The artwork creates a social environment in which people come together to participate in a shared activity. Bourriaud claims “the role of artworks is no longer to form imaginary and utopian realities, but to actually be ways of living and models of action within the existing real, whatever scale chosen by the artist.”^[15]

In Relational art, the audience is envisaged as a community. Rather than the artwork being an encounter between a viewer and an object, relational art produces intersubjective encounters. Through these encounters, meaning is elaborated collectively, rather than in the space of individual consumption.^[16]

I find ideas in relational art relate a lot with complex systems thinking.  Some of the connections I describe in the Complexity and Creativity talk for Research Club. Basically, the study of complex systems focuses on the relationships between parts, how such interdependence can affect the dynamics of the whole and how properties of the whole system can give rise to complex interactions.

An artist who works in relational aesthetics considers similar systemic properties in designing a piece of art.  In thinking of the audience as a community (producing meaning collectively), an artist explores how the audience may interact with one another, and how the environment can cultivate particular interactions.  Instead of designing an individual experience, the artist designs that collective experience. Such a design can both empower the audience members by giving them some creative control, and also produce unique and personal experiences for each member.

In Portland I used relational aesthetics to inform a couple projects.  pdx i love you was a collaborative public performance piece, a platform to encourage individual creativity as well as collaboration. hexagon followed a structure of composition to encourage the process of discovery in the composition, exploring how simple rules could produce complex behavior.  The performance sought to decentralize a musical experience, by surrounding the audience.

Currently the relational aesthetics approach informs soup night and whirl.  People’s interpersonal interactions can be used as a medium of art, and as a way for an artist to produce meaning.

I love this sort of work because it focuses on design for group interactions and builds connections between academic research and social business opportunities.

Two Minds

Part of Pentland’s inspiration comes from the Two Minds model of Kahneman and Simon.  We have  the Habitual mind, which is fast, parallel, and automatic.  It is far older than the Attentive mind, which is slow, serial and rule based.  Research has found that for simpler problems, with fewer factors, the Attentive mind is immensely successful.  Whereas for complex problems, the Habitual mind is able to make better decisions.  We are not aware of all of the decisions and processes of the Habitual mind, and Pentland’s work seeks to reveal its influence.  What are factors that may be taken into account by the Habitual mind, which significantly affect individual and group behavior?

Primitive Economies

To understand group dynamics, the researchers go ‘back to the beginning’.  Initial research in primitive economies focuses on simplified models of myopic individuals (considering only one interaction at a time) who interact only in local exchanges- approximating primitive society.  How do these local exchanges shape the larger-scale structure of society?

What Pentland found is that the global structure is stable and efficient, with a system that truly includes as much opportunity for the rich and the poor (very different from our current system) and the myopic dynamics converge to a stable local exchange network.  Maximization of the group and indiviaul in this scenerio are the same, so the invisible hand actually works.  Such a stability allows for trust- people will see each other often- which provides the infrastructure for an information economy and language.  Individuals can now trade tips and valuable information.

Backchannel Comments

In a paper on collective intelligence and information exchange, researchers find that important factors in predicting the success of a group include commenting, contributions and also backchatter comments.  Backchatter are phrases usually less than a second long, but can help indentify how engaged individuals are.  Backchatter is identified in this research through analysis of audio recordings, so it is an automated procedure.  The same process can be used for an individuals cell phone to determine health factors, such as signaling depression.

While individuals can learn skills to work better in groups, Pentland finds that such skills are difficult to use for manipulation.  For example, if people learn how to show more empathy, they will actually become more empathetic.

Informal Communication

In analyzing a call center, the density of informal communication predicts productivity.  Informal communications include face to face conversations in passing, as well as emails or phone calls between groups.  Such information can be retrieved from badges that can sense where individuals are, and from such data one can determine who is communicating with who.

This was a Harvard Business Review idea of the year.  Informal communication patterns account for 40%-60% of group performance.  For the example of the call center, in the informal communication conduit, people trade tips on what works for customers. For this call center case, $15 million in savings results from changing the coffee break structure.

In constructing a successful business environment, if there are strong lateral and informal communication channels, information will clear better.  (Consider the connection between this informal communication, and the sidewalk ballet of Jane Jacobs.  They are the same in their function, in creating healthy productive groups.)

Who You Spend Time With

Another research project outfits a dorm of students with cell phones with sensors that reveal proximity and communication patterns between students.  The research takes place during the 2008 presidential election, which allows researchers to see what factors contribute to shifts in political opinion.  Surprisingly, the findings suggest that it is not your friends who influence you, but the people you spend time with.  A very small proportion of people you spend time with are your friends.

Previous techniques in network research ask individuals to name their top three friends, and a network is constructed.  The problem with this sort of work is that it fails to take into account other social influences- it assumes the outcome within the method.  Remember the Habitual mind in the Two Minds theory- that factors people are not consciously attentive of can have a large influence in behavior.

