Last week, I finished *“e”: The Story of a Number* by Eli Maor. It explained a lot of familiar and unfamiliar math. It covered hyperbolic, exponential, and imaginary functions. Exponential functions generate geometric progressions. Exponential functions when graphed present you with the same log scale doubling we talked about in the the previous blog post, Cognitive Models on the Efficiency Frontier.

When you move to imaginary numbers, you end up with an exponential polar graph. These graphs had me thinking about how the concentric circles represent the gaps generate by discontinuous innovation. The arc around the circles represents continuous innovation.

Polar forms deal with a radius or magnitude in some direction like vectors, but the book got here from imaginary numbers.

A discontinuous innovation becomes disruptive if investment in it (price) generates a performance reflected in a price-performance curve has a slope greater than the technology being replaced. I discussed this in The Word is Discontinuous.

S-curves or price-performance graphs, increase to an inflection point and decrease beyond the inflection point. Immediately before the inflection point, the slope of the S-curve is at it’s maximum. This maximum slope has meaning in a polar representation. Innovation slows down beyond it. And, it serves as the boundary that the next technology must exhibit before that technology is disruptive.

I’ve graphed two disruptive innovations, Technology A, and Technology B, followed by continuous/sustaining innovations.

On the left, two S-curves are shown along with their inflection points and maximum slopes. On the right, we have a exponential polar graph depicting the serialization of the discontinuous innovations.

For Technology A, shown in red at r=1. We transfer the maximum slope found on the S-curve for Technology A to the polar graph using the x-axis as the base of the angle. The base of the angle represents the discontinuous innovation. The vector at the given angle from the base is the disruption threshold. As continuous/sustaining innovations sweep the arc at r=1, it brings the technology to (before) and beyond (after) the disruption threshold.

For Technology B, shown in green at r=2. We build the representation as we did for Technology A, except that we use the disruption threshold for Technology A as the base of the angle to the disruption threshold for Technology B.

The tan areas represent the innovations before the disruption thresholds. The orange areas represent the innovations after the disruption thresholds. Notice that even a disrupted technology might be improved while it is being replaced.

As the technologies are improved the innovations are serialized over counterclockwise positions along the arcs.

The distances between subsequent radii, represent the new cognitive model of the discontinuous innovation. A continuous innovation extends the current cognitive model. A discontinuous innovation replaces the current cognitive model entirely. The basis of a discontinuous innovation is far removed from that of the current cognitive model. Examples of such cognitive model conflicts, include Newtonian and Einsteinian physics, traditional cost accounting and ABC cost accounting, Quicken’s single entry accounting system and standard accounting’s double entry accounting system. Such wholesale replacement of cognitive models are called paradigms. Paradigms likewise involve mutually exclusive populations of adopters, and Moore’s technology adoption lifecycle.

Each paradigm is represented by its own circle. The gaps between the circles represent cognitive models in the form of ontologies and, later after the ontologies are realized, taxonomies. Note that ontologies are represented independent of implementation considerations represented in UML. The SemanticWeb, Web 30, will bring more attention to ontologies. Information architects construct ontologies, but they need to be captured during requirements elicitation through ethnographic research.

The radius of each circle ignores commoditization. This implies that the radius is never achieved in reality before the vector of differentiation must change. The technology related to the new vector of differentiation would be a new circle, but the base of the new disruption threshold would be independent of those already depicted on the graph, shown by the blue lines. See slides 53-59 of “So you don’t have a market? Great!.

On slide 57 should have annotated Points of Parity as PoP, rather than PoC, Points of Contention. This language was first describe in *Value Merchants*. In this vocabulary, points of party provide no competitive value beyond market participation.

Please comment. Thanks!

April 9, 2011 at 9:31 am |

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