Evolving Fractal Drawings


Jon Bird, PhD

Centre for Computational Neuroscience and Robotics, University of Sussex, U.K.



Dustin Stokes, PhD

Centre for Research in Cognitive Science, University of Sussex, U.K.





We are using an evolutionary robotics approach to generate minimal models of creativity. Our preliminary simulation results demonstrate that this methodology can produce robots that mark their environments and interact with the lines that they have made. These simulated robots possess a ‘no strings attached’ form of agency and some of their behaviour can be described as novel relative to their individual behavioural histories and to the behaviours of other members of the evolving population.  Arguably, they thus satisfy two conditions necessary for creativity: agency and novelty.  Open questions remain:  Are the robots’ behaviours creative?  Will they become creative as we incrementally increase the complexity of the robot controllers?  Can their mark making be classified as drawing?

A number of common criticisms of the project fall under the broad category of value.  In the preliminary model the robots neither evaluate the process nor the product of mark making.  The robots can only detect the presence of a mark in a 2mm x 2mm region underneath them. How could an agent, from such a ‘myopic’ viewpoint, have any sense of the global pattern of the marks made across a large arena?  And how could such agents have any sense of when to stop?  Finally, the present results fall short of our artistic goal of producing a gallery exhibit, since currently the products of these robots are unlikely to engage audiences (without considerable knowledge of the methodology involved).

Here we outline a fractal framework that addresses each of these concerns: our robots will be endowed with a ‘fractal detector’; and they will acquire fitness for making marks with a self-similar structure.  The robots will be able to interact with their products in a way that involves a kind of judgment of the product as it is being produced.  Although their viewpoint will still be limited to a local region, the robots will be able to generate a coherent self-similar pattern across the arena without requiring a global perspective mechanism, such as a bird’s-eye view camera or topographic memory.  The framework will also provide a natural finishing criterion: once a self-similar pattern covers the arena the robots will no longer make marks.  Finally, fractal images do engage people.  And knowledge that these agents have a ‘fractal preference’ should enhance that engagement, since audiences will be able to watch the development of a global self-similar pattern through the simultaneous mark making of a group of robots.   



Minimal Robotic Creativity        


Philosophical analysis of creativity does not come easy.  Evolving artificial agency comes no easier.  We are doing both at once. Our research team thus comprises artificial life researchers, philosophers, cognitive scientists, and artists, all of us motivated to evolve some kind of creative behaviour.  We here outline a theoretical framework for extending our evolutionary robotics (ER) approach to generating models of minimal creativity. The focus is on the role of evaluation in creativity and how that role might be accommodated in our robotic models using a fractal framework[1].


Philosophical analysis

Our assumptions about creativity are minimal.  We start with only two conditions, each of them necessary but non-sufficient, for creativity.  A creative behaviour must result from agency.  Agency requires autonomy.  Our sense of the term does not require, as the philosophical sense does, intentionality, deliberation, or cognition.  It simply requires behaviour that is not imposed by an external agent or programmer.  A remote controlled robot would thus not qualify, while many of the systems that populate evolutionary robotics would.  We sometimes refer to this as ‘no strings attached agency.’

Intuitions also tell us that novelty is a condition for creativity: creative artefacts or processes are novel ones.  Here too we err towards barely minimal assumptions.  Following Boden [3], we distinguish absolute and relative forms of novelty.  As Boden argues, relative novelty is sometimes as theoretically interesting as absolute novelty.  For example, one may have a novel thought which, although others have had it before, is novel relative to one’s own mind.  We broaden the latter—which is what Boden calls ‘psychological’ novelty—to include non-cognitive behaviours.  This is done in two ways.  A behaviour of some agent R may be novel relative to the behavioural history of R.  Or a behaviour of some agent R may be novel relative to a population of which R is a member.  Call the first ‘individual-relative novelty’; call the second ‘population-relative novelty.’

The choice for conceptualizing agency and novelty so thinly is motivated both by our particular research goals and a general methodological assumption we share with much of cognitive science. Our interest is to see what lessons can be learned about creativity and cognition though the use of synthetic, bottom-up modelling techniques. We may, after all, be working with overly thin notions, but the working supposition that weaker instances of agency and novelty may be near what’s necessary for creative behaviour enables fruitful experimentation and hypothesis generation.  This is often how cognitive scientists begin, that is, by asking what might some minimal conditions be for some phenomenon, and what can we learn from attempting to satisfy just those conditions?