Social Networks from Behavior Patterns

Other techniques to construct social networks focus instead on patterns of behavior and locations. In San Francisco for instance, the movement of cabs create a network of the city between locations.  The network reveals groups of nodes with few connections to other nodes, the locations of two groups could be right next to each other, but people didn’t mix between them.  Based on one’s behavior, the group of nodes they are most likely to move within can be determined.  People within groups also have similar characteristics, which can be useful in understanding purchasing patterns, as well as disease spread.

Influencials can be identified within social networks by having high centrality.  These individuals have particular behaviors.  They are information harvestors, and become the conduit to connect other people in their social network with useful information.

Successful group structures consist of individauls who are able to go out of the group, harvest information, and bring it back to clear with the group.  Such a theory is represented in Nathan Eagle’s work on network properties and socioeconomic status, which correlated each individuals ability to clear information within communities and between communities with a socioeconomic indicator of their neighborhood.

Further research shows that to predict the success of a business plan, for instance, one can measure the number of information harvesters in the group.

On Privacy

Monitoring workplace interactions, cell phone calls, and interpersonal interactions delves into serious privacy issues, especially given that such information can be used to influence individual’s behavior. Currently, individuals data are collected and centralized- in many cases the data can be harvested and sold to other companies for purposes including advertising.  Pentland’s insight into the issue is that information flows are most stable and efficient in systems where individuals own their own data.

I am excited about further research exploring the benefits of these personal-owned-data systems, where people are in charge of their own data.  Such research will encourage applications built on the decentralized infrastructure, and these applications will take advantage of improved information flows.  Once a few successful models come out with their documentation, the programming architecture will spread.  The norm will follow that individuals control the dissemination of their data.

Such a process elucidates some exciting principles regarding the design for longevity.  By interviewing experienced operators, designers have an idea of what functions are important and how they can be compartmentalized.  Once there is a platform- a structure to hold modules- the design begins to account for an uncertain future.

The design is broken down to scales.  What will still be around in 100 years in the environment, what are we certain of? What will exist on the 50 year scale, the 10 year scale, the 5 year scale, etc.  The materials and flexibility are chosen with an idea of time scales and necessary flexibility.  A longer time scale function will require less flexibility than a shorter time scale function, but would require robustness. Think of Stewart Brand segmenting the long term dynamics by scale:

We can approach design for longevity and sustainability by focusing on the processes at each time scale as Brand illustrates.  A cool effect of the modular design for longevity is that for processes that need to be more flexible, users are encouraged to participate.  There is choice of what modules to use- what functions to feature, and how to adapt to local environments.

The modularity allows for collaborative design- not only between a design and current users via conversation on needs and observations, but also between the designer of the present and users of the future.  Through this collaboration, the designed objects can function successfully for far longer periods of time.

Controlling Nature: Determinism and Disenchantment

Kauffman begins with a brief historical review, describing our relationship with Nature and the role of deductive thinking. Aristotle explored the mechanisms of how something came to be, through his four causes: four causes: material, formal, efficient, and final.  The process of deduction gave formal way to understand the world around us.

Bacon extended the power of deduction claiming that by the force of logic, you can command nature.  The empiricists looked at the world to gain knowledge, which would then become power- over nature.  Bacon is quoted to have said “we put Nature on a rack”.

With Newton came the techniques to formalize deductions.  Netwon mathematized efficient cause: the prior conditions considered to have caused something.  If you set up the initial conditions, you can deduce via the equations the development of a system.  If I know how fast the ball is when I release it, I can deduce how far it goes.

Such a framework provided for the clockwork conception of the universe, and eventual disenchantment in entering modernity (Weber), valuing scientific understanding and orienting processes towards rational goals. The Enlightenment included the desire to wield command over nature for the progress of society- just the idea of checks and balances is very Newtonian.

The 20th century saw the beginning of a highly commoditized world, with value tied to being consumers or producers.  Why are people making poolside purple penguins? Living in a late capitalist society where banks are gaining more and more power, we don’t know what else to do.  We are disenchanted.

The Adjacent Possible,  Darwinian Pre-Adaptations, and Unpredictability

Now consider more complex sorts of interactions. For example, all of the molecular interactions and mutations that give rise to a particular organism.  If any of those reactions and combinations were changed, the organism would be different.  Consider the possibilities of the organism today, as you look into the future.  This space of possibilities of the organism is the adjacent possible (a concept created by Kauffman).  But this need not relate only to biology.  Just the possible future states of a system given its conditions today is the adjacent possible.

Consider the functions of a particular biological form. The set of all possible functions into the future is the adjacent possible.  A Darwinian pre-adaptation is when something is evolutionarily selected for in the current environment that serves an additional function in a future environment.

There are plenty of examples of this, including the flagellar motor which can be found in bacteria, but serves a function other than locomotion.  Pther examples include the swim bladder, adn the three bones in the inner ear.

Is it possible to predict all of the pre-adaptations of humans?  Definitely no.  The difficulty lies in listing all of the possible features of everything within humans, since features are relations and depend on the environment- we must also consider all possible environments.

