One of the first fractal discoveries was made in pursuit of conflict resolution.

Now, these realities are being translated back from mathematics into our understanding of conflict.

*The Koch Snowflake: One of the earliest described fractals. (Wikimedia Commons.)*

Let’s dig in with a quick quiz.

What’s the length of the coastline of the United States? (No googling.)

a.) 12,383

b.) 18,924

c.) 29,093

d.) 95,471

Make your best guess- write it down if you like- and hold onto it.

THE GOOD NEWS ABOUT COMPLEXITY

A few months back, I attended a conference for a conservation organization I work with in my capacity as a peacebuilder and facilitator. This is an established nonprofit working to preserve a functionally extinct species against a two-hundred-year backdrop of disease, human interference, and failed intervention. To make things even more challenging, they can now add climate change to the mix.

If I had to sum up their work in a sentence:

They’re working tirelessly in human time to solve an as-of-yet-unsolvable problem that exists in tree time.

A science panel was held at the conference in which members and media were free to interrogate the ranking scientific minds of the organization and its board. A kind and well-intentioned attendee, who appeared to lack some background information on the challenges of the problems the organization faces, raised a hand, and asked, in essence, “Why don’t you just try [this most obvious possible thing] to solve your complex problem?”

The panelists squirmed visibly. All of the obvious things have been tried in every conceivable iteration for decades. Nobody spoke.

Finally, a quantitative geneticist on the panel took a quiet breath, leaned gently toward the microphone, and spoke something that I wrote down on the spot in my favorite Hilma af Klint notebook (slightly redacted here):

If you can’t read my handwriting, you’re in good company. It says:

“The good news about complexity
is we might be able to do a lot of things,
and they could have an effect.”

My primary job that day was to show up as a supportive partner to the organization. But my ulterior motive was to translate some of their own wisdom, espoused in a jam-packed agenda ranging from panels to happy hours, back to the organization.

When we next met, we reflected on their learning from the conference and community gathering by writing 5/7/5 haikus, because that’s way more fun than sharing a sentence or two.

Using the quote above, I rewrote their insight as follows:

the good news about
complexity is that more
becomes possible

They felt so encouraged by this that I sat with the insight for a while, toying with the idea of how many places it could ring true.

I was brought up, professionalized, and trained in my Western culture’s dominant interest-based negotiation paradigm as an investigator and later, a mediator, supporting the litigation process and community. The field of narrative conflict analysis was at first like a siren song to me. Alluring, seemingly true to life- but potentially distracting.

Meanwhile, the English undergraduate in me perceived the relevant narrative undergrowth beneath these ostensibly interest-based negotiations. All the plot patterns were there. But I hadn’t given myself the bandwidth to map them out in a way that made sense.

Amanda Ripley, a journalist focused on repairing our cultural conflict discourse, has written in book and essay format about why complexity is a boon for conflict. In her essay Complicating the Narratives, Ripley tapped into the wisdom of many conflict resolution experts in her research, including GMU’s own Sara Cobb. Ripley ultimately determined:

The lesson for journalists (or anyone) working amidst intractable conflict: complicate the narrative. First, complexity leads to a fuller, more accurate story. Secondly, it boosts the odds that your work will matter — particularly if it is about a polarizing issue. When people encounter complexity, they become more curious and less closed off to new information. They listen, in other words.

Yes- there are places where that traditional, Western interest-based/positional bargaining format works. But it finds its limits in intractable conflict. It often gets stuck when material facts are in dispute, when the numbers and dates don’t line up, and when the values underlying the need for a resolution are out of synch.

While all the anecdotal evidence is there, and even some good data on difficult conversations, the persistently analytical part of my brain wanted to be able to stake a claim for narrative complexity against the backdrop of my positional bargaining-focused training environment.

Enter: a WWI-era weather forecaster.

