Algorithmic logic

The self-organisational principle that forms the basis of the method Self-organising Bezier Curves corresponds with some of the investigations concerning self-organisation, described by Paul Coates.¹ In some of his examples, very basic agents, represented by a point in a two-dimensional space, self-organise to form geometric patterns of varying complexity. The basic behaviours that force the agents to shift position are repulsion and attraction. For instance, when repelled by a single point with a certain force, the agents move away. If the agents are simultaneously attracted to the same point,

1 Paul Coates, Programming Architecture, Routledge, Oxon, 2010, page 13.

Figure 1: Computer simulation of E.

Coli bacteria. Eshel Ben-Jacob.

as carried out by Philippe Block and colleagues.¹⁰ However, the examples in this chapter help to raise relevant topics on different levels. It is illustrated how physical simulation can be an integrated part of the design process, and how generative techniques can help to establish feedback loops between construction aspects and design intentions. A new sustainable technique for concrete casting was implemented in this case study through the use of digital production. Subsequently, material properties have influenced the method development. Both examples demonstrate how an iterative collaboration can inform the design process, and improve the result.

These methods are concerned with morphology based on a priori topology. Self-organised topology with respect to the overall form is discussed in Chapter 4. First, we take a look at self-organised pattern distribution.

10 L. Lachauer, m. Rippmann, P. Block, ‘Form Finding to Fabrication: A digital design process for masonry vaults’, Proceedings of the International As-sociation for Shell and Spatial Structures Symposium 2010, Shanghai

Figure 2: Patterns formed with self-organising agents.

Figure 4: Self-organising agents form-ing a Voronoi pattern. Paul Coates.

they will begin to form a circle and remain at the perimeter where the attraction overrules the repulsion. This basic behaviour is also used in scientific algorithms that simulate bacterial behaviour. An example is illustrated in Figure 1, where the behaviour of E. Coli bacteria has been simulated by a team led by Eshel Ben-Jacob at Tel Aviv University.² The model simulates how complex radial patterns emerge from basic behavioural rules, reproducing patterns recognised in natural bacteria colonies. Figure 2 illustrates a series of experiments where agents self-organise and form complex patterns, as performed by the author. The agents are governed by rules of repulsion and attraction. All patterns were generated in a three-dimensional environment, influenced by gravity and a ground plane.

The two results at the top of the chart are two-dimensional, whereas the others outline spatial formations. The simple mechanisms of repulsion and attraction are sufficient to generate order on a level that is different to the rule set that governs the form-generating process.

In this sense, the pattern formation is an emergent property. Within a system of fixed agents, representing centre points in cells and a logic based only on repulsion and attraction, it is possible to make a population of agents that self-organise into a Voronoi pattern. A two-dimensional example by Paul Coates is shown in Figure 4, and as he documents in Programming Architecture, the code for generating a self-organised Voronoi pattern is far shorter and less complicated than a method constructed from computational geometry. ³

As indicated, the method developed for constructing a surface pattern of self-organising Bezier curves is based on these basic behaviours. Figure 5 shows how the first step of the process manages the distribution of a field of centre points, only from a rule of repulsion. The image on the left shows the initial disorganised field of circles. The centre of the image demonstrates how after 25 generations the circles have moved away from their neighbours, but not far enough to create the specified distance between them. The

2 Phillip Ball, Shapes, Oxford University Press, Oxford, 2009, page 147.

3 Coates, op. cit., page 15.

Figure 3: Cress seeds on wet cotton self-organise into a homogenous pattern.

Figure 5: Self-organising circles.First step of the method Random Bezier Curves.

image on the right shows the pattern in an equilibrium state after 200 generations. The circles have stopped moving because they all have the specified minimum distance to their neighbours. The result is a homogenous pattern. The mechanism can be compared with a simple self-organising behaviour that occurs when cress seed are placed on wet cotton. The seeds form a gel, which forces them to rotate and distribute evenly, as shown in Figure 3. These patterns are aperiodic since there is no precise repetition. Furthermore, they display a nonlinear form of organisation because the position of each element is the result of continuous local interactions. Each element cannot be positioned independently from the adjacent parts, but only through iterative negotiations between the element and its immediate neighbours. When one element is moved, it affects the entire system. In a sense, the field can be compared with a cellular automata, where the state of each cell depends on the current state of its neighbours.⁴ The difference here is that the state of the cell is represented through its positioning, and more importantly, the relations between cells are not a priori, but dynamic.