Any evolutionary adaptation changes the evolution of the biosphere.  Consider the adjacent possible of the biosphere- we do not know all of the possibilities of the adjacent possible. We don’t know what will happen, or even fully what can happen.

Using probability mathematics we can make predictions.  But for any probability measurement, you must know the number of possible states (ex. probability of choosing the correct door of N doors).  Without knowing the size of the adjacent possible, we don’t know the number of possible states, so we cannot calculate probability.

Some may liken the unpredictability of the adjacent possible to the halting problem in computer science (that given a program there is no way to write another one to tell if it will finish or run forever).  Though they both contain the idea of the unknown given initial conditions, they are very different problems.  One can tell all possible states of a machine given its description, you can calculate its sample space.  But for the adjacent possible, you cannot.

The same unpredictability exists for the adjacent possible of the technological world. Consider a simple technological timeline- mainframe paving the way to PCs, which created the platform for word processing, leading to files and folders, which led to sharing, then the internet, then Google.  Each development gave rise to a new set of possibilities.

Now contrast this with the clockwork universe- where everything is describable by the initial conditions.  This is a very different picture from what was described before- now because of the indeterminate size of the adjacent possible, everything can never be fully determined from the present.

Empty niches can be thought of as adaptive opportunities.  Filling these niches gives rise to new species, which enlarges the adjacent possible, which then provides for more adaptive opportunities, and the cycle continues.

The biosphere builds that which it will become.

To return to the first idea of Aristotle’s causes, is an empty niche a cause?  It is not, it is something new, it is an enabling condition.  A difficult concept to wrap our minds around because we are very much still in the Newtonian mode of thinking.  The evolution of the biosphere is not totally causal.  And this applies not only to the biosphere.  We live in a world where the adjacent possible affects the predictability of the economy and also technology.

Understanding their behavior is essential- everyday decisions are made for the future of cities, and future of companies.  To have the tools to understand their growth and behavior leads to better models for prediction.  City planning and organizational design can be well informed by findings within complexity research.

Cities: Designing for efficiency on a runaway treadmill

Geoffrey West provides some exciting insight regarding the scaling laws within cities and corporations in a recent NYTimes article.  He is an eminent mathematician at the Santa Fe Institute, researching scaling laws within complex systems.  As mentioned before, understanding scale plays a huge role in modeling the behavior of complex systems.

To me, the most interesting parallel is that with Jane Jacobs, where West describes:

“What the data clearly shows, and what she [Jane Jacobs] was clever enough to anticipate, is that when people come together, they become much more productive.”

To interpret these equations as quantifying the productivity of human interactions reveals their relevance, though I don’t believe that these equations alone come close to accomplishing what Jacobs has in respecting individual potential and understanding the collective dynamics in cities.  But it is incredible- to have developed a metric that can give an approximated quantified measure of collaboration.

The crux of West’s research in cities lies in the statistical findings that

Many diverse properties of cities from patent production and personal income to electrical cable length are shown to be power law functions of population size with scaling exponents, β, that fall into distinct universality classes.

Which basically means that as the population of a city changes, one can predict quantities like the level of innovation (by measuring patents) the income for individuals, and the amount of infrastructure development.

Supercreative employment increases with population

So one can develop a framework to see all cities as spanning the axis of population size of the same fundamental city.  It is a startling finding, not without its exceptions though.

On average per person, an individual uses less energy in a city than they would if they lived in the suburbs, and that is due to economies of scale.  In developing a world model for sustainaiblity, cities are an essential tool, West describing them as “one of the single most important inventions in human history”.

In looking at the long term dynamics of a city, feedback phenomena are essential to understand.

As a city grows, the larger population requires more resources (though less per capita). The city moves faster, the pace of life becomes faster.  As resources deplete, the city requires innovation.  Innovations cause the city to grow.  And the cycle moves faster and faster- so lifestyle changes faster, and  innovations are required at a faster rate.

What this means is that, for the first time ever, people are living through multiple revolutions. And this all comes from cities. Once we started to urbanize, we put ourselves on this treadmill. We traded away stability for growth. And growth requires change.

Companies: The urge for growth towards probable failure

The difference with corporations is that as size increases, innovation can’t keep up.  The corporations become tied down with bureuacracy, too much time is spent on just maintaining the structure, and there is little flexibility to adapt.  So the corporations fail.  Every corporations goal is to earn a profit, and profit is growth, but growth leads closer to failure.  West’s research supports this.  But, I believe this is due to an old school model of corporate organization- people can rethink the way an organization operates as it scales up, in such a way to improve innovation.

Lately its been common knowledge that hierarchies don’t work in fields that require adaptive organizations, that a relatively flat structure is the way to go- ideas evangelized by IDEO and Google.  But you can’t sweep an elephant under the rug- scale will still become a problem, and Google has recently shown serious action on that issue- how can a company scale in an industry as competitive as internet tech.

