## The Cones of Normal Cores

A few days ago, I drew a quick sketch about constraints, symmetries, and asymmetries. Discontinuous inventions break a physical constraint, change the range of a physical constraint, weaken a physical constraint, or bend a physical constraint. That discontinuous invention goes on to become a discontinuous innovation once they escape the lab and business people build a business around it.  Asymmetries present us with the necessity of learning.

So we start with a rotational symmetry out in infinite space. This is the space we seek in the economic sense, the theory yet faced with the realities of practice, the desired, the sameness, the undifferentiated, the mythical abundance of the commodity. We could rotate that line in infinite space and never change anything.

Reality shows up as a constraint deforming the  infinite space and the symmetry into an asymmetry, an asymmetry we are not going to understand for a while. Not understanding will lead any learning system through some lessons until we understand. Not understanding makes people fear.

The symmetry generates data supporting a normal distribution. When the symmetry  encounters the constraint, the density is reflected at the boundary of the constraint. That increases the probability density, so the distribution exhibits skew and kurtosis.

The normal distribution of the symmetry is shown in light aqua. The skewed distribution is shown in a darker aqua.

The skewed distribution exhibits kurtosis which involves a maximum curvature at the shoulder between the core of the distribution and the long tail of that distribution, and a minimum curvature at the shoulder between the core of the distribution and the short tail of that distribution.

With a discontinuous innovation, we enter the early adopter market via a series of Poisson games. The core of a Poisson distribution, from a top down view, would be a small circle. Those Poisson distributions tend to the normal, aka become a normal distribution.

In the previous figure we annotated these curvatures with circles having the given curvature. The normal distribution gives us two circles with the same curvature as the circle is symmetric . The tail of the normal can be considered to be rotated around the core. The skewed distribution gives us a circle representing the curvature on the long tail side of the core larger than the normal , and a circle representing the curvature on the short tail side shorter than the normal.

These curvature circles generate conics, aka cones. Similarly, the Poisson distribution is the tip of the cone, and the eventual normal is the base of the cone. The technology adoption process generates a cone that gets larger until we’ve sold fifty percent of our addressable market. The base of the cone gets larger as long as we are in the early phases of the technology adoption lifecycle. Another cone on the same axis and using the same base then gets smaller and comes to a tip as the underlying technology is further adopted in the late phases and finally is deadopted.

The early tip represents the birth of the category, the later tip represents the death of the category. The time between birth and death can be more than fifty years. These days, the continuous innovations we bring to market in the late mainstreet phase of the technology adoption lifecycle lasts only as long as VC funding can be had. Or, no more than ten years beyond the last round of funding. All of that occurs inside the cone that shrinks its way to the death of the category.

We innovate inside a polygon, so we involve ourselves with more than one constraint. We will look at the distributions involved from the top down looking at the circles that constitute the distributions involved. The normal distributions are represented by circles. Poisson distributions are represented by much smaller circles. Technology adoption moves from a small footprint, a small circle, to a large footprint, a large circle.

Notice that as time pases on the adoption side of the technology adoption lifecycle, the distribution gets larger. Likewise on the deadoption side, the distribution gets smaller. Smaller and larger would be relative to sample size and standard deviations. The theta that is annotated in the diagram indicates the current slope of the technology associated with that constraint and the productivity improvement of the technology’s s-curve, aka price-performance curve, and by price we me the invested dollars to improve the performance.

Notice that when we pair adoption and deadoption, we are looking at a zero-sum game. The Poisson distribution would represent the entrant. The circle tangent to the Poisson distribution would represent the incumbent in a Foster disruption. The
s-curves of both company’s competing technologies is still critical in determining if a Foster disruption is actually happening or not, or the duration of such disruption. Christensen disruptions are beyond the scope of this post.

I annotated a zero-sum game on the left, earlier in time. The pair of circles on the right, are not annotated, but are the same zero-sum game. There might be five or more vendors competing with the same technology. They might have entered at different times. Consider Moore’s market share formula he talked about in his books. The near monopolist gets 74% and everyone else gets a similar allocation of the remainder.

Notice that I used the term Core and orientation in the previous figure. The orientation would have to be figured out relative to the associated constraint. But, the circles in each zero-sum game represent curvature of the kurtoses involved that drive the length of the tails of the distribution relative to a core.

That core is much wider than shown in all but the weak signal context of a Dirac function that indicates some changes to conditional probabilities.

The arrow attached to each kurtosis indicates the size of each as the distribution normalizes.

The core is usually wider. As it gets wider, the height of the distribution gets lower. The normalization of the standard normal or the fact that the area under the distribution will always equal zero is what causes this. I did not change the kurtosises in the figure, but the thicker core implies progress towards the normal and less difference between the two kurtosises. The width of the range should stay the same throughout the life of the distribution once it begins to normalize. Remember that it takes 36 to 50 or so measurements before a sample normalizes. Various approximation methods help us to approximate the normal when we lack adequate data. Skewness and kurtosis will be present in all samples lacking sufficient measurements. Look for Skewness and kurtosis in the feedback collected during Agile development efforts. The normal, in those circumstances will inform us as to whether the functionality is done and deliverable.

Core width will change over the adoption lifecycle.  I drew this figure thinking in terms of standard deviations. But, the Poisson distribution is what we have at the early adopter phase of the lifecycle. In the vertical, we tend to the normal. In the horizontal, some complex data fusions give us a three or more sigma normal and in the late phases we are in the six or more sigma range. The core width is correlated with time, but in the lifecycle, time is determined by seats and dollars, and the lifecycle phase rather than calendar time. Note that I correlated the underlying geometries with time as well. Our financial analysis tells us to pass on discontinuous technologies, because the future looks small in the hyperbolic geometry we don’t know that we are looking at. Euclidean is easy. And, the spherical geometry that leaves us in banker numbers, in information (strategy) overload, aka 30 different approaches that all work. No, he wasn’t lucky. He was spherical.

Enjoy.