Fractals, Self-Organizing Principles, and Self-Segregation

Fractals are all around us

Came across this study on planting patterns, “Fractal planting patterns yield optimal harvests, without central control” that bears some closer review.

What’s fascinating about this is that it seems to present a real-world solution to a problem commonly referred to as the “tragedy of the commons,” in which the self-interest of individuals leads to the destruction and overuse of shared natural resources.

Spatial patterning often occurs in ecosystems as a self-organizing process caused by feedback between organisms and the physical environment. “The centuries-old Balinese rice terraces are also created by feedback between farmer’s decisions and the ecology, which triggers a transition from local to global scale control,” explains Lansing. “Our model shows for the first time that adaptation in a coupled human-natural system can trigger self-organized criticality.”

This is absolutely fascinating, and it’s interesting to consider this in relation to other ideas on a “self-organizing criticality” that we’ve explored here before, such as in brains and sand piles.

For proponents of ed reform who argue against centralized control, this seems like it could be worth digging into further.

The aspect of fractals here is also tantalizing. After chatting with a colleague about the article, he referred me to Ron Eglash, a mathematician who has studied fractals in African history and culture. Do yourself a favor and watch his TED talk, it will blow your mind.

Eglash raising an interesting point about self-organizing principles: they can be wonderful, as in Google search, or our brains. But there is also a dark side, such as in the spread of HIV or the damaging effects of capitalism. He suggests that the fractal algorithms employed in Africa could present us with “robust” “ways of doing self-organization — of doing entrepreneurship — that are gentle, that are egalitarian.”

I’d love to explore more about how the fractal design of school structures and systems could be utilized for a productive purpose. Please share if you’ve got more on this.

Another interesting angle on the Balinese rice farmers is suggested in an earlier study reported also on, “Phase transitions of rice farmers may offer insight into managing natural resources.

Their study and modeling seems to suggest that smaller self-segregated communities within a society are desirable in the long-run.

They found that the downside of the segregation is that it increases the social disharmony throughout the society as a whole. The upside, however, is that the social disharmony within each community becomes very low. In some communities, individuals are more likely to keep cooperating with each other—using the shared resource fairly—compared to the situation without segregation. These results were very similar to what the researchers observed in the segregated society of the Balinese subak.

This seems to be a dark side unmentioned in the more recent study on how the Balinese farmers exemplify a self-organizing society in harmony with nature. Or perhaps this isn’t a dark side — it’s a suggestion that some self-segregation can be positive.

But I don’t know anything about the Balinese subak, either, so not sure how much to read into this. Certainly worth going deeper into this. If any readers have knowledge of African fractal algorithms or Balinese subaks, please share!

The Resiliency of a Snowflake

By Amada44 (Own work) [Public domain], via Wikimedia Commons
Just came across an interesting addition to the idea of redundancy in networks (via Slashdot via NewScientist) which suggests a three part hierarchical structure of partial loops can be the easiest to repair. The structure looks akin to a snowflake.

They found the best networks are made from partial loops around the units of the grid, with exactly one side of each loop missing. All of these partial loops link together, back to a central source. These have a low repair cost because if a link breaks, the repair simply involves adding back the missing side of a loop. What’s more, they are resistant to multiple breaks over time, as each repair preserves the network’s fundamental design.

These networks have three levels of hierarchy – major arms sprouting from a central hub that branch and then branch again, but no further. When drawn, they look remarkably like snowflakes, which have a similar branching structure.

We’ve explored the idea of redundancy in networks, as well as the concepts of loop networks and anastomosis. Such structures demonstrate greater resiliency in the face of chaos.

The now relatively mundane notion of distributed leadership via grade-level, departmental, and other teams throughout a school seems to align with the concept of a resilient network structure. The challenge for district leaders, however, seems to be how to connect those in-school structures to a wider external network that will support them without introducing more chaos.

In NYC, word is out that the non-local network structure which the NYCDOE under Klein instituted to “break up the fiefdoms of the districts” may be on its way out. So the big question here is whether Fariña can reintroduce localized power while avoiding the corruption and silos that plagued some districts.

Not an easy thing to achieve in a system as vast as NYC, but a looped network, snowflake-like structure may be worth bearing in mind.

Loop Networks and Anastomosis

By Rosino (vascular) [CC-BY-SA-2.0
via Wikimedia Commons

In my last post, we looked at the geometric features of resiliency, which included diversity, web-network structures, distribution across a range of scales, and the capacity to self-adapt and self-organize.

We can learn more about the characteristics of web-network structures in a fascinating article in Quanta Magazine, “In Natural Networks, Strength in Loops” by Emily Singer.

The article describes how complex networks form into an architecture of a random series of interconnected, hierarchically nested loops. Such nested loops increase the resiliency and robustness of networks. This resilient architecture can be seen in leaves, insect wings, cerebral vasculature, fungi networks, and the Eiffel Tower.

The cheapest network to operate is a simple branching tree structure, which is employed by some ancient plants. Though efficient, this structure is not very resilient. When a link is damaged, parts of the system suffer loss of fluid and die.”

This is a theme that you can also hear reverberating throughout business and management literature. What seems to be most efficient on the surface—simple hierarchies with clearly delineated pathways of power and resources—is also most fragile.

“[Researchers] found that an architecture of hierarchically nested loops — meaning loops within loops within loops — is most resistant to damage. “Loops make the network redundant.””

“The researchers also found that loop networks can better handle fluctuations in fluid flow as environmental conditions change.”

The world we live in today is increasingly volatile. Communities, markets, and states are increasingly subject to external or internal turbulence. This is why modern leaders increasingly talk about agility and adaptiveness, rather than domination and control.

“In a paper published in Nature Neuroscience in July, the researchers showed that the capillaries form a continuous network. “This means that the microvessels — capillaries — are fully connected among each other,” said Kleinfeld. “There are no regions of isolated vessels, no gated communities in real estate terms.””

Isolation increases fragility. This principle made me think of another excellent article, “The Social Life of Genes” by David Dobbs, which I will dive into further in a future post. In his discussion of researcher Steve Cole, he explores the idea that loneliness and social isolation results in greater illness and premature death. As Cole puts it, “Social isolation is the best-established, most robust social or psychological risk factor for disease out there. Nothing can compete.”
So how can we apply the principles of naturally resilient networks to man-made structures, such as school communities?
Let’s consider of how an architecture of nested loops can apply to leadership, curriculum, and infrastructure. We must seek opportunities to build connections between:
  • Distributed leadership throughout every department and grade level team (teams here equating with “loops”)
  • Interdisciplinary practices and content
  • Learning spaces that create varied opportunities for social interaction and solitary study
There’s a great term that we can use here which a commentator introduced on the original article: anastomosis. Anastomosis refers to connections made between adjacent channels in a network. Anastomosis is what we want to create in a school community. By seeking opportunities for anastomosis in our curriculum, infrastructure, and teacher teams, we can create more resilient schools.