A recent Businessweek article describes Google’s recent deliberate actions to ensure the company stays ahead of its competition, and maintain itself with a competitive edge of innovation.  In the article, Larry Page says

“We do pay a price for [shared decision-making], in terms of speed and people not necessarily knowing where they go to ask questions.” His elevation to CEO, he says, “is really a clarification of our roles. I think it will help with our speed.”

The article highlights the company with a series of profiles on individuals- emphasizing a shift in the organization of the company,

Page says one of his goals is to take the decisive leadership style they have shown within their product groups, spread it across the company, and apply it to major decisions. “We’ve been inspired by a lot of the people who have been operating with more autonomy and clear decision-making authority,” he says.

Its hard to exactly predict what changes may come in such a volatile industry, but what is clear is that given the wide range of technologies Google invests its time in, there is no way a single individual inhabits the capacity to effectively manage all of them.  Instead, now the focus is on cultivating masters in a particular sector, and giving these people autonomy within the organization.  From reading the profiles, it is clear that the individuals are some of the most capable in the world for their sector, and being in an environment to cultivate their freedom within their sector is what will ultimately lead to the success of the organization as a whole.  As West describes with cities, innovation is key, and creative freedom can cultivate innovation.

From recent developments in understanding cities and corporations, results have yielded the inevitable failure of both from growth- but there is a potential from collaboration that is unpredictable even with the best models. There will be ways to design cities and organizations effectively, but such designs will need to empower individuals- be it citizens, or consumers- in a way that allows them to be creative individually and collectively.

I had the pleasure of giving the Patterns and Meaning lecture at the 2011 NECSI winter school.  It was my favorite lecture when I took the course, so preparing a talk on the subject was an exciting opportunity.  I’ll begin with the contents of the lecture, in particular how they unify many important ideas regarding meaning, and then describe how these ideas can be applied to designing experiences for groups and individuals.  I will follow that with a summary of my own explorations on meaning.  The lecture was originally developed as an exploration of complex systems as applied to art.

Complexity and Meaning lecture

Throughout history there has been a lot of thought devoted to understanding meaning.  Take a couple of approaches: semiotics and postmodernism.  Semiotics describes meaning as produced from that which signified and that which is signified.  We use these mappings to develop an understanding of things around us.  Thus, there is a way to draw the meaning out of anything perceived.

On the other hand the postmodern concept of meaning stresses that there is an infinite number of meanings for anything.  Each individual has a unique understanding of things, and thus there is no absolute meaning for anything- it is arbitrary.

These are two very different perspectives on meaning, most notably on how arbitrary meaning is.  How can we reconcile this?

Begin with a pattern, and use the term pattern in the most general sense- to be anything (in a similar way as Christopher Alexander describes a pattern). How do we ascribe meaning to that pattern?

We use a neural network model of the brain.  There are two characteristics in particular to focus on.

• First, every input corresponds to a particular imprint onto the network.
• Second, a similar input to a previous input will have a similar mapping.

This leads to the concept of basins of attraction where similar input will be mapped to the same point.

Applying this simple model to the mind: (1) every perception is a pattern that is mapped to a particular neural pattern, and the neural pattern is a representation of that perception.  (2) Similar patterns in the world correspond to similar neural patterns.  The implication of these two are that proximate neural patterns recall proximate patterns of the world.  Because of the nature of the network, there exist basins of attraction where similar input map to the same concept.

A concrete example: if I see an apple, it recalls the idea of an apple I have developed over my experiences with apples.  Conceptually there is a basin of attraction that is apple.

The Mona Lisa is not in this basin.

From these two relationships, a powerful model of meaning emerges.  Everything you perceive affects your conceptual map, so the meaning for one pattern is affected in some way by the meaning of all other patterns.

Also, our ability to communicate our experiences and interpretations of patterns can affect others conception of patterns with whom we communicate.  An intuitive example: because of the segmentation of populations around Earth throughout history, many cultures developed unique ways to represent the same underlying ideas.

The social component of meaning among groups, as well as the interaction of all perceived patterns in an individual provides for a dynamic, context-dependent model of meaning.  Meaning can change based on the topology of interactions, as well as the time frame.  Meaning is both constructed by the individual, and a society.  It is not entirely arbitrary, and it is not entirely absolute.

A Network (upon Network) of Interactions

So you can think of it as a web of connections occurring on two scales: the individual, and society.  In the individual there is the web of all perceptions, and regions of clustered patterns which form concepts in a concept map. Every time you see something new it imprints and can affect related (and even unrelated) concepts in your map, since all of the patterns have meaning based on their relationship with the other patterns.

And the second scale is that of society- the web of communication and interactions among people who can share patterns with each other.  Patterns seen over and over can develop meanings by recalling other elements from the context in which they occur, or meaning through explanation and communication.

