Full Article: Curves that Matter

Here is a link to a full version .pdf of the Curves that Matter article.

You can also find prior articles on my website.

Cheers... Rob

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.

Curves that Matter: Normal and Rhythmic Curves

Normal Curve

Shape and characteristics of the curve. The normal curve, or bell curve, is one of the most recognizable distributions and describes many natural patterns and phenomena. The curve is symmetrical around a middle or median with 50% of the values to the left of the midpoint and 50% of the values to the right. The curve can lean or skew to the left or right or it may appear flatter or taller (kurtosis). When applied to probabilities the curve is a valuable predictor of natural distributions from human traits, to where flying balls land, to just about any kind of random variation in outcomes of a system, either simple or complex.

The central limit theorem further explains the normal curve by suggesting that the distribution of any combination of a large number of generally random activities or variables will produce a normal distribution with an expected value for the mean and predictable variation around the mean. For example, rolling a pair of dice 100 times or so will produce a lot of 7s and far fewer 2s and 12s.

The curve allows many kinds of behaviors to be explored with good estimates of expected results. The area under the curve between any two points is interpreted as the likelihood or probability of events occurring within that range. The curve is divided into segments called standard deviations, positive to the right and negative to the left. Approximately 68% of the distribution lies within one standard deviation of the mean; 95% of the distribution lies within two; and 99.7% within three standard deviations.

Some examples. There are many applications and examples where the normal curve is enlightening. Here are a few:

Natural and biological traits - Early in the formulation of the normal distribution were studies based on the growth patterns of pea plants. In 1809, Gauss used the then new normal curve to better understand astrological data. In the years that followed, it was discovered that when considering a large group of humans, nearly all basic traits like height, weight, strength, or intelligence are normally distributed. Today, the normal distribution is fundamental to learning about how natural and biological systems and traits are understood.

Test scores - Like human traits, the tests that measure them produce scores that are normally distributed. But testing gets a double whammy from the normal curve. The tests themselves are not perfect and have errors in their scores as estimates of the traits. Those errors themselves are also normally distributed and when planning to use scores, measurement professionals account for both the normal distribution of the traits and the normal distribution of the errors.

System errors - No machine or system is perfect and like tests, the errors that are made are normally distributed. When machines that cut nails, for example, are inspected for variation, the lengths of the nails that result form a normal distribution. The tails of the distributions, those extremes on either the higher or lower end can serve as cut points for when a nail is too long or too short. The normal distribution helps us understand and mitigate system and machine errors, both in simple machines like nail cutters and more complicated systems like medical testing where the stakes are higher.

Population and organizational behavior - We know that large groups are normally distributed along any number of group and individual characteristics. But these groups, whether in populations or in organizations, have a unique behavior known as regression to the mean. Upon repeated measurement of a single trait or behavior, individuals who exhibit extreme traits, slowly regress or move toward the mean.

Insights for strategy crafting. The normal curve yields trustworthy insights for crafting strategy. The consistent patterns of symmetry, distribution and variance around the mean, predictable deviation, and regression to the man apply in universal ways.

Standard scores, performance, and selection systems - The normal distribution can be standardized, or set to a mean of 0 and standard deviation of 1. This allows for distributions of many kinds of traits or characteristics to be compared and used in analytical ways. One such application that yields insight to strategy is performance measurement and selection systems. A critical part of strategy execution is establishing and managing the resources necessary to drive implementation. For large organizations, tracking the performance of human resources and selecting the right resources for the execution path is a critical matter. The normal distribution of performance outcomes gives key insight to managing the resource necessary for success.

Predictable outcomes in the environment - Like human performance internally, many kinds of environmental outcomes are normally distributed and wise strategy crafters can use these expected or predictable distributions to their advantage. As if making a bet, investing strategy with an expected outcome near the mean yields the best likelihood of success, and expecting results between -1 and 1 standard deviation will result on average with two-thirds of the results realized. The biomedical and agricultural industries build their core strategic models on these behaviors. On the other hand, less than 1 percent of outcomes fall outside the extremes of -3 and 3 standard deviations. Insurance companies use knowledge of the normal curve in their strategy crafting to both protect customers during extreme events and consistently generate profit margins.

Game theory and simulations - Modeling future scenarios is a key activity for strategy crafting and advances in computing have allowed gaming and simulations to become help tools in building strategies. By combining normal distributions of multiple forces, both current and future, likelihood scenarios can be developed to help uncover which strategic choices may have more advantage over others.

Diffusion of innovations or new ideas - One classic use of the normal distribution was developed by Rogers in the Diffusion of Innovation. The graphic below shows traits of adopting new ideas across the normal curve.

Adapted from Rogers Diffusion of Innovation, 1962

Rogers suggested that individuals adopt new ideas with the largest proportion being the majority, or 68% between -1 and 1 standard deviations, where the first group is the early majority and the second group the late majority. Interesting are the groups outside of one standard deviation. Only 2.5% of a normal distributed population could be considered innovators and 13.5% the early adopters. On the other side of the distribution, the tails beyond one standard deviation are the laggards, the last ones to adopt the news ideas - if they ever do.

