Curves that Matter: Golden Spiral and Fractals

Golden Spiral

Shape and characteristics of the curve. Geometry is one of the magical disciplines of mathematics. The numerical representations of natural phenomena are made visible by geometry. Kepler studied nature and motion in the 17th century and suggested the golden spiral (actually the golden ratio on which it’s based) and the Pythagorean Theorem were two of the most magical laws observed in mathematics and in nature.

The golden spiral is an expanding curve that starts from a center point and grows outward. The growth factor of the spiral is the golden ratio, an irrational number approximately 1.618. This ratio is most simply defined when you divide a line into two parts so that the whole length divided by the long part (a+b/a) is also equal to the long part divided by the short part (a/b). These lengths also produce the side of the golden rectangle.

The curve can be constructed by continuing to segment the rectangle into smaller and smaller portions using the same ratio and connecting the corners of the squares with quarter circles.

The golden spiral is also related to the Fibonacci numbers. The Fibonacci sequence is constructed by summing the two prior numbers to create the next number. It is simply written 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 and so on. As the sequence gets larger and larger, the ratio of the larger number to the smaller of any pair approaches the golden ratio or 1.618.

Some examples. But the magic of the golden spiral goes well beyond the mathematics. What makes it a curve that matters is its recurrence in nature, aesthetics and beauty, psychology and large social systems.

Building pretty things - It’s been long believed that the golden ratio, especially geometric designs built to those proportions are pleasing to the eye. There are countless ancient and modern structures that use the golden spiral and rectangle both at large scale for the entire structure and at small scale for the details. There are also numerous analyses of classic works of art and painting showing the ratio at work. Many of today’s logos, graphics, web pages, and 3D designs have the ratio in their primary structures, whether built that way intentionally or done intuitively. When you see something striking, look a bit closer for the spiral and proportions at play.

Natural patterns - The list is very long of objects in nature that adhere to the golden spiral to include seashells, spiral galaxies, hurricanes, biological cells, flower petals, pinecones, etc. Da Vinci’s Vitruvian Man shows the relationships of human anatomy cast over a pentagon, where we know the golden ratio is hard at work managing the relationships among the angles and segments. In fact, all across the human body the ratio is evident, from the ratio of one connected bone to another to the width of teeth to facial features to less obvious internal structures and processes. In fact, nearly all life forms that we know about show an amazing adherence to the golden ratio in their proportions and in other ways.

Examples in complex social systems - From the obvious appreciation of beauty and human expectations to more complicated relationships and human processes within the areas of learning and psychology there are many examples of the golden ratio and spiral in social systems. There is evidence that some organizational processes are optimized along the proportions of the golden ratio like the balancing of innovation portfolios, managing risk and investments, even the stock market. Perhaps nearing the borders of our understanding, theorists like Ken Wilber and Don Beck have suggested that all of human evolution, growth and change of cultures, and the universe itself is fundamentally locked into golden spiral growth.

Insights for strategy crafting. There is obviously something fundamental in our universe about the golden ratio and associated spiral and its permanence across scale and size from tiny vibrating atoms to the organization of galaxies. We can use this knowledge as we craft strategy.

Making strategy beautiful - An obvious but not frequently employed insight is to make strategies themselves attractive. More than just the wisdom or intelligence in the content of the strategies or the thoughtful engagement in the process, the balancing of the words, images, and proportions of the strategies should be attractive to people. The better the strategies look and feel, the more likely individuals are to pay attention.

Be attentive to spiral growth patterns - Whether it’s tracking market growth or planning organizational growth patterns, spiral growth proportional to the golden ratio is a natural expected result. Knowing this ahead of time, we can plan to take advantage of this awareness at all levels of scale. Changes in economic models and stock prices reflect human opinions and expectations all along proportions of the golden ratio.

Human and organizational development - Learning patterns and human and organization development followed the Fibonacci sequence and golden spiral. By setting expectations, limits, and goals for individual and organizational learning, strategy crafters can help to gauge the development of capacities necessary for successful execution. Timing and sequencing of strategies should follow the building of the capacities with demands balanced with individual and organizational competencies and skills.

Leading large scale change - Large scale systems change, barring the most catastrophic kind, requires consensus building and aligning a lot of people. It has been suggested that leadership, consensus building, and large scale change follows a golden spiral growth pattern. With this knowledge, strategy crafters can plan for waves of growth in the new ideas and aspirations of the strategies. Layer by layer, strategy leaders and champions can build consensus with natural pattern of growth approximating the golden spiral.

Fractals… more than just a curve

Shape and characteristics of the curve. Fractals are complex geometric patterns that arise from simple, recursive behavior. Importantly, many patterns and behaviors within living systems are inherently nonlinear and take on a fractal nature. Fractals are patterns that repeat across a system on varying scales. While linear formulae and quantified components can be depicted as simple shapes made of lines such as squares, rectangles, and circles, a very different geometry exists, however, in nonlinear systems. Fractals can be mesmerizing and at the same time take the form of common yet complicated forms found in nature.

Pascal’s triangle provides a simple way to understand how a fractal pattern might arise. The image below shows the triangle with 1 at the top and numbers below in a triangular pattern where each number is the sum of two numbers directly above it.

If you generate a large triangle and color the even numbers white, and the odd numbers black, the result is an approximation of the Sierpinski triangle, a very simple fractal.