The Internet affecting Interconnections

Imagine now how the internet deeply affects who we are able to communicate with- we are not restricted only by proximity.  The possible speed of our communications are faster, the length of them further.  We now have interactions and patterns that are purely virtual (norms in web design and web navigation), others that can span cultures (Google, Facebook, international blogs, memes), and still others that remain dependent on geographic location(Twitter dialects, interpersonal norms)- and all of these which affect the individual’s  concept map.  None of these maps and meanings associated with these patterns are fixed either, they are all dynamic in some fashion- some changing far more rapidly than others. This poses a challenge for designers seeking to create a specific experience- but the same time there is a lot of data to explore how meaning can change, opportunities for collaborations and conversations, and content revolving around design processes.

Application to Design

An example: individuals from different geographic regions will have some differences in their concept maps due to the fact that the geography creates significantly more links of interactions between individuals in each region than between them.  Between the separate groups, certain norms, patterns and associations can develop, some of which are very subtle. It is hard to design for a culture that one hasn’t lived in- unintended meanings can be triggered (even taboos!), or one can just miss the point of what is most useful for that group.  This is why conversation is essential- to build a model in the designers mind of how the user sees the world, how they move in it (D-Lab, Human Centered Design).

This may seem intuitive, it is not unusual to have a design project fail to see from the point of view of its users- and thus create a useless (or even harmful) product.  A few examples are in the Postcolonial Computing paper .

As a designer, this is an exciting model because it combines a pattern (and function) conception of things in the world, to symbology through the idea that a particular pattern can represent those experiences of the pattern (which includes function).  It is a way to reconcile two versions of meaning I found as I researched producing meaning through creation, described below.  In creating, a good designer takes into account that any meaning created will be related to past experiences of individuals (basins) , as well as communication among the individuals (social component).

Additional Readings:

Dynamics of Complex Systems

The Timeless Way of Building

Man and his Symbols

A Thousand Plateaus

Some background on my own exploration of meaning

I never had the experience of a serious art critique until taking Temporality and Narrative in New Media at GWU years ago. Things can be interpreted in many ways, but there are ways to direct interpretations. And that is where meaning comes in.

From the class I understood it to be something of semiotics-most simply understanding that which signifies and that which is signified.  Every element you include in a piece is some metaphor for something else.  And the way in which elements interact matters as well (in a way this is the seed of what eventually led to my fascination of complex systems).  If you are thoughtful of the elements and their interpretations, you are making progress.

A reading of Jung’s Man and His Symbols gives a strikingly similar development of meaning through metaphors.  He described man as something like meaning makers, “Man positively needs general ideas and convictions that will give a meaning to his life and enable him to find a place for himself in the universe.”  We can place meaning into inanimate objects, things to represent the self in some way, or part of ourselves.  It is through these objects or concepts that we can construct a narrative and gain a deeper understanding of ourselves and place in the universe.  An artist creates an experience through their work that taps into our meaning making abilities, but for the most part, it is the projection into ‘higher’ state that produces meaning.

However, after reading Christopher Alexander’s The Timeless Way of Building, I found it hard to connect his method of production to meaning as described above (through symbols and metaphor).  For the most part, his process revolves around function- what things do and  how to create a living space with that understanding.  Patterns can be connected in a harmonious way that we can intuitively understand.  But where are the metaphors to higher concepts? Where is the role of decoration and style?  I found his method for deliberate creation incredibly compelling, and as an aspiring designer I wanted to connect it to my current process of creation (via symbols).

There are definitely answers, but at the time I thought a lot about the different approaches to creating experiences.  If design is about creating a targeted experience for the user, meaning as described through symbols would inform one to create after understanding the metaphors in the culture and how they create an imaginative higher form significance, whereas meaning as described through patterns would inform one to create by understanding the functions of physical things with each other in a space that is living.  Its much deeper than this, but that was the distinction (higher significance vs. physical, grounded experience) that was most clear to me.

The complex systems approach to understanding meaning- by taking into account the dynamic interplay of individuals as well as input patterns in a neural network model of the brain- gave an exciting way to bridge the distinction.

Tensors and Growth

The growth is modeled using a growth tensor of rank 2.  Think of it as a matrix, or an extension of a vector.  A vector has magnitude and one direction.  A tensor of rank 2 has magnitude and 2 directions associated with it.  If we have a vector and we want to obtain another vector, we can multiply it by a scalar or a matrix.  The matrix can change the direction, while the scalar cannot (except for transforming it in the opposite direction).  Here is more info on tensors if you’re curious, they’re pretty wild.  We can solve for the velocity of growth at any point from this matrix by solving for the velocity components of the following set of linear equations, where T is the growth tensor.  The velocity components can then be calculated by a given position.

So for any point on the surface, we can get a growth vector, and this gives us a field of velocities.

The above is from the thesis, and shows you different types of growth that can occur on a surface. Isogonic growth is growth where the rate of growth is the same in all directions at all locations.  If I apply this to a circle, it will still be a circle.  Uniform anisotropic growth is that the growth is the same at all locations, but changes given a direction.  If I apply this to a circle, it will become deform into another shape.  Note that the growth operates on the shape such that the shape is at the origin.

In this example we’ll simply use growth along the x axis, where R is the rate of expansion.