PEACE MATH PIONEER

While I was stewing over making a case for complexity, I first heard of Lewis Fry Richardson, a Quaker mathematician who tried to forecast conflict, from James Bridle in their book Ways of Being - The Intelligence Singing All Around Us. I listened to the book on a road trip and had to pull over on the spot to transcribe some of Richardson’s story into my notes.

As a physicist, Richardson first pioneered weather forecasting with differential equations that he worked out by hand. Pen and paper were the only computational structures available for such complex math when he published his first forecast-related work in 1922. To put Richardson’s pen-to-paper accomplishments into perspective, the first computer calculations made in 1950 that were comparable to Richardson’s took nearly 24 hours for machines to complete.

Richardson’s weather calculations were part physics and part poetry. He described the orchestration of weather events on a theatrical scale, correctly calculated a theory of turbulence, and created other weather calculations that have proved to be accurate. But the weather wasn’t Richardson’s only interest. He was an ardent pacifist, even though it cost him dearly in his career prospects. (He resigned from a career-fulfilling academic post when his department merged with military interests.)

As a conscientious objector in World War I, Richardson volunteered for a Friends Ambulance Unit (FAU) that cared for both civilian and military victims of war. So close to the fighting, Richardson pondered a scientific analysis approach to conflict.

Image of Lewis Fry Richardson’s Personnel Card from his FAU (Quaker ambulance) unit. From Scientific Quaker.

A pioneer in peace research, Richardson used linked differential equations (much like his weather system calculations) and probability theory to generate a couple of hypotheses around conflict. According to the lore, he worked out this approach in the trenches.

First, he looked at escalation patterns, essentially utilizing the arms race as his litmus, creating linked differential equations to calculate the possibility of conflict. He hypothesized that a nation’s armament build-up was calculable and proportional to the weapon stores of rivals, existing grievances, and current weapon stores.

He also proposed that the likelihood of war between two nations was a function of the length of their common border, presenting war data from 1815-1945, and hypothesizing a base 10 logarithmic scale for conflict.

To initiate the research on his shared border/ conflict hypothesis, he wrote to countries to obtain records about border lengths. What he found, termed the Richardson Paradox (sometimes called the Coastline Paradox) gave shape to the future of mathematics and our understanding of conflict narratives.

Let’s cut to the chase- this was not an exact science, but it was a unique pioneering enterprise. Richardson’s calculations were incredibly innovative for their time. They were meant to provide a forecast- not for the weather systems of the world, but for the conflict systems. In Richardson’s estimation, if we could calculate the purchases of armaments in relation to known grievances, we could produce a war forecast. Border relationships could be mathematicized into a conflict outlook.

This statistical method, though not without its faults, is still relevant:

As Bryan Hayes wrote in Computing Science in 2002:

“We tend to see all wars through the lens of the current conflict, and we mine history for lessons convenient to the present purpose. One defense against such distortions is the statistical method of gathering data about many wars from many sources, in the hope that at least some of the biases will balance out and true patterns will emerge. It’s a dumb, brute-force approach and not foolproof, but nothing else looks more promising.”

Richardson sent off to many countries to request data on their coastline lengths and launch into a series of differential equations. He received many replies to his inquiries about the borders of other countries, but none of the replies contained information that matched. Portugal and Spain, for instance, reported conflicting lengths of their shared border. The measurements of borders varied so widely that it became clear to Richardson that something was awry in the measurement methods.

Richardson properly observed that the biggest differentials in measurement corresponded to coastlines, so he tuned into the specific features of coastlines and how people were measuring them.

What Richardson discovered was staggering. The unit of measurement employed by the measurer affected the recorded length of a coastline. The irregularities and natural features of coastlines resulted in alterations to the human ability to measure.

Picture a rocky coastline with bumps and jutting edges. If you used a meter stick to measure, you’d miss some of the coast’s contours. If you went down to a foot-long measurement, you’d catch some more contour and your measurement increases. If you go down to a six-inch ruler, your measurement increases yet again.

