Controlling Chaos: a Simple Deterministic Program for Drawing Complex Organic Shapes
Kevin McGuire, MCS
Object Technology International
Email: KMcGuire@acm.org
Abstract
It is difficult
and frustrating to create complex organic shapes using the current set of
computer graphic programs. One reason is because the geometry of nature is
different from that of our tools. Its self-similarity and fine detail are
derived from growth processes that are very different from the working process
imposed by drawing programs. This mismatch makes it difficult to create natural
looking artifacts.
Drawing
programs have a limited palette based on Euclidean geometry. It is difficult to draw clouds, rivers, and
rocks because they are not lines or circles. Paint programs provide interesting
filters and effects, but require great skill and effort. Always, the artist
must arduously manage the details. This limits the artist’s expressive power.
Ideally, the
artist should have macroscopic control over the creation while leaving the
computer to manage the microscopic details.
For the results to be reproducible, the system should be deterministic.
For it to be expressive there should be a cause-effect relationship between the
actions in the program and change in the resulting picture. For it to feel organic, the details should
be rich, consistent and varied, cohesive but not repetitious; it would be
fitting if the way we drew was more closely related to the way things grew.
We present a
simple drawing program which provides this mixture of macroscopic control with
free microscopic detail. Through use of
an accretion growth model, the artist controls large scale structure while
varied details emerge naturally. Its algorithms are simple and deterministic,
so its results are predictable and reproducible. The overall resulting
structure can be anticipated, but it can also surprise. Despite its simplicity,
it has been used to generate a rich assortment of complex organic looking
pictures.
1. Introduction: Our Cold Geometry
“Why is geometry often described as “cold” and “dry”? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightening travel in a straight line.
More generally, I claim that many patterns of Nature are so irregular and fragmented, that, compared to Euclid… Nature exhibits not simply a higher degree but an altogether different level of complexity.
The existance of these patterns challenges us to study those forms that Euclid leaves aside as “formless”, to investigate the morphology of the “amorphous”. [1]
In 1983, Mandelbrot wrote about a world of fascinating and rich behaviour, the visualization of which we know as fractal geometry (fractals). The “cold geometry” that Mandelbrot referred to above is Euclidean geometry, where a line describes the exact shortest path between two points, where a circle is a perfect infinity of small lines.
These pure forms are only idealizations of what we find in nature. In reality there is no perfectly straight line. The qualities we enjoy in nature, its variety and richness of shapes, structures and details, are extremely difficult to express using Euclidean geometry. As a result, there is a fundamental mismatch between Euclidean geometry and the geometry of nature. This makes the former feel cold, overly precise, the opposite of a walk in the woods, or a summer day on a beach.
Mandelbrot clearly demonstrated that completely new forms exist in an alternative geometry, one that was more closely linked to the behaviour of the physical world. Yet our design tools are trapped in a stifling Euclidean box. This reduces the breadth of our design language by precluding the creation of organic form. In turn, our built environment ends up being sterile and uncomfortable, unrelated to our ancestral natural home.
2. The Quality of Nature
What are the qualities of nature?
Nature is messy, complicated. While you can approximate a circle with an infinite number of straight lines, that’s only true in calculus. Nature doesn’t actually do that; you won’t find a perfect circle. A stone buffed by centuries of ocean currents is not perfectly smooth. Even raindrops are not round.
Nature is detailed. These details are difficult to express in Euclidean geometry. Expressing something as complex as a branching structure requires the assemblage of a very large number of line segments. Similarly, attempting to render a cloud using intersecting spheres would be extremely difficult, if not futile.
Yet nature has simplicity, structure. Branching patterns are structures; they consist of recursively self-similar patterns repeated over different scales. Furthermore, the same patterns emerge in seemingly unrelated places: the branching of a tree is essentially the same as the branching of lungs because they both solve the same problem [2].
Thus nature does have a geometry, we just don’t understand it well enough. We are stuck in an Euclidean mind set, making it difficult to imagine anything else.
3. Qualities of Our Ideal Drawing Program
What qualities should our ideal drawing program have in order to attain these goals? Most importantly, it should be natural and expressive.