And the corresponding growth tensor is simply

Creating the Surface

We begin by creating a triangulated surface with semi-random triangles.  Because of the nature of FEM, uniform triangulation and random triangles can have undesirable effects.  With random triangles, some points can be too close.  Place the nodes on the surface randomly, and repel them.  I used the separation function we made in the Boids algorithm, adding a drag force proportional to velocity.  Run this until all of the nodes are in equilibrium.  You can implement this as running the loop until all of the velocities are zero, or you can choose to run the loop for a certain amount of time as the particles separate (but this does not guarantee the particles will fully separate evenly). Now triangulate the nodes using a Delauney triangulation.

This gives you a triangular mesh on a plane.  Project this plane in the normal direction to get a second layer.  Each triangle now becomes a wedge element.  Each wedge element has six nodes.

We are going to focus on this element for now. Then we will see how to combine all of them to calculate the stress on the nodes.

Finite Element Method – Qualitative

The finite element method can provide approximate numerical solutions to a wide range of engineering problems.  This is most useful when encountering problems where an analytic solution is difficult to find, or simplifying assumptions cannot be made. We begin first with something familiar: finite differences.  These are also used to find approximate numerical solutions.  If we have a shape, we can approximate it using a grid to cover the shape.  More points (a finer grid) produces a more accurate covering.  The finite element method is similar, only it uses triangles instead and can much better approximate the region.

The above figure is from here, which provides a good introduction to the ideas of FEM. When we created smooth particle hydrodynamic simulations, we approximated continuous functions and properties over a region using kernel functions and interpolation.  Similarly with FEM we divide the region into nodal points and use interpolation functions to approximate the property of interest over the region.

The FEM allows one to divide a region into elements, calculate the solutions for individual elements, and then combine the element solutions to form a soltuion for the region.  In our case we find the stiffness of each element, and then combine them to obtain the stiffness of the entire material.

There are a variety of ways to calculate the properties of the individual elements.  We choose the direct approach.  Other approaches include the variational approach, and the weighted residuals approach.  We divide the region into elements, calculate the element properties with an interpolation function with the element nodal points, then combine each of these element matrices into a giant matrix to represent the entire system. We impose the boundary conditions (any values we know about the system and the nodes) and then solve the matrix equation for the system.

Finite Element Method – Quantitative

So from the perspective of an element, the amount of force generated is proportional to the displacement of the nodes.  Sound familiar?  In the matrix form is looks similar to F=kx, Hooke’s law to determine force for a spring.

Where F is the force of the element, K is its stiffness, and q is its nodal displacements.  For each element we have six nodes.  Each node has three degrees of freedom, in the x direction, y direction, and z direction.  This gives us a total of 18 degrees of freedom.  [F] is an 18 x 1 vector with F_x, F_y, and F_z for each node.  Similarly [q] is an 18×1 vector with the corresponding q_x, q_y, q_z for each node.

If we want to determine the value of some function over the entire element, it is helpful to define a coordinate system within the element.  These coordinates would be called isoparametric coordinates. We can define three variables to integrate over in the wedge.

We can now use interpolation with the six nodes to determine the value of any point within the element, as a summation of the value at the node times a weighting function N.  The weight is called a shape function, and is some value that depends on the isoparametric coordinates, and is inbetween 0 and 1.  Mathematically this looks like

And if we have a matrix of N_i * I and concatanate them for i = 1, …, 6, where ‘I’ is the identity matrix, we have an 3×18 matrix for N.  Given V is the 1×18 vector of concatanated values for each of the nodes, we can represent the above as a matrix equation [v] = [N][V].  If Q is the 1×18 vector of position points of all the nodes, we have [x,y,z]=[N][Q], converting the coordinates in the element to cartesian coordinates. Given a displacement [u,v,w] in an element, we can interpolate it over the nodes in an element using the shape function N.

Where u, v, w, and N are dependent on the isoparametric coordinates.

Before contintuing onward to calculate the element stiffness matrix, it is helpful to have a better understanding of the shape function.  It should be pretty familiar in concept, and fortunately it is much simpler than the analog we used in SPH.

For any of the shape functions, you can vary the coordinates and see how it changes within the element. We can integrate over these coordinates to find [K_e], the element stiffness matrix.  We can approximate the integral using the Gaussian integration technique.  Instead of computing the continuous integral, we compute the discrete sum over select points that are weighted.

Now that we know how to go about integrating, we need to determine what exactly to integrate over.  To find [K] we integrate over two other values, the material stiffness matrix [D], and the matrix to relate displacement with strain, [B].  The material is uniform throughout, so the stiffness matrix is I * constant.  We can derive strain in each direction as a summation over the shape fuctions, since we can relate the shape functions and the displacement. There is also the Jacobian matrix [J] that transforms between coordinate systems (cartesian and isoparametric). The final equation for [K_e] is

where we use n Gaussian integration points.