Thus, The Richardson Paradox is born: the length of the coastline increases without limit as the unit of measurement shrinks. The more you zoom in on the unique features of a coastline, the more the inconsistencies show themselves. [Not only that, but we can also consider that coastlines are ever-evolving with longshore drift and the geologic processes that shape our shores- even if we reached an accurate number (and we can’t), it would change after the next big storm.]

Here’s an animation of the coastline paradox.

So- back to your best guess about the coastline of the U.S.?

Congratulations! 🎉You were right! No matter which choice you made, your answer aligned with some governing body or other.

According to the Congressional Research Institute, the coastline of the U.S. was 12,383 miles, then later made a jump to almost 30,000 miles.

The CIA pegs our coastline at 19,924 km (12,380.2 miles), while the NOAA records it as seven times the length of the CRI, at a whopping 88,633 miles.

When I said the length increases without limit, I meant it:

“No matter how far down the rabbit hole you go, you’ll eventually hit the molecular level.
If you zoom in far enough, you’d measure the beach by counting atoms.”

Richardson’s posthumously published work on conflict forecasting based on shared border length never amounted to a reliable body of mathematical theories about conflict, but it did demonstrate that mathematics can contribute to a better understanding of conflict, and people are still working with his hypotheses.
And more importantly for this Rabbit Hole, the completely counterintuitive paradox he uncovered while studying conflict opened the door to an understanding of fractals.

“MORE SPLENDID DETAIL AND VARIATION’

Benoit Mandelbrot (1924-2010) built off of Richardson’s coastline paradox work with the development of his understanding of fractals. Mandelbrot, born in Poland, was a French-American mathematician who worked for IBM and developed what’s known as the Mandelbrot set [start this Numberphile video at 8 minutes to see the visual that first clued Mandelbrot in].

Mandelbrot, in the Ted talk below, explains the nature of “roughness” and fractals by illustrating the difficulty of measuring the surface of a Romesco cauliflower or the area of a human lung.

“Humanity had to learn about measuring roughness… very few things are very smooth.” -Benoit Mandelbrot

(By the way, try as they might, scientists haven’t yet isolated the gene that makes Romesco flower into a fractal shape.)

But wait- why are we talking about scientists measuring cauliflower? What’s the relationship between fancy vegetables and understanding conflict through a narrative lens?

James Bridle, in their exploration of ecology and “more-than-human intelligence,” sums up for us the effective result of Richardson’s paradox:

“Instead of resolving into order and clarity, ever-closer examination reveals only more and more splendid, detail and variation.”

In other words, Richardson’s revelation was that upon closer inspection, nearly everything on Earth is more complicated than it seems. Fractal math can find order within patterns that seem impossibly complex. And fractals, used as a mirror of the complexity around us, are instructive. And they’re everywhere.

Let’s look at the four types of fractals:

Infinite intricacy
Zoom Symmetry
Complexity from simplicity
Fractional dimensions

Here’s one fun example from the Fractal University account on Instagram. The Sierpinski triangle is a famous illustration of a fractal that utilizes a simple, repetitive set of two actions, which generate a complex arrangement.

You start with a triangle, then repeat two simple rules over and over, plotting midpoints.

In the first image, the first point is plotted, apart from the three initial triangle points.
Second, the artist has begun plotting random dots and corresponding midpoints:

Here’s what this project looks like after 1000 points, and then after 25,000 points.
If done correctly, this final image will be generated *every time.*

Repeating the right two actions, even at random, triggers a bigger, observable, and rather intricate pattern, ad infinitum.

There are untold numbers of things in our midst that employ fractals. Trees have fascinating fractal aspects, and so does the Eiffel Tower. Whether aware of it or not, humans have been employing fractal design using the Fibonacci numbers, which far predate our modern (Mandelbrot-era) understanding of fractals.

We’ve utilized fractals extensively in art and music. We rely on the fractal nature of our lungs. Our very cells appear to arrange themselves in a fractal pattern.