The agency and novelty conditions give us two necessary conditions for creativity.  The weakness of this definition is easy to see.  I can right now place my head in the oven and utter ‘We need milk, butter, and bread.’  This is novel behaviour for me, and indeed behaviour that depends upon my autonomy.  But would anyone count it creative?  Novelty and agency, even of a very rich, cognitive sort, are thus not enough for creativity.  We recognize that, and the point of this paper and the project stage it outlines is to determine what is enough, even minimally.  Nonetheless, we have to this point been proceeding with this incomplete definition in hand: creativity requires agency and novelty.  So if we are going to build creative systems, we at least have to build systems that possess these two properties.  This, as it turns out, is hard enough as a start.


Evolutionary robotics methodology

We use an evolutionary robotics methodology for two main reasons. First, it is an established technique for developing situated robot controllers (and to a limited extent their morphology). Second, ER can potentially generate models of minimal creativity that overcome the limitations of our understanding of creativity. This is possible because we aim to minimise the constraints that we place on the controller architecture by artificially evolving artificial neural network (ANN) controllers from low-level primitives (processing units and the connections between them). The evolutionary process is also free to exploit any constraints that arise from the interaction of the robot and the environment that may not be apparent to an experimenter.  ER can therefore generate controller architectures to solve problems that are not well-defined.


Initially, we have carried out our experiments in simulation using a model of a Khepera robot based on empirical measurements, a standard ER platform (Figure 1a). This approach has advantages over doing evolution on physical robots: it is far quicker; and it avoids damage to the robots during early evolutionary stages when the controllers often crash into the arena walls. Bill Bigge, a researcher on the DrawBots project, is developing a custom robot for testing our controllers in the real world (Figure 1b).


Figure 1: a) a Khepera robot modelled in our initial simulation experiments;

b) prototype DrawBot for testing controllers in the real world.

In simulation, each robot controller consists of seven sensors (six frontal IR sensors and one line detector positioned under the robot) and six motor neurons (a pair of motor neurons controlling the left wheel, right wheel and the position of the pen - up or down). At each time step in the simulation, the most strongly activated neuron of each pair controls its associated actuator. Each of the seven sensors connects to each of the six motor neurons. A genetic algorithm is used to determine the strength of each of these connections and the bias of each of the motor neurons.

An initial population of 100  robots controllers (phenotypes) is encoded as  a string of 0s and 1s (genotypes). Every generation each genotype is decoded and the performance of the robot controller is tested and assigned a fitness value. A new generation of genotypes is then generated by randomly selecting genotypes, with a bias towards fitter ones, and mutating them (flipping 0s to 1s or 1s to 0s with a probability of 0.01 per gene). Our experiments were carried out for 600 generations.



Figure 2:  a high fitness individual from an initial experiment – it does an initial loop of the arena with its pen down and on the second loop makes line segments parallel to the line it initially made.


We aim for fitness functions that minimise our influence on the resulting robot behaviour. We do not specify the types of marks that a robot should make, rather, we reward controllers that correlate the changes in state of their line detector and pen position. For example, if a line is detected and the robot’s pen is then raised or lowered within a short time window, the robot accumulates fitness. This fitness function resulted in robots that followed the walls and made marks around the edge of the arena (Figure 2). When the fitness function also rewards robots for making marks over the whole area of the arena then different behaviours evolve (Figure 3) and robots turn away from the walls at angles and mark the central parts of the arena as well. In all our experiments crashing into walls is implicitly penalised by stopping the evaluation and thereby giving the robots less time to accumulate fitness. It is important to note that although we, via the fitness function, evaluate the mark making behaviour of the robots, the robots themselves do not assess the marks that they have made.

Figure 3: when the fitness function rewards making marks over the whole arena, the robots no longer follow the walls but turn away from them at angles and mark more central regions.


On What’s Missing: Value

As we stated at the outset, our working assumptions about creativity are minimal; we do not purport to have offered a complete analysis of creativity nor to have evolved any richly creative behaviour.  “Fair enough”, one might respond, “but can you evolve rich creativity?  Creative behaviours involve evaluation, and creative artefacts, for example, artworks, are things we value.  So if your research does not yet address these facts, can it ever address them?”  We give two responses to this general worry.  One, we allow that some kind of value condition may be what’s needed for a complete analysis of creativity.  That is, perhaps value plus agency and novelty will get you creativity.  We are willing to take this as a plausible suggestion, without committing to the claim that the conjunction of these three properties is sufficient.  Rather, we merely accept that value is a good general area to mine in the search for richer models of creativity.  Our second response is more straightforward.  We believe we can address the concerns about value by extending our robotics framework. We now distinguish four such concerns.     