Note we can rewrite the matrix equation for stiffness as [F] = [K][P]+[R] if we notice [q] = [P] – [Q] , where [Q] is the undeformed coordinates.  So to obtain the global stiffness equations we must find

For a system of n nodes, F_g is a vector of length 3n, since each node has three force components.  Similarly the coordinate vector [P_g] has length 3n. The global stiffness matrix [K_g] is simply adding the appropriate values at the correct indices of the element stiffness matrices.

In our simulation the bottom layer is fixed and the top layer is free to move.  We set some initial strain (the growth of the bottom layer), insert the boundary conditions (force on all nodes is zero), and solve the equations (calculate the nodal coordinates of the free nodes.  We now solve the familiar equation

Our set of linear equations is pretty sparse, since we created the global stress matrix by simply combining the element matrices.  Because of this, we can use the conjugate gradient method to solve the set of linear equations.

We’ll take a brief moment to discuss a few concepts.

And here is an algorithm for it, from Wikipedia.

repeat

if r_k+1 is sufficiently small then exit loop end if

end repeat

The result is x_k+1

We calculate the eigenvalues and eigenvectors of the stress tensor on the mesh to determine the principle stresses at given points.  If the largest of these stresses exceeds the material’s stress, a fracture is formed.

There are a few ways to represent the fracture.  One way is to delete the element.  This can produce fractures, though it is not realistic. Another method is to subdivide the elements along the principle stress plane.

To optimize we can use dynamic subdivision. If a given element is a candidate for fracture, we subdivide and recalculate the stresses.  Once an element is small enough, a fracture can form on element relaxation.

More info will be posted soon!

Google’s search algorithm, to a heavy extent, depends on the nature of linking between websites.  On the scale of the website and what links to it, Google is able to provide an answer the global problem of findability on the internet.

Muhammad Yunus developed a micro loan system that re-frames loans to take into account collective dynamics of a group.  Through challenging the assumption of who one can give a successful loan to, and by tapping into group trust, he has created an effective method of alleviating poverty.

What is Emergence?

Simply put, emergence is when behavior at a smaller scale of a system produces global behavior that is not entirely intuitive given the behavior at the smaller scale.  The global behavior is not directly programmed in the behavior at the local level. Something surprising happens at the larger scale.  From simple local interactions, we can arrive at global behavior that is immensely complex, sometimes seemingly random. Think of the classic example of Conway’s Game of Life. The rules to define the simulation are simple, but the behavior can be pretty unpredictable. Or on the other hand, we can think of systems which have a lot of randomness at the smaller scale, but order emerges at the larger scale. This underlies the concept of self-organization, a process which is widespread in the development of biological and societal form.

Another way to think of emergence involves the notion that the whole is greater than the sum of its parts. The behavior of the system is a result of not only its parts, but their interactions. The actions at the level of the individual do not imply that of the behavior of the system. We need to understand not only the system at the individual level, but at the level of multiple individuals to observer the effect of their interactions. One of the most exciting examples of emergence involves evolution. At a smaller scale we have natural selection, species in a population surviving or dying over time based on selection pressures from the environment, with possible species variations through reproduction and sexual selection and the random chance of mutation. At a larger scale we have the development of biodiversity, a wide range of organisms from bacteria to whales, and an environment with so many different species able to thrive within different niches. Global biodiversity is not programmed in the local interactions; it is an emergent property.

Why is it important?

Understanding emergence is essential when working with complex systems. For the most part, complex systems exist because they are grown, i.e. they have evolved to be what they are. Think about what makes up a complex system. They are composed of many parts that interact across scales, from individuals to groups to the whole. When you think of a city, it grows through the interaction of all of the individuals and organizations which comprise the city. The city is not formed by a single individual, it is formed by its population. Emergent behavior can be difficult to understand, since by its very nature it is unintuitive and unpredictable (the very reason why it has received so much attention). In understanding global behaviors of a system, it helps to understand how individual interactions led to the global behaviors. In this way, one can attempt to reproduce the global behaviors through recreating the individual interactions.

Example of Emergent Behavior: Fireflies

A popular example which illustrates emergent phenomena is that of fireflies which can spontaneously synchronize. There is no head firefly with the ability to communicate with all of the others, so how is it possible that they can flash in synchrony, when no firefly can see all of the rest? In their paper ‘Synchronization of pulse-coupled biological oscillators’, R. Mirollo and S. Strogatz seek to answer this question by developing a model that introduces a mechanism of simple interactions between the fireflies. Through this local interaction, the global behavior is that of synchrony. The key concept with respect to emergence is that the local interactions do not mention anything about synchronizing. Each firefly can be thought of as an oscillator which fires at time T, and resets to zero only to fire again at time T. All of the fireflies have the same period T, but start at different times in their period. At each step, all of the fireflies increment in their period. A firefly has the ability to see neighbors within a certain detection radius. When one of the neighbors of a firefly flashes, the firefly increments in its period, leading it closer to flashing. The nature of this firing response is important. Say that a firefly is at time t’ in its period. The firing response is defined as follows: the amount of increment increases as t’ increases. So if a firefly is later in its period, the increment will be higher. If its just beginning its period, the increment will be lower. The next is not so important for the concepts, but mathematically it can be more accurately stated as such:

Here, t” is the new internal time if a firefly experiences a flash at time t’. As defined in the model, f is the firing function and epsilon is a small constant < 1 .
From these specifications, the fireflies will synchronize. The synchronization can be viewed in the following applet. I approximate the increment function as described in the paper ‘Firefly-Inspired Sensor Network Synchronicity with Realistic Radio Effects‘ Werner-Allen, et. al. as

Where epsilon is 0.1 .