Why? It appears that the universe plays favorites. And it seems, so do we.

THE FRACTAL NATURE OF CONFLICT

Fractals have entered the chat.

Popping up in the work of movement activists and theorists like adrienne maree brown, we’re seeing the rise of fractals studied within the emergent strategy framework, as well as in traditional academic settings.

Look at the delta around New Orleans, and then look at how these veins and artery systems move through your system and your heart and your lungs. Look at the spiral shapes on your fingertips, and then look at the shape of galaxies.
And in that way, we can begin to see there are no isolated patterns. The universe has some favorites, and they repeat and they repeat, at every scale.
- adrienne maree brown in an interview with Krista Tippett.

A couple of details about fractals:

Fractals are self-similar at all scales. Large to small, small to large, and in relation to the whole.
Fractals are generated from the repetition of processes (aka recursive algorithms). Here’s the process: You start with an initial condition and apply a process that creates a new condition. Then you apply that same process to the new condition.
Fractals can generate infinite complexity. The complexity increases with the repetition of processes. In the mathematical realm, fractals are ever-unfolding.

Apart from mimicking our bodily shapes (lungs as trees, fingerprints as spiral galaxies), we can begin to see fractals in our behaviors, in our histories, and in our conflicts.

Staying with adrienne maree brown- in her book Emergent Strategy, she explains the nature of the universe’s patterns being reflected down to the smallest of structures:

“How we are at the small scale is how we are at the large scale. The patterns of the universe repeat at scale. There is a structural echo that suggests two things: one, that there are shapes and patterns fundamental to our universe, and two, that what we practice at a small scale can reverberate to the largest scale.”
—adrienne maree brown, Emergent Strategy (p. 54).

The places where we find, create, and resolve conflict reverberate across the dimensions of scale and to the whole.

A small conflict will be felt across larger dimensions, and a larger conflict will be felt at the smallest scale.

Anyone who has ever mediated a divorce for people raising toddler twins far from family in a village-less society during a pandemic can tell you– these larger dimensions can cut deep at the personal level. Conversely, the small conflicts also affect the whole.

Conflict, as an aspect of human nature, is experienced simultaneously on multiple scales, often repeated, and can bear seemingly infinite complexity.

FRACTALS IN STORYTELLING

Structure, stories, in essence, are fractal in nature.
- Bryce McNabb

As Mandelbrot said, we humans had to learn about roughness. If conflict is the inevitable bumping up against the rough edges of the human experience, then maybe these rough and variable edges, these places so inevitably scalable that our tools of measurement no longer apply, could be where the human story goes to grow. As Neil Theise explains in his book Notes on Complexity, “the boundary at the edge of chaos is fractal.”

What prevents chaos at the boundaary? How do we jump scale? How do we share meaning? How do we grow through conflict?

In stories.

It’s been eloquently advocated that the three-act plot structure is a representation of fractals.

In a graduate level narrative-focused conflict resolution course, Professor Simmons spelled out the fractal nature of storytelling and episodic structure using Yorke’s five-act symmetrical model.

Not only do individual episodes use this structure, but seasons of shows also have it. To further zoom out, five successive seasons of a show will utilize this structure.

Why?

Because it works.

Okay… but why does it work?

I’m guessing for the same reason the Sierpinski triangle (above) works.

There is an inherent fractal world, a scalable geometry of change, growth, and understanding, that we can see reflected back in our storytelling modes.

As for the implications for conflict resolution, growing the story in these directions can be messy, time-consuming, and out of step with our culture’s dominant interest-based negotiation paradigm.

Can things actually resolve into complexity? And more importantly, can we be okay with that?

I’d venture that narrative conflict analysis, as a mirror for fractal reality, makes room for the multi-scale, multi-dimensional makeup of our experience in conflict and, correspondingly, the multi-scale, multi-dimensional makeup of potential resolutions.

