bridges and
aerial roots (red).
The algorithm thus allowed the creation of second
order bridges, that is, bridges that rested on another
existing bridge and a bar or on two existing bridges.
In the same way, aerial roots could also be added to
branches or bridges without distinction. This condi-
tion showed how by introducing these strategies, the
topology of the DLA pattern acquired greater com-
plexity and showed further levels of emergence.
Building the physical
The “Arbol de la lluvia” (Rain Tree) was a small instal-
lation that was built at the gardens of Medellin Fac-
ulty of Architecture (Universidad Nacional de Colom-
bia) between 2nd and 4th of May 2018. It was pre-
designed at the University of Alicante and made with
GENERATIVE DESIGN - Volume 2 - eCAADe 36 |109

Figure 13
Parametric model
of the “Árbol de la
lluvia”.
Figure 14
“Árbol de la lluvia”
in front of Faculty of
Architecture in
Medellín University.
Bars (RDE 21 PVC
1.1/4”); knots (PLA
1.75mm with
printing time 5-10
hours each one)
the collaboration of Medellin’s UNAL Fab-Lab. It is
the exact replica of the digital model explained in 4.1.
section, and specifically corresponds to iteration 500
shown in Figure 11, reaching a total height between
4 to 5m. It was thus generated by means of DLA ag-
gregation, to which bridges or inosculated bars were
added. It follows the line of the research that started
with an artificial tree mounted a year before by the
Universities of York (Toronto), TU-Delft and Alicante
(Serrano et al. 2017). Reactive artificial plant species
were added to this installation: they played the part
of bird feeders and emulated exchanges between liv-
ing and inert materials thanks to devices controlled
by Arduino. A previous thought-provoking work had
consisted in looking at what kind of native tree was
located next to the installation. A previous thought-
provoking work consisted in looking for the native
tree to host the installation.
Finally, a monumental Ceiba pentandra that
looked like a set of bifurcations produced by a DLA
growth pattern was chosen. At the discrete design
level, the skeleton was formed by only three types of
knots (see Figure 13), so the difficulty lied not in the
variability of the manufactured knots but in the con-
trol of the exact orientation of each part and of the
whole. The supports were arranged on a horizontal
plane and started with a certain angle with respect
to the vertical angle, which, in fact, made the system
more dynamic. Due to the time it took to execute the
knots (each know took several hours of 3D printing),
only three quarters of the total were printed in the
end.
110 |eCAADe 36 - GENERATIVE DESIGN - Volume 2

CONCLUSIONS
Branched structures have aroused the interest of ar-
chitects and engineers due to their efficiency at trans-
mitting loads to supports. Standardizing knots in
free structures is rarely implemented because current
production tools such as numerical control make it
possible to manufacture structures formed by an un-
limited number of different elements. Nevertheless,
non-standard constructive elements manufactured
specifically for a particular design are hardly useful in
other configurations. This paper suggested going in
the opposite direction, and follow the guidelines of
combinatorial design (Sánchez 2016).
A virtual model was proposed to design free-
form branched structures with a reduced number of
unique elements, based on the work of Busch (2011).
An algorithm was programmed and applied geomet-
ric constraints to the DLA model permitting assem-
bly based on discrete units. As a stochastic pro-
cess, the DLA was subject to a succession of random
events that resulted in a different form each time. In
this context, the algorithm has been shown to be ef-
fective in preventing incompatibilities, avoiding the
self-intersection of the structure’s bars. Extra ele-
ments were incorporated that reinforced the stabil-
ity of the DLA clusters, inspired by strategies found
in plants. This constitutes a novelty. The benefits of
these changes in the topology of branched structures
can be inferred from the work developed by Klemmt
(2014).
The work presented in this paper constitutes a
first approach to the application of inosculation topo-
logical strategies, and it can be understood as a fer-
tile field of research. As future work, the capacity of
cooperation between inosculated artificial trees and
other structural systems can be explored. The as-
sembly of “Árbol de la lluvia” installation has demon-
strated the potential of combinatorial design, simpli-
fying the manufacture process and the prevention of
mistakes associated to multiple construction compo-
nents.This experience invites us to speculate on what
other typologies of elements, small enough to be 3D
printed, deserve to be considered from this philoso-
phy of combinatorial design.
And finally, what kind of architecture is ap-
proachable thanks to the performative condition of
the virtual and physical model described in this pa-
per? Probably, one in which the contingency and
biomimetic predominate over the cartesian and pure
formal; one that tries to learn from intelligence of col-
lective behaviours (agents); one ready to imbricate
bodies, machines, code, discourses and space.
ACKNOWLEDGEMENTS
The “Árbol de la lluvia” was performed by students
from the University of Cuenca (Ecuador) and Medellin
(Colombia). The initiative took place within the
framework of the 7th International Seminar on the
Representation of Projects dedicated to Landscape
Perception at the University of Medellin. The authors
would like to thank Graphic Methods, Theory and De-
sign Department in University of Alicante, to Escuela
de Medios in Facultad de Arquitectura and to Fab
Lab team in Universidad Nacional de Colombia Sede
Medellín for hosting and funding the workshop.
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