The simulation is done in Processing. I encourage you to open it up and play with the number of fireflies and the period.
It is difficult to really see where in the period each firefly is, so I created another model which simulates the fireflies as if they are standing in a single line. They bounce up and down periodically, flashing when they hit the ground. If a neighbor flashes, they increment in their period, evident in a sharp jump. This one has an interface to allow you to change the number of fireflies, the period, and the detection radius (how far a firefly will look to its left and right to detect a neighbor.

Its pretty exciting to see the simulation converge to synchrony. From a simple rule given a neighbor flashing, organization emerges as the fireflies flash as a group. The group is able to act as a whole through local interactions.

Cool! … now what?

Think about environments that your are in or have seen, and reframe them in the systems perspective. How do the parts combine to form the whole. Is the emergent behavior obvious given local interactions?

What if there is a particular global quality you want to achieve. What are ways in which local interactions can achieve those global qualities? Because the global behavior can be unpredictable, it can become a difficult problem to determine the local interactions. But other times, those interactions can be found through the ingenuity of a group and global behaviors can be achieved. The question then also becomes, how do you create and environment that will promote such creativity? This is a question of emergence, in designing local interactions to promote global creativity.

Design and Dissemination

An example of the utility of thinking in terms of systems and emergence is in the world of creativity and design. If a designer and the consumer have similar environments, it is very likely the designer can find successful solutions for the consumer. In cases where the designer and the user are in completely different environments (including different experiences and customs), it will be immensely difficult for the designer to make something useful. In some cases, products will be developed, only to suffer completely different uses in the field (which is not always bad, but it is a gamble). So if instead the designer works with the user, together they can design something successful. From the complex process of improving experiences through design, the collaboration leads to the emergence of successful, effective solutions. This may appear intuitive, but it has not always been the case that those who believe they have solutions have taken the time to fully understand to conditions those who they are trying to help.

With respect to dissemination, the goal is to have emergent behavior that spreads information, or a product. The question becomes: what sort of local interactions will result in this global goal? There are many ways to approach this problem, but a few methods include understanding the topology of the society in disseminate within. Certain people are more likely to spread information or a product, so if one targets those people, the product will spread faster and further. Another approach is to focus on the role of the individual. If one is empowered to be included in the product design and construction, they will not only help in characterizing the problem, but will have a deeper connection with the product. It is more likely they will, in their enthusiasm and success, spread the information and product to others. The task becomes including those who use the product to become part of the design process, be it through business, construction, or design, and they will in their own interests spread the product.

An apt analogy is to think of the process of gardening and growth. The designer is a seed, and the soil are those who will use the product in whatever environment they are in. The soil contains the nutrients, that which the seed needs to form into a healthy plant. From the interaction of the two, a product is formed, the plant. Though the seed may be the same, in two different environments, the plant will be different. From the environment of collaboration, many solutions are possible. Such solutions may spawn more seeds that in turn develop more solutions.

This process is not too difficult to grasp. What becomes harder to fathom is the type of solutions developed in the aggregate. The variety of solutions and even ideas for possible solutions of the group far surpass that of the individual. It becomes difficult to predict the creative capabilities that emerge in a group.

Collectively we are capable of much more than any of us alone can imagine.

The above article describing emergence was created as a supplement for Eric Reynold’s presentation on creative capacity building and systems science at the 2010 International Development and Design Summit in Fort Collins, Colorado.

Social systems are becoming more complex from technological advancements and increased connectivity. Individuals are further empowered with the capability to augment their memory and communication through computers, the internet, and cell phones. Every society has structures which influence collective behavior, and with all of the possible configurations of people in a population, the question emerges for designers of how to implement a method to use the collective information and create a successful design solution [1].

Cities have been shown to have fractal geometry. In this paper we show how the fractal shape can emerge from a generative process that takes information on the scale of individuals or groups, and uses it to design a permanent infrastructure on the scale of a city. In this sense, we grow cities consisting of individuals and roads, starting from just individuals. [from introduction of paper, see PDF for rest]

A Generative Method for Infrastructure Emergence from Kawandeep Virdee on Vimeo.

A talk I gave at the December 2009 Research Club brunch at Tribute Gallery on applying ideas of complexity to creative projects.  Applications of complex systems ideas in the projects HEXAGON and PDX I Love You are described in the presentation.  The talks were limited to five minutes each, and a transcript is included below.

Brunch #1 / Lecture #3 / Kawandeep Virdee Talks About Complexity from Research Club on Vimeo.