Joshua T. Fisher and Peter Coleman of Columbia University suggest that intractable conflict, in particular, is a representation of fractals:

“In the language of applied mathematics and dynamical systems theory, these conflict systems revolve around attractors that give rise to strong, coherent patterns that draw people in and resist change.”

They conclude:

“Our attempts to manage conflicts must also be self-similar; scaling to accommodate the precise manifestation of an attractor at different scales. We need to examine the patterns at multiple levels, in order to gain a more comprehensive understanding of the entire pattern of the attractor in a conflict. We likewise need to understand how and why conflict dynamics and attractors scale in order to understand the fundamental component or structure of an intractable conflict.”

Though approaching the idea from a different angle than emergent strategist adrienne maree brown, they land in much the same place, meeting Amanda Ripley there. Complexity isn’t an aberration.

CARRYING COMPLEXITY WITH A NARRATIVE APPROACH

Conflict, like our lungs, is a reflection of our fractal nature. We tell stories of conflict in a fractal method. And any approach to resolution will need to adequately reflect the fractal complexity of the conflict itself.

What does this mean for peacebuilders? If we accept the fractal nature of conflict as a reflection of the fractal order of the universe, and we understand the scale of the attractors as a singular factor in our ability to comprehend a conflict, then our work has to carry that complexity.

Fractal Stained Glass; Wikimedia Commons

We have to engage our conscious attention at different scales all at once.

We have to remain practical and solution-oriented while recognizing that there are deep forces and patterns at work that may transcend our understanding.

This means we need to be as attuned to the small dynamic shifts as we are to the colossal, societal structures whose seismic force is felt at every level.

Revisiting the wisdom of adrienne maree brown one last time:

“every single large system or structure or network or political protocol, all of it is made up of small things: of humans either having or not having necessary conversations, and humans being willing to stand up for what is right and stand up against what is wrong.”

What else besides a narrative approach to conflict analysis can capture the action, the momentum, and the ever-shifting sands of conflict?

If the undercurrent of intractable conflict is fractal in nature, then capturing the conflict’s essence in a story structure that, in itself, mimics the fractal nature, isn’t some distracting siren song.

Quite the opposite, actually. It may be the most logical approach.

Narrative complexity in conflict isn’t the diversion.
It’s a natural, fractal reflection of conflict itself.

Richardson’s hope of forecasting conflict like the weather never came to pass. But he opened a window (a fractal patterned, stained glass window, I assume) into our understanding of conflict complexity that can’t be closed.

Against that traditional positional bargaining, interest-based backdrop, and whether for the use of conflict or quantitative genetics, we can hold a fractal reality, and I can keep the “good news” haiku in my back pocket:

the good news about
complexity is that more
becomes possible

*Postscript:*
I didn’t realize it until I wrote this Rabbit Hole that the note I jotted down at the beginning was written in a Hilma af Klint notebook.
Guess what she was exploring in her work?
An innovative artistic geometry informed by….
*You guessed it.*
“Beautiful, damn hard, increasingly useful. That’s fractals.”
-Benoit Mandelbrot

References:

Bridle, J. (2018). Ways of being. Verso.

brown, a. m. (2017). Emergent Strategy: Shaping Change, Changing Worlds. AK Press.

Coleman, Peter & Vallacher, Robin & Nowak, Andrzej & Bui-Wrzosinska, Lan. (2007). Intractable Conflict as an Attractor: Presenting a Dynamical-Systems Approach to Conflict, Escalation, and Intractability. SSRN Electronic Journal. 10.2139/ssrn.1066810.

Hayes, B. (2002). Computing Science: Statistics of Deadly Quarrels. American Scientist, 90(1), 10–15. http://www.jstor.org/stable/27857587

Ripley, Amanda. (2019, January 29). Complicating the Narratives. Solutions Journalism Network. https://thewholestory.solutionsjournalism.org/complicating-the-narratives-b91ea06ddf63

Investigating the Mathematical Core of Narrative Conflict Analysis