Figure 6 shows a more complex type of simulation directly related to biology. Here, the behaviour of ‘true slime mould’ Physarum polycephalum is simulated through the use of a dynamic agent based system. This type of slime mould is a single celled organism that can form complex structures in order to distribute nutrients and to find food sources. As Jeff Jones explains, ‘the true slime mould Physarum polycephalum, exhibits a very wide repertoire of pattern formation behaviours used for growth, movement, food foraging, nutrient transport, hazard avoidance, and shape maintenance.’⁵ The behaviour of slime mould, Physarum in particular, is a well-studied phenomenon, and has demonstrated how complex transportation systems can be simulated through experiments with slime mould.

The agent-based simulations are driven by basic behavioural rules, and the pattern formations are generated spontaneously, that is, they emerge from the agent’s interactions. The agents represent particles of the Physarum plasmodium gel-sol structure, and move forward or randomly change direction, depending on a ‘chemo-attractant gradient’. Essentially, local interactions within the organism result in gradually changing formations that are optimised in terms of supporting the survival of the organism. In other words, this is an explicit example of self-organisation. As the figures of simulations demonstrate, behavioural rules are sufficient to generate a variety

4 Wolfram, Stephen, A New Kind of Science, Wolfram Media Inc, USA, 2002, page 28..

5 Jeff Jones, ‘Characteristics of Pattern Formation and Evolution in Approxima-tions of Physarum Transport Networks’, Artificial Life, Spring 2010, Vol. 16, No. 2 , pages 127-153.

Figure 6: Simulation of behaviour of

‘true slime mold’ Physarum. Differ-ent populations size. Illustration: Jeff Jones.

of complex patterns.

The method Self-organising Bezier Curves was not developed for creating patterns that resemble slime mould patterns.

Interestingly, the appearance of the patterns in Figure 6 and those generated with Bezier curves in Figure 7 share similarities. However, there is a fundamental difference in the way they are generated. The behavioural rules that direct the self-organising curves are unrelated to natural phenomena but based on geometric relations. With regards to the slime mould simulation, dark holes occur in passive areas, surrounded by the dynamic structure of the slime mould organism.

With regards to the Bezier Curves, neither the interior of the curves or the area between them is ‘active.’ Only the borders that separate the two area types is generated and manipulated. As described in detail in Chapter 8.2, the curves are inscribed in polygons and

Figure 7: Self-organising Bezier curves. Top: Different adjustment of distance parameter. Below: shapes grow from centre points outwards.

Left: underlying polygonal geometry.

Right: Resultant Bezier curves.

Figure 8: Self-organising Bezier Curves, volumetric geometry. Left:

CNC milled sample of three-dimen-sional pattern. Right: Rendering of adaptable surface pattern.

are constructed from straight lines. The pattern formation emerges through an interaction between the polygon vertices and the lines connecting them. Figure 7 shows the underlying polygon geometry and the visualised pattern of the Bezier curves. The illustrations also show how the shapes are initially formed near the centre of the polygon, and then grow outwards until a certain distance from their neighbouring shapes is reached. The self-organisation takes place in two steps. First, distribution of the circles, or centre points occurs.

Then follows generation and expansion of the polygons and Bezier curves. In order to address the issue of surface depth, the different states of curve generation are related to different depths. The curves generated initially are placed on the interior of the surface thickness, and the proceeding curves are placed on the exterior. Subsequently, they are connected by new surfaces forming the inner sides of the openings. On the left-hand-side of Figure 8, a CNC milled sample of the ‘deep’ pattern is shown. The black background is seen through the penetrated surface. On the right-hand-side, an example of how the method was further developed towards a self-organising adaptive facade system is illustrated. In Figure 8, the orientation of the surface affects the distance between the curved penetrations, the initial size of the polygons, and how far they can expand. As such, the facade pattern is nonlinear, self-organising and adaptive.