By natural, we mean that the images should have those non-Euclidean qualities mentioned above: the details should be rich, consistent and varied, cohesive, structured without monotonous repetition.
By expressive, we mean that the tool should provide control over the macroscopic structure. Control is important so that you can get what you intended. It follows that there must be a predictable cause and effect relationship between your actions as the artist and the resulting picture.
In addition, the process must be reproducible so that you can explore in a controlled and consistent manner. Thus the system should be deterministic (no random number generators).
Finally, we want the tool itself to fill in the details, thereby freeing the artist to focus on the overall shape. We should leave the computer to manage the complexity. Hopefully richness and variety will result from the interactions in the generative process. This promotes creativity by reducing drudgery.
4. Problems with Current Tools
We now analyze the state of popular computer assisted drawing technology and show why they do not meet our criteria.
4.1 The Geometry of Nature is Different than that of Our Tools
Current drawing and paint programs are almost all based on Euclidean geometry. Such perfect shapes do not correspond to what we see in nature. Thus our ability to render natural looking scenes, and indeed our ability to design and mass-produce natural looking artifacts, is greatly limited by this geometric incompatibility.
We can divide popular tooling into two categories, drawing tools and paint tools.
Drawing and architectural drafting programs provide a limited palette of shapes typically consisting of lines, circles, squares, ellipses, splines, etc. Since natural phenomena do not follow this geometry, organic form is very difficult to express.
Paint programs on the other hand provide a variety of interesting filters and effects. You can do almost anything if you are good enough, and patient enough. Unfortunately, the skill involved is tremendous, the effort for achieving the required detail arduous. While some artists can become adept at using them to create organic form, it is really more a statement about their skill as artists in spite of the tooling, rather than a statement of the expressiveness of the tool as a medium.
Also of importance is the role of modern manufacturing methods and the loss of the craftsman’s hand in softening our geometry. However, this topic is beyond the scope of this paper.
4.2 Automation vs. Innovation
Our tooling is failing us because we have simply automated that which we were already doing on a drafting table.
Consider the evolution of word processing programs. At first, they were only direct automations of the typewriter. Over time however a progression occurs whereby we move from simple automation to paradigm shifting innovation. Word processors have been tailored to the true problem of document creation and bare little resemblance to their mechanical ancestor.
This in turn has affected the way we express our ideas.
Our computer design tools have not made this same transition. All we’ve done is taken the drafting board and recreated it on the computer (automation), but stopped there (no innovation). The geometry of the drafting table is based on a mathematics that can be easily managed by hand calculation. Despite the incredible computational power now available to us, we retain the same geometry and hence the same design language. Like the QUERTY keyboard[2], our design tools continue to be constrained by an outdated problem. Although the engineering task has seen advances in the form of physical modelling and finite element analysis, the creative design task has not.
4.3 There is Nothing Under the Hood
Drawing and painting programs do not model the relationship between the elements of the drawing, nor between the elements and the whole. Put another way, the structure of the image does not bear relation to the structure of the thing being drawn. For example, you could draw a tree using multiple line segments, but it would be difficult to then make the tree bend in the wind: you would likely have to reposition most of the line segments.
The problem is that the process of creating the image is unrelated to the way things grow. Thus you can’t grow another one a little different from the first. In our example the picture’s treeness is wholly a function of the skill of the artist. Changing the image requires the reapplication of this skill down to the minutiae.
5. Our Build Environment is Uncomfortable
The language of our design affects the quality of the outcome [3]. So too does the language of our tooling. It is difficult to imagine a sensuously curved Gaudi doorway resulting from popular design technology. The difficulty in expressing such a complex 3D curved form discourages both its ideation and its execution. As evidence, one need only walk down the average modern city street.
It is our belief that the unnaturalness of the geometry of our tools has created a built environment that is uncomfortable and unsatisfying.
We are not kin to this geometry. It is a comparatively recent phenomena in our evolutionary life as a species. It suites the needs of our design and manufacturing technology, but not the needs of we as human animals. There is no mistaking a city for a forest. Our concession to nature consists of the planting of the occasional tree, the well-groomed patch of grass. We have dispossessed ourselves of our heritage.