Rhythmic Curves

Shape and characteristics of the curve. Rhythmic curves, periodic (or sine) waves are most often observed in the behaviors of stable systems where there is motion (things are not completely at rest) but the system has settled into a pattern. Rhythmic curves are centered around a midpoint and oscillate over time returning the midpoint at regular intervals. These curves exist because energy is moving through a system.

Rhythmic curves can be described by at least four different kinds of characteristics: frequency, period, wavelength, and amplitude. Frequency is the how often the rhythm cycles in any given time segment and a related characteristic, period is how long it takes for the pattern to repeat. Like the tempo of a song, these curves can have rapid frequencies vibrating extremely fast or very slow frequencies lasting thousands of years. Wavelength is the distance between peaks of the curves. For curves representing physical phenomena, this can be seen as an actual distance. Finally, amplitude is the maximum deviation of the curve from the midpoint and can be understood as the height of a wave.

Some examples. Almost all stable, dynamic systems are characterized by one or more rhythmic curves. Some notable examples include:

Natural and manmade rhythms - Simple mechanical systems display nearly perfect waveforms. From the swinging pendulum, to the rhythms of clockworks, these systems have clear and predictable cures. Biological systems are equally driven by rhythms like heartbeats and the well-recognized sinus rhythm and more complicated biorhythms that drive our waking and sleeping, hormonal, and hibernation cycles.

Flow of energy through matter - Like a ripple in still water, most forms of energy flow through states of matter (liquid, gas, and solids) in wavelike patterns. Whether it be the rise a fall of an earthquake, waves lapping on the shore, or the winds rising and falling on the mountainside, there are constant examples of energy in our environment.

Community and organizational rhythms - More complicated systems do display rhythms as well. Most of our organizations are driven by management induced rhythms and there is a long history of organizations being viewed as machines. We see annual and quarterly reports and cycles, monthly and weekly meetings, and daily and hourly metrics all as evidence of rhythmic curves. Communities as well display rhythms. They go through both cycles of birth, growth, and decline, as well as rhythmic cycles of productivity, energy levels, and convening and communicating.

Cultural, economic, and societal rhythms - At the broadest level, we see societies and economies cycling in rhythmic patterns. Many cultural rhythms have their source in current or historical seasonal variations. From agronomic economies and religions to more modern fashion trends and educational academic years, we observe a large number of cycle inducing causes and effects. But there are other long-term rhythms and cycles. For example, it can be observed through history that periods of creativity and romanticism have cycled with periods of science and technological advances. And on the grandest natural scale, we are aware of epochs and ages that last millions of years in length.

Insights for strategy crafting. Most strategy crafters, whether intentional or not, count on either stability, straight lines, or rhythms for success. Uncovering, understanding, and anticipating rhythms are key insights for those crafting strategy.

Capturing the energy contained in waves - Rhythmic patterns are in essence energy driven and many waves both contain energy in themselves or in the behaviors that they induce. Strategies can be crafted with the intention of capturing this energy in a sense and using it to the strategy’s advantage. From the simple approach of selling lemonade in summer to the more subtle insight to synchronizing communications with naturally occurring rhythms in a new markets cultural behavior, strategists should seek out the rhythms at play in their environments of concern as well as hope to capitalize on their permanence and characteristics as they move into the future.

Making waves - Often, we find that an organization or its environment is lacking the necessary rhythms to make change, create stability, or strength along the lines of our strategic intent. The insight here is that rhythms can be intentionally induced. Whether it be a slow quiet drumbeat or a high energy noisy pattern, new rhythms serve to enhance flow of energy through organizational systems and their environments.

Changing the scale of rhythms - There are two ways in which strategies can be crafted to acknowledge the major forces acting on an organization or its environment so as to benefit from the naturally occurring waves; one by amplifying and another by dampening. Amplifying a rhythm as a strategy adds energy to the wave by either increasing the primary characteristics frequency, period, wavelength, or amplitude. Acting on the same characteristics, dampening takes energy away from the rhythm and decreases its impacts and effects.

Laminar flow versus turbulence – A final insight about rhythms for the strategy crafter is about the stability in the system that allows the waves to remain intact and persist. Rhythms keep up and waves move along well when the environment is smooth. Those that study how energy transmits through substances call this smooth flow as laminar flow. Its opposite is turbulence. During turbulent flow, unsteady vortices appear on many scales and interact with each other, disrupting energy in the system and it’s existing rhythms. Strategies for intentional turbulence can be used to change cultures, break bad habits, or prepare for new ideas to have impact by breaking down rhythms of stability. This also gets us ready for more complicated curves and their inherent nonlinearity. More to come.

Curves that Matter: S-curve

S-Curve

Shape and characteristics of the curve. The S-curve generally shows growth over time and can be broken down into three (or maybe four) key parts. Early growth is slow with modest gains in the earliest part. Once the effort or activity gains momentum, there is a period of substantial and increasing growth. This is maintained until certain limits are reached and growth begins to slow in the third part. Finally, in the last part serious upper limits are researched and growth stops where consistent inputs maintain a high level of output but no additional output.

