## The Shape Of Innovation

In the past, I’ve summarized innovation as a decision tree. I’ve summarized innovation as divergence and convergence, generation and tree pruning. So I drew this figure.
The generative grammar produces a surface. The Constraints produce another surface. The realization, represented by the blue line, would be a surface within the enclosed space, shown in yellow. The realization need not be a line or flat surface.

In CAD systems, the two surfaces can be patched, but the challenge here is turning the generative grammar into a form consistent with the equations used to define the constraints. The grammar is a tree. The constraints are lines. Both could be seen as factors in a factor analysis. Doing so would change the shape of the generated space.

In a factor analysis, the first factor is the longest and steepest. The subsequent factors are flatter and shorter.

A factor analysis produces a power law.

A factor analysis represents a single realization. Another realization gives you a different factor analysis.

When you use the same units on the same axes of the realizations, those realizations are consistent or continuous with each other. These are the continuities of continuous innovation. When the units differ in more than size between realizations, when there is no formula that converts from one scale to another, when the basis of the axes differ, the underlying theories are incommensurate or discontinuous. These are the discontinuities of discontinuous innovation.

The surfaces contributing to the shape of the enclosed space can be divided into convex and concave spaces. Convex spaces are considered risky. Concave spaces are considered less risky. Generation is always risky. The containing constraints are unknown.
The grammar is never completely known and changes over time. The black arrow on the left illustrates a change to the grammar. Likewise, the extent of a constraint changes over time, shown by the black arrow on the right. As the grammar changes or the constraints are bent or broken, more space (orange) becomes available for realizations. Unicode, SGML, and XML extended the reach of text. Each broke constraints. Movement of those intersections moves the concavity, the safe harbor in the face of gernerative risks. As shown the concavity moved up and to the left. The concavity abandoned the right. The right might be disrupted int he Foster sense. The constraints structure populations in the sense of a collection of pramatism steps. Nothing about this is about the underserved or disruption in the Christensen sense.

The now addressible space is where products fostering adoption of the new technology get bought.

The generative grammar is a Markov chain. Where the grammar doesn’t present choice, the chain can be thought of as a single node.

The leftmost node is the root of the generative grammar. It presents a choice between two subtrees. Ultimately, both branches would have to be generated, but the choice between them hints at a temporal structure to the realization, and shifting probabilities from there.

New gramatical structures would enlarge the realization. Grammars tend to keep themselves short. They provide paths that we traverse or abandon over historical time. The realization would shift its shape over that historical time. This is where data mining could apply.

When the constraints are seen from a factor analysis perspective, the number of factors are few in the beginning and increase over time. This implies that gaps between the realization and the factors would exit and diminish over time. Each factor costs more han the factor before it. Factors add up to one, and then become a zero-sum game. For another factor to assert itself, existing factors would have to be rescaled.

Insisting on a factor anlaysis perspective leaves up with trying to find a factor designated as the root constraint. And then, defining the face offs. This subgrammar vs this collection

of constraints. Each would have rates, thus differential equations. Each would be a power law. So in our system there would be four differential equations and four power laws. There would also be four convergences. These would be reflected in the frequencies of use histograms.

Notice that nowhere in this discussion was innovation based on an idea from management. The ideas were about enlarging the grammar, aka ontological sortables, and the breaking or bending of constraints. When a constraint built into a realization breaks, Glodratt told us that the realization moves some distance to the next constraint.These efforts explore the continuities and discontinuities of the possible innovations. Produtization is the next step in fostering adoption.

As always, enjoy.

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