6. Algorithmic Approaches
Expanding our overview outside the graphic tools domain, we now analyze a few of the current approaches to creating complex visual forms.
Fractals [1,4] were for a time a dominant source of new visual computer imagery. Although beautiful, they do not help us much as a design tool. As an artist, it is difficult to express an idea using a fractal because you really can’t do much with them. You are basically limited to exploring the fractal landscape, zooming and panning, and changing colour mappings. You can’t design a fractal to look the way you want. You cannot modify its details or its shape, it is what it is. If you are a clever mathematician, perhaps you can devise new recurrence relations to map out new fractal spaces, but unfortunately most of us are not clever mathematicians.
Genetic algorithms [5] on the other hand are great for generating scores of variations. Their usefulness however is intrinsically tied to our ability to come up with the right fitness function. Without it, we are left with the task of manually culling a large population of possibilities. Since fitness functions are notoriously tricky to encode, our ability to design with them is limited. This reduces their expressiveness.
GA’s on their own do not suggest a geometry for the forms they generate. Rather, this is a function of what the gene was designed to encode and what process is used to generate the outcome. Thus they do not intrinsically help with this aspect of organic shape creation. However, it is possible that GA’s make it easier to generate more complex forms, perhaps bringing about greater organic detail in our artifacts.
Finally, of great interest in organic form creation are the Evolutionary Art work in [6], and the graph grammar plant generation in [7]. Each use rewriting systems to generate their images. In [7], one is primarily limited to branching structures, restricting expressiveness. In both, it’s not clear how one arrives at the image one intended.
7. The Splat Generator
We now introduce the Splat shape generator.
Splat is based on an accretion growth model. Accretion is the mechanism by which coral and other structures grow: the accumulation of micro debris, each particle placed by the Brownian turbulent motion of the water. This is similar to a process called diffusion-limited aggregation [8]. With it, natural structures such as metal leaf patterns have been generated. However, diffusion-limited aggregation’s drawback is that you cannot control the actual growth of the shape. This is due to its reliance on random Brownian motion of the particles.
In order to control the accumulation dynamics, we introduced a gravitational model to our accretion growth. This allows one to shape the path that the particles travel, which allows one to control where the particles accumulate, which in turn controls the final shape produced.
Our system consists of the following elements that can be placed by the user:
· Point mass particles capable of movement.
· Emitters which release particles into the system with a given speed and direction.
· Fixed location gravitational masses to attract the point masses. Negative masses (repellers) can also be used.
The process of shape generation proceeds as follows:
1. The artist places at least one emitter and one fixed mass on a canvas.
2. The emitter ejects a particle which moves under the gravitational pull of the fixed mass(es).
3. When a point mass hits a fixed mass, it sticks to it.
4. The emitter changes the initial direction and speed, fires another particle. Eventually the particle comes in contact with either the fixed mass or a previously (now motionless) particle. As with accretion growth, the shape eventually emerges from these accumulations.
Figure 1 demonstrates one result. The circles at the ends of the elliptical growths are emitters and the squares are repellers. The attracting masses are buried within the particles and are no longer distinguishable. Figure 2 demonstrates the result of a less symmetric configuration. Further demonstrations can be seen in Figure 3 and Figure 4.
Figure 1: Eight Emitter Generation
Figure 2: Less Symmetric Generation
Figure 3: “Whale Bone”
Figure 4: “Wreath”
8. Control
Control is important for intentionality: the problem of producing a result that somehow matches what one intended. In Splat, the artist is able to control the process of his creation in the following manner:
· Number and placement of the masses, and their weight. This affects the envelope of the trajectories. Masses can be given a negative weight, which makes them repel particles instead of attract them. This allows further shaping of the space.
· Initial velocity of the particles. This also affects the envelope of the trajectories.
· Number and placement of the emitters. Each emitter tends to correspond to a pair of branches or growth areas. A symmetric placement of emitters will produce a symmetric picture.
