When we move physical constraints, we innovate discontinuously. When we innovate discontinuously, we create economic wealth as a sideband to making a lot of cash, and we create institutions and careers. We haven’t been doing that lately. Instead, we innovate for cash alone, and we cash in our economic wealth for cash and never replace that economic wealth.

The discontinuity at the core of a discontinuous innovation cannot be overcome by iterating beyond current theory. We need a new theory. That new theory has its own measures and dimensions. These at the invention layer of innovation. These cause a discontinuity at the commercialization layer. That discontinuity is in the populations being served. The existing population says no. The nascent adopting population says or will come to say yes. Polling is fractured by discontinuities.

When we do our financial analysis, the discontinuous innovation generates numbers that fail to motivate us to jump in and make it happen. Why? It’s a question that I’ve spent years looking at. I’ve blogged about it previously as well. My intuition tells me that consistent underreporting is systematic and due to the analysis. My answer revolves around geometry. We do our analyses in terms of a Euclidean geometry, but our realities are multiple, and that Euclidean reality is fleeting. Our Euclidean analysis generates numbers for a hyperbolic space, underreporting the actual long-term results. Results in a hyperbolic space appear smaller and smaller as we tend to

We do our analyses in terms of a Euclidean geometry, but our realities are multiple, and that Euclidean reality is fleeting. Our Euclidean analysis generates numbers for a hyperbolic space, underreporting the actual long-term results. Results in a hyperbolic space appear smaller and smaller as we tend to infinity or the further reaches of our forecasted future. Hyperbolic space is the space of discontinuous innovation.

Once a company achieves a six-sigma normal or the mean under the normal we use to represent the technology adoption lifecycle, or in other terms, once a company has sold fifty percent of its addressable and allocated market share, the company leaves the Euclidean space and enters the spherical space where many different financial analyses of the same opportunity give simultaneous pathways to success. This where a Euclidean analysis would report some failures. Again, a manifestation of the actual geometry, rather than the numbers.

Maps have projections. Those projections have five different properties used in different combinations to generate a graphical impression. Explore that here. Those projections start with the same numbers and tell us a different story. Geometries do the same thing to the numbers from our analysis. Our analysis generates an impression of the future. The math is something mathematicians call L2. We treat L2 as if it were Euclidean. We do that without specifying a metric. It’s linear and that is enough for us. But, it’s not the end of the story.

The technology adoption lifecycle hints at a normal, but the phases decompose into their own normals. And, the bowling alley is really a collection of Poisson distributions that tend to the normal and aggregate to a normal as well. So we see a process from birth to death, from no market population to a stable market population. Here as well, the models change geometries.

I’ve summarized the geometries in the following figure.

We start at the origin (O). We assert some conditional probability to get a weak signal or a Dirac function. We show a hyperbolic triangle, a Euclidean triangle, and a spherical triangle. Over time, the hyperbolic triangle gains enough angle to become Euclidean. The Euclidean triangle then gains enough angle to become spherical. The angle gain occurs over the technology adoption lifecycle, not shown here, parallel to the line through the origin.

When we look at our numbers we pretend they are Euclidean. The hyperbolic triangle shows us how much volume is missed by

our assumption of Euclidean space.

Here I drew some concentric circles that we will come back to later. For now, know that the numbers from our analysis report only on the red and yellow areas. We expected that the numbers reported the area of the Euclidean triangle.

The green triangle is the Euclidean triangle that we thought our numbers implied. In a six-sigma normal, the numbers from the analysis would be correct. Less than six sigma or more than six sigma, the numbers would be incorrect.

In the spherical geometry, the problem is subtly different. We keep thinking in Euclidean terms, which hides the redundancies in the spherical space. Here, competitors have no problem copy your differentiation even to the point of coding around your patent. You have more competition than expected and end up with less market as a result. The risks are understated.

To reiterate the problem with the hyperbolic space, we can look at a hyperbolic tessellation.

In a Euclidean tessellation, each shape would be the same size.

The differences in impressions generated by the hyperbolic view and the Euclidean view should be obvious. We’ve been making this mistake for decades now.

In a spherical tessellation, the central shape would be the smallest and the edge shapes would be the largest.

Here, in a hyperbolic geometry, the future is at the boundary of the circle. Numbers from this future would appear to be very small.

In a factor analysis view, the first factor would be represented by the red polygon. The second factor would be represented by the orange polygons. The third factor would be represented by the yellow polygons. The edge of the circle lays at the convergence of the long tail with the ground axis. The edge would be lost in the definition of the limit. The convergence is never achieved in a factor analysis.

Building a factor analysis over those tessellations tells us something else. Factor analyses return results from hyperbolic space routinely. The first factor is longer and steeper. The hyperbolic tessellation would do that. Neither of the other spaces would do that. So where you do a factor analysis, you may be engaging in more geometric confusion.

Notice that the spherical geometry of the typical consumer business is, like most business people, biased to optimism. The future is so big. But, to get to those numbers, you have to escape the Euclidean space of the very beginnings of the consumer facing startup.

With a discontinuous innovation and its hyperbolic space, the low numbers and the inability to get good numbers to arrive in the future usually convinces us to not go there, to not launch, so we don’t. But, we’d be wrong. Well, confused.

Economists told us that globalism would work if we didn’t engage in zero-sum thinking. But, that is what we did. We, the herd, engaged in zero-sum thinking and doing. We innovated continuously, which has us ignoring the economic wealth vs cash metric. We, in our innovation songs, confuse the discontinuously innovative past of the Internet with the continuously innovative present. Or worse, disruption. Thinking we’d get the same results. This even when the VCs are not confused. They deal smaller, much smaller now than back then.

Wallowing in cash doesn’t replace the economic wealth lost to globalism. We can fix this in short order without the inputs and errors from our politicians. But, we have to innovate discontinuously to replace that lost economic wealth. It’s time to say yes in the face of the hyperbolic no. We can create careers and get people back to work.