Just to show one of the strange relationships, Pascal’s triangle can also generate the Fibonacci sequence (which was discussed in the Golden Spiral) by adding the diagonals.

There are an infinite number of fractals, but they all have certain characteristics:

• They are generated from simple, recursive patterns. The key here is iteration and feedback. When the output of a very simple operation is fed back in as input and the process is run thousands of times, fractal patterns emerge. Fractals are found in nature, like in the shapes of trees and leaves, because organisms grow using a large number of iterations following simple rules.
  
• They have self similarity at all levels of scale. Fractals are never-ending patterns that have detailed structures at very small scale. If you look at a coastline from space you may see a series of patterns caused by large scale forces, but as you look closer and close, those same patterns repeat. Even zooming in on the microscopic scale, there are similar patterns.
  
• Their irregularities are not explainable by traditional geometry. Most of geometry (and the curves that matter I’ve discussed so far) use standard mathematical functions and are approximations of patterns that exist and much of nature. Fractals are nonlinear, generated from living processes in many cases, and lie outside the explanatory power of Euclidian geometry.
  
• They display emergent properties that are more complex than the simple patterns that created them. An example here is the swarming of birds. Each bird is following a simple set of rules to remain in flight in the flock, but the swarm itself takes on a set of behaviors observable only as a complex system.
  
• They contain attractors that appear in surprising places. Attractors are the special bounded portions within a fractal. For example, when mapping the fox and rabbit populations, the system is attracted to the form of an oval when the two population quantities are measured and graphed - a simple example of an attractor in a dynamic system. As complexity within the system increases, so does the complexity of its attractors, to the point that in the most complex systems, fractal patterns begin to emerge in the attractors. Attractors in many complex living systems cannot yet be measured and mapped precisely, but they do exist.

Some examples. Complexity scientists have shown that patterns replicate at varying scales within the complexity of systems. Fractals have been shown to exist in the most surprising places in nature from plant growth to DNA to organizational structure. Examples are found in the pattern of frost on a window, the shapes of leaves, plants, and trees, and in the gyrations of the stock market. Rarely are simple lines and squares found in natural phenomena.

Fractals in nature - Fractals show similarity across seemingly independent and unrelated systems. For example, the structures of the branching air passages of the human lung are rather similar to the branching structure of broccoli.

Organizations as dynamic, nonlinear systems - Fractal characteristics often appear in large organizational systems. For example, the management style of the CEO is often replicated and reinforced in junior managers. Fractals also appear in organizational processes as, over time, certain units within an organization induce their way of doing business in other units.

Eddies or attractors - Nonlinear systems display patterns in less easily measured phenomena where behaviors are attracted to a repeating form or attractor, like an eddy in a stream. Examples include social norms within organizations, good or bad habits for individuals, and annual weather patterns in certain regions. The complexity of forms and attractors makes it important to discern patterns within systems.

Insights for strategy crafting. Strategy crafting in environments that display complex nonlinear behavior can be quite a challenge, but the reality is the most of our organizations are embedded in nonlinear environments and themselves act as nonlinear systems. There are insights that suggest fractals are curves that matter.

Change and dynamic equilibrium - Nonlinear systems are defined by the flow of energy through the system. This flow of energy creates a dynamic situation, but there are points of stability, attractors, and dissipative structures. This stability within the system acts as an equilibrium but dynamic rather than static. Change can occur in the system through bifurcation that is induced by changing the flow of energy or by modifying system components. As the energy flow through a system increases to the point of transformation, the system will self-organize to a higher level of complexity to absorb the energy flow. This may work in both directions, in that reducing the energy flow within a system could cause the system to move to a state of lesser complexity. These complex system dynamics give insight to strategy crafters and suggest that transformative change is a function of energy flow.

Nonlinear system aperiodicity - In the linear system, the clockwork changes at a regular rate. Like a metronome, the system moves with a regular and periodic beat. When a deeper look is taken at these systems, however, few phenomena are truly periodic. Aperiodicity is more frequently found. Even in the one of the most regular systems, the human heartbeat, scientists have found complex patterns and chaos. So there is a little bit of aperiodicity even in the most rhythmic curves. The lesson for strategy crafting is to account for aperiodicity and balance the dependence on regularity by increasing flexibility and adaptability so that when extremes in aperiodic patterns and turbulence occurs, the system has the ability to respond successfully.

Difficulty in predicting - A key insight from fractals and related complex curves is that there is a significant degree of unpredictability in natural complex systems. The failure of linear models to adequately explain many phenomena led informed strategy crafters to employ complexity and chaos theory to model natural systems such as weather, the behavior of chemical and physical systems, and changes in populations and behaviors in ecosystems. The curves I have described so far tend to outperform straight lines when predicting the future. Fractals result from the complexity and unpredictability and suggest we favor forecasting in terms of probabilities rather than predicting precise destinations.

Emergence - Strategy is consistent, focused behavior over time adapting in response to emerging conditions. As we enact our intentions and move toward the envisioned future the environment continues to change, sometimes in surprising ways. Often our intentions do not become reality, they are unrealized elements of strategy. When new strategic elements emerge, they may not be as intended; good or bad, they are facets of what the future may hold. Knowing that complex, dynamic systems show emergent properties, we should expect our strategic path to be more of an expedition and less of a simple station to station train ride.