1.  The non-evaluative process worry

If creativity is a process, that process must involve some kind of evaluation: the agent needs to make judgments about the behaviours it is performing, and those judgments must in turn play some important role in the dynamic process of creation.  There are many ways of developing this thought further, but the basic idea is just that agents must make choices of some sort in acting creatively, preferring one option over another, this action over that one, and so on.  Without this evaluative feature of the process, we seem to have purely reactive behaviour.  Our robots seem to suffer from this very problem.  There is nothing like evaluation in their mark-making behaviour.  They simply react in a way that at most depends upon sensory motor morphology, the arena boundaries, previous engagements with that environment, and (if the agent is of a later generation) the performance of agents in previous generations.  Nowhere in that causal chain is there anything that looks like judgment or evaluation.  The robot’s mark making processes are thus non-evaluative.


2. The myopic worry

This worry is intimately tied with the one just canvassed.  In fact, it is partly explanatory of the non-evaluative problem.  As a simple feature of their physical structure, our robots can only “see” marks that are underneath their 2mm x 2mm line detector.  Their viewpoint is thus myopic. This is problematic if we think of artistic creation (or, analogously, of creativity in non-artistic realms).  A painter, for example, will often focus on a small component of her painting, but will return to the larger work of which that component is just one part.  Without such capacity, we would never get pictorial representation.  And the point generalizes to non-representational paintings (and artworks generally): Rothko, Mondrian, Pollock and the like took a step back from their work to “see the picture” even if it wasn’t picturing anything.  This is part of the creative process and, what’s more, it is essential to the artist’s evaluation of her own work.  Without a more global perspective of the work and how its parts constitute the whole, the artist has little to evaluate, scrutinize, and change.  Our robots are stuck with a local perspective of the marking surface, and no memory to conjoin each of these perspectives for something more global.  This partly explains why the process is non-evaluative.  It is blocked from being evaluative, since if you don’t see the whole picture, you certainly cannot evaluate it. 


3.  The never ending worry

If the processes of our agents have an end point, it is at best arbitrary, unrelated to whatever marks have been made on the marking surface.  This is at tension with how we think about artworks.  Artworks, excepting a very small number of cases, are spatially and/or temporally bound.  They have distinct stopping points.  We can see that a portrait or sculpture or film is finished.  We can hear when a musical performance or recording concludes.  And the artists in question make this decision: they decide when the work is done and where its spatial and temporal boundaries lie.  Our robots have no such stopping mechanism.  Or better, whatever stops them—namely, either they crash into the arena boundaries, or the trial comes to an end when they complete a specified number of time steps—it has nothing to do with there being some product which that agent decides is finished.  This is a problem.  We are never going to get gallery displayable images out of these systems if there is no mechanism which encourages a non-trivial stopping point.


4. The aesthetic merit worry, or the “You say a robot did that?” worry

In addition to our motivations to learn about creative processes, members of our research team would ultimately like some results that can be exhibited.  More precisely, we would like some results worthy of aesthetic appreciation, where that appreciation does not stand or fall with knowledge of the robotic systems that produced those results.  At present, our results would at best warrant appreciation of the latter sort.  That is, perhaps if one were informed about the artificial life and robotics techniques responsible for evolving the mark structures, one might attribute some aesthetic merit to those patterns.  Perhaps.  But one is very unlikely, if lacking such knowledge, to look at Figures 2 and 3 and say “how interesting”, “lovely”, “beautiful” and so on.  The patterns themselves are, quite frankly, not particularly aesthetically interesting.

One might respond, of course, by invoking the same feature of much of modern art.  Contemporary art museums are full of conceptual works, found artefacts, and performances, the appreciation of which requires knowledge of art history and theory.  So if our images require contextual knowledge, that puts them in no worse a position than lots of artworks.  We acknowledge this (and will certainly keep it in our back pocket should such a defence be needed), but we also acknowledge that much of art is not of this sort.  One needn’t know much, if anything, to “just see” the beauty in a Rodin sculpture or a Vermeer painting.  In fact refusing to see the merit in such works would likely indicate that you were either a tasteless fool or an elitist attempting some kind of snobbish irony.  Some artworks, on their own, just are aesthetically valuable.  And although our hopes are humble ones, we would like some mark patterns whose formal properties alone warrant aesthetic appreciation.  So far we are a long way from reaching this goal.

Towards a Richer Robotic Creativity: A Fractal Framework

The above are all significant worries.  They identify a general evaluative constraint on theories of creativity that many seem to endorse.  And they reveal how the current state of our research falls short of that mark.  These challenges are not, however, insurmountable.  We think they can be addressed, and without compromise of our overall theoretical and methodological preference: minimal assumptions and bottom-up modelling. 