· Sequencing the firing of emitters.
· Algorithm determining next initial direction/speed of the particle. This effects the pattern of particle placement. For example, the result can appear stockier, or bushier.
The net effect of these various control axes is the ability to generate a variety of unique pictures. The combinations can be quite subtle.
9. Controlled Chaos: Analysis
The resulting pictures exhibit a curious blend of macroscopic structure and seemingly random detail. Due to the deterministic nature of our system, one would assume that an exactly symmetric initial configuration of say two emitters and one mass (Figure 5) would produce an exactly symmetric picture. Surprisingly, this was not observed. For example, Figure 6 displays numerous asymmetries when comparing left and right branch ends.
Figure 5: Macroscopic Symmetry
Figure 6: Microscopic Assymetry
The following are some factors we propose to explain this behaviour:
· Sensitivity to initial conditions: the hallmark of chaotic systems, sensitivity to initial conditions describes the phenomena by which two near identical systems diverge over time due to the cumulative effect of their minute differences. Our system exhibits analogous dynamics by virtue of each particle emission changing the state of the system in a small way.
· Floating-point inaccuracy: floating-point numbers are used to compute the net gravitational force on a particle, which in turn determines its speed and direction. It is likely that inaccuracy in the representation of floating-point numbers produces slight differences in otherwise perfectly symmetrical configurations.
Particles can take long circuitous paths before encountering a stationary mass (Figure 7). Even for perfectly symmetrical initial conditions, due to the above factors it can and will occur that a path that was available to one particle would now no longer be available to its mirror twin. Since particles are not fired in parallel, the effect of the first one changes the state of the world for the second, eliminating the initial symmetry in a small but potentially catastrophic manner. Although macroscopic symmetry may still exist, it no longer will microscopically. These small changes build up over time, producing small scale variety.
Dramatic loss of macroscopic symmetry can also occur. In some cases, the particle can block a large series of paths, stalling the growth of its symmetric twin.
Between these two extremes, small scale dissimilarity and large scale asymmetry, lies a large solution space filled with surprise and unpredictability. This surprise is felt to be a healthy quality for a creative process.
10. Conclusions
We’ve achieved many of our initial goals. We’ve produced organic looking shapes by modifying a simple natural growth model. They exhibit macroscopic symmetries yet contain a richness of fine scale variety. This enhances the organic quality by providing structure without the sterile exactitude of repetition. The system is deterministic with several axis of control. This is important for expressiveness and for intentionality.
The system demonstrates a new language of geometry and a novel avenue for expressing organic shapes. Remarkably, random number generators weren’t required. This suggests an alternative approach to variety, whereby systems have sufficient simplicity to be controllable but sufficient complex interactions to produce variety.
The system does have several shortcomings. For one, although over time one is able to develop an intuition about how to craft a picture, it is nonetheless daunting to predict the outcome. Subtle variations can lead to considerably different pictures. Second, the vocabulary of shapes appears somewhat limited. Finally, the time required to produce a given picture is somewhat slow (on the order of several minutes), which harms exploration.
For these reasons, we believe the expressiveness of the system is still less than we desire. However, we consider it to be a demonstration of an idea: that we may find new organic form generation, and new geometries, by looking to nature to teach us how to create.
11. Reference
[1] Mandelbrot, Benoit, The Fractal Geometry of Nature, W.H. Freeman and Co., 1983.
[2] Stevens, Peter S, Patterns in Nature, Little, Brown, and Co, 1974.
[3] Alexander, Christopher, The Timeless Way of Building, Oxford University Press, 1979.
[4] Peitgen, Richter, The Beauty of Fractals, Springer-Verlag, 1986.
[5] Holland, J.H., Adaptation in Natural and Artificial Systems, University of Michigan Press, 1975.
[6] Todd, Steven and Latham, William, Evolutionary Art and Computers, Academic Press, 1992.
[7] Prusinkiewicz, Lindenmayer, The Algorithmic Beauty of Plants, Springer-Verlag, 1990.
[8] Feder, Jens, Fractals, Plenum Press, 1988.