Logistic functions are inherently nonlinear. Many forecasting and projection techniques make assumptions about linear growth. What we actually experience, especially in the world of strategy and living systems, is varying periods of quick and slow growth and decline. A key characteristic of the S-curve is two cycles of rapid then slow growth and decline.

Some examples. There are literally thousands of applications and examples where the S-curve shows its characteristics. Here are a few:

Growth in living systems – most living systems, from the smallest cell, bacteria, and viruses to larger populations of animals to humans in cities for example show growth S-curves where initial populations grow slowly, then gain momentum and grow quickly, then slow growth and level out at a maximum population.

Carrying capacities – from a similar but opposite perspective, we see the S-curve at work in the carrying capacities of systems. Whether it be trees on an island or reactions in a beaker, closed and defined systems have an upper limit in their ability to sustain growth of their components.

Individual and organizational learning – individual skill and knowledge acquisition often follows the S-curve as well. It takes a while to learn early concepts but once the general patterns of a new discipline are acquired, learning proceeds at a rapid pace. Advanced topics are more difficult and tend to proceed more slowly until the bulk of the discipline or skill has been understood. This applies to both individual learning as well as organizational learning when new processes, technologies, or other ideas are introduced.

New products & technologies – a very well known example of the s-curve is the introduction and adoption of new product and technologies to a market. Adoption of new products starts slowly and as word spreads, growth accelerates until the market starts to saturate. Growth slows dramatically during saturation until desire for the product or technology wanes and eventually slows to a trickle or stops. One special adaptation of the S-curve and new technology is the Hype Cycle popularized by the Gartner Group. This curve combines two S-curves of different scale and has five distinct phases where expectations change as the new technology makes it way through the population.

Insights for strategy crafting. Awareness of the nature of individuals, groups, organizations, markets, and other phenomena yield key insights for those concerned with crafting strategy. The general pattern of slow growth - rapid growth - and declining gains applies universally in predictable ways.

Planning and execution effort for outcomes – It takes considerable resources to plan for change and then implement and execute. Knowing ahead of time we can gauge the appropriate amount of resources to be applied to support launch and change efforts. Early efforts need a lot of vision and support energies and often the outcomes we experience are underwhelming at first. When the new ideas take hold, we sometimes experience gains that are well beyond expectations or the resources we’ve committed to support growth. Flexibility in matching and optimizing natural growth or adoption patterns is helpful. And finally, as a natural part of any process, we reach a point in planning or execution where our inputs and resources experience diminishing returns. There are a few things we can to keep efforts moving forward.

Linking multiple curves together (jumping) – In most cases, a set of environmental (or sometimes internal) forces or conditions act on effort late in the S-curves growth to limit further growth. If these forces can be changed or disrupted, there is the possibility to reset the growth pattern and experience a second or third period of exponential growth or gain. In essence this reset is like linking two S-curves together - some call this curve jumping. Strategy can be intentionally crafted to both observe S-curve growth patterns and employ curve jumps at appropriate times.

Planned disruption – One way to do this is with planned disruption. Highly stable environments have many S-curve phenomena operating in the saturated growth stage. One way to stimulate growth is to intentionally destabilize the environment and direct energies and efforts to opportunity areas where exponential growth may next occur. But rather than random disruption the idea of targeted or planned disruption yields more expected results. Like positive turbulence, planned disruption is a tool for the strategy crafter; perhaps one not wielded in excess but with cautious prudence.

Adoption of ideas – Strategy, particularly when it’s focused on innovation, is often about putting new ideas into action and creating value. Idea generation and absorption follow S-curve growth patterns. When organizations or environments are continually barraged with new ideas, changing technologies, or redirected strategy, they shift toward slow growth as they are both saturated and grappling with the efforts of new growth. Keeping the organization in peak output relative to input requires a balanced timing of pushing change forward. We know that groups of individuals respond in predictable ways across a wide distribution of types. I will discuss this further along with the next curve, the normal curve.

Curves that Matter

I recall a conversation I had a few years ago with a colleague. We were attending a meeting of the Association for Managers of Innovation in Celebration, Florida and the group was leaving our meeting space to take a group photo. I can’t recall the spark that set us off, but we were talking about the S-curve (sigmoid curve or logistic function) at length. The curve essentially suggests that things grow rapidly at first until the environment is saturated until things grow more slowly and then eventually growth stops. We started to give examples to each other where the curve helped us understand something complicated, or how we used the curve to help us do something, and moved on to surprising places the curve explained something mysterious. Over the course of the 12 minutes or so that it took us to walk from the meeting room to the photo location and get organized for the picture, we uncovered a fair number of uses for the S-curve and joked about how it might be fascinating to do the same thing with the handful of ubiquitous curves that gave insight to so many processes. This conversation has stuck with me, somehow not fully resolved. Well, now four years later, never having done just that, here I am writing about these curves and how they all matter to Innovations in Strategy Crafting. I will explore different curves and give insight to a few applications of each for those crafting strategies and working with organizational change. More to come.