I’ve been trying to convey some intuition about why we underestimate the value of discontinuous innovation. The numbers are always small, so small that the standard financial analysis results in a no go decision, a decision not to invest. That standard spreadsheet analysis is done in L2, a Euclidean space. This analysis gets done while the innovation is in hyperbolic space so the underestimation of value would be the normal outcome.

In hyperbolic space, infinity is away at the edge at a distance. In hyperbolic space, the unit measure appears smaller at infinity when viewed from Euclidean space. This can be seen in a hyperbolic tiling. But, we need to keep something in mind here and throughout this discussion, the areas of the circle are the same in Euclidean space. The