One solution, in a word, is fractals.  Fractals, understood broadly, are patterns which display self-similarity at different magnifications [4].  We intend to use them in the following ways.  Endow the agents with a ‘fractal detector.’  Endow the agents further with a ‘fractal preference’, such that they will acquire fitness for making fractal patterns on the arena surface.  In the next section we outline how we plan to evolve robots that make and evaluate fractal patterns. We then discuss how this approach is sufficient to address each of the four value worries discussed above.   

How can a robot measure fractal dimension?





















Figure 4: different degrees of pre-processing of a camera image before an ANN measures its fractal dimension.


There are various options for endowing the robot with a fractal detection capacity. The simplest and most widely used approach for measuring the fractal dimension of a structure is the box counting method. A binary image of the structure is divided into a grid of uniform cell size. The number of cells or boxes in the grid which contain one or more black pixels (assuming the image is black) are counted. The size of the grid cells is varied, generally ranging from larger than 1 pixel to less than the size of the image. For each cell size, the number of cells containing parts of the image is counted. The log of the box size is then plotted against the log of the number of boxes containing part of the image. If an image is fractal then the data points fall on a straight line and the slope of this line gives the fractal dimension. There are other approaches to measuring fractal dimension, such as decomposing the image into its power spectrum. However, because of its conceptual simplicity, our initial approach is to deconstruct the box counting algorithm and consider which steps we will pre-process and which parts we will leave open to the evolutionary algorithm to configure (Figure 4).


The robot will be fitted with a camera and one option is to do no pre-processing on the image and supply the controller with an array of grey level values. It is an extremely challenging task to artificially evolve an ANN to use these raw pixel values to identify fractals.  The ‘no pre-processing’ option appears untenable given the time scale of our project. At the opposite extreme, we could use a box counting algorithm to process the camera image and provide the controller with a hardwired ‘fractal detector’ unit whose activation is 0 if the image is non-fractal or a positive value (< 1.0) that is proportional to the fractal dimension of the image. As we have done all of the processing up front, this approach is open to the criticism that the robots are still not evaluating their mark making.


We are therefore initially implementing a ‘count unit’ approach to provide information about structure in the camera image at different scales. Each unit is associated with a different box size and their activation is dependent on the number of boxes which contain marks. We will explore the effect of using different transfer functions for these units. For example, we could use a logarithmic function, analogous to the box counting algorithm and leave the ANN to compare the activation of the different count units to determine whether the mark structure is fractal.



Steps towards evolving fractal drawing


We want to leave the evolutionary algorithm some freedom in how it uses the camera activation and configures the ANN to detect fractals. First, the less pre-processing we do on the image, the stronger our claims that the robots are determining, to some extent, the evaluation criteria. Second, dynamically estimating the fractal dimension of a changing structure is not a well-defined problem and we are currently unclear about how we should extend our ANN primitives to enable the robot to solve this problem. Unlike most applications of fractal dimension analysis, the robot will be making an estimation of a dynamic structure: it will be both moving across the arena floor and have the ability to change the mark structure with its pen. This is a non-trivial task that is at least approaching the order of complexity of some of the most challenging behavioural tasks that have been accomplished using an ER methodology [6]. One open question concerns the extent to which the controllers will require some form of memory and whether this could be implemented with ANN primitives such as recurrent connections. ER is a discovery methodology that can potentially evolve controllers that can solve this problem.


We plan to carry out a series of experiments of increasing behavioural complexity. Initially we will focus on getting a robot to discriminate between fractal and non-fractal structures. The camera will be pointed forwards so that it can view patterns on the arena walls and robots will gain fitness for staying in a region in front of a wall area that has a fractal pattern; the other walls will have random patterns which have the same pixel density as the fractal pattern[2]. The next step will be to determine whether robots can discriminate between wall patterns with different fractal dimensions. These two experiments will be important to clarify the neural network primitives that are required and to test the sufficiency of the ‘count units’ approach.


Assuming we accomplish the above, the next step is to get robots to draw fractal patterns. In our initial experiments the fitness function will reward the area of arena covered with self-similar mark structures. It is an open question whether the robots have sufficient degrees of freedom to generate fractal marks. It may be that we have to add another degree of freedom to the pen and enable it to move from side to side as well as up and down. Another option that we might have to explore is building reactive drawing behaviours into the ANNs. There are also a number of issues that have to be explored concerning the camera: where should it point – in front of the pen?; what size image should we use?; and over how many scales can we expect the robots to generate self-similar marks?




Even though there are many challenges to be solved before we evolve robots that make self-similar patterns, we are keen to pursue the fractal framework we have outlined as it addresses all four value worries that we described earlier in the paper.


Addressing worries 1 and 2: Fractal evaluation

The behaviour of our artificial agents lacks an evaluative component for a rather simple reason: our agents aren’t looking for particular mark structures. They simply respond to any marks under their line detector.  And given the small size of this sensor (2mm x 2mm) they also have a myopic perspective.


Consider the myopic worry first, since addressing it will contribute to a solution to the non-evaluative process worry. Obvious solutions to the worry might involve incorporating a bird’s eye view camera in the overall system, or endowing the ANN with a topographic memory of some sort.  These may well be viable options.  But they may also be unnecessary if fractal patterns being detected and constructed.  A self-similar pattern can be completed in the agent’s local surface area.  When a fit agent moves across the surface, it will continue to implement this pattern, making marks on parts of the surface which are not self-similar.  These adjustments contribute to the overall pattern of self-similarity, but without the need for any topographic memory or global view of the surface.  In a sense then, by looking for a fractal pattern in any given local region, the agent is working on the bigger picture without having to actually look at the bigger picture and its myopia is thus rendered harmless with respect the worry at hand.


How does all of this help with evaluation?  The proposed framework neutralizes the myopia of our earlier agents not by giving them a global viewpoint, but by taking advantage of the nature of fractal patterns.  If a region of marked surface isn’t self-similar, then a fit agent will detect this and add marks  to make it self-similar. The agent thus has something to look for and a preference for making things that look a certain way.  This capacity is admittedly not a sophisticated aesthetic or artistic one.  But it is an evaluation technique, which results in the agent making choices: it will prefer some marks over others, and will change some and leave others.  Moreover, fractals are a broad enough pattern category that the agents have considerable freedom in the marks they can make.  


Addressing worry 3: Done!

The never ending worry, recall, was that the mark making behaviour of the agents has no non-trivial stopping point: if the agent does stop, it has nothing to do with the completion of some pattern.  The fractal framework makes quick work of this worry: at some point, the arena surface will be with a self-similar pattern and the robot will no longer add any more marks.      


Addressing worry 4: The aesthetic appeal of fractals

People generally like fractals, or at least that is what experimental studies show us [5].  There is a lot to say here, but here are just a few intuitive reasons to think that fractal patterns created by our agents would (self-sufficiently) be aesthetically interesting.  Fractal patterns are detectable.  That is, they are identifiable patterns, and so part of the engagement when viewing them is finding the self-similarity at different magnifications.  Second, the range of self-similar patterns that the robots can produce is potentially very broad and the resulting marks may surprise us. An element of surprise is an aesthetic merit, and thus a potential benefit of the fractal framework.


To be clear, none of this is intended to show that our agents are or would be producing artworks.  What makes something an artwork is an extremely deep and rich issue, and one that likely depends upon a number of factors: context, theory, and artistic intention for starters.  Our robots might be behaving in minimally creative ways and making marks in aesthetically interesting ways, but we remain agnostic on the question of whether they are making art. However, we are confident that the fractal framework that we have outlined in this paper is a promising approach for investigating value issues with minimal models of creativity.


5.1             Acknowledgements

The Computational Intelligence, Creativity and Cognition project is funded by the AHRC and led by Paul Brown in collaboration with Phil Husbands, Margaret Boden and Charlie Gere.




[1] J. Bird, D. Stokes, P. Husbands, P. Brown and B. Bigge, Towards Autonomous Artworks. Leonardo Electronic Almanac, 2007 (in press).


[2] J. Bird and D. Stokes, Evolving Minimally Creative Robots. In S. Colton and A. Pease (Eds.) Proceedings of The Third Joint Workshop on Computational Creativity (ECAI '06), 1 - 5, 2006.


[3] M. A. Boden, The Creative Mind. Routledge, 2004.


[4] B. B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman and Company, 2007.


[5] R. P. Taylor, Fractal expressionism - where art meets science. In J. Casti and A. Karlqvist (Eds.) Art and Complexity, Elsevier Press, 2003.


[6] D.Floreano, P. Husbands, S. Nolfi, Evolutionary Robotics. In B. Sicilianoand  O. Khatib (Eds.) Springer Handbook of Robotics, Chapter 63, Springer, 2007 (in press).


[7] T.M.C. Smith, P. Husbands, A. Philippides and M. O'Shea, Neuronal plasticity and temporal adaptivity: GasNet robot control networks. Adaptive Behavior, 10(3/4), 161-184, 2002.



[1] For a detailed technical description of our experimental set up, please refer to [1] and [2].   



[2] This ER paradigm is based on visual discrimination experiments carried out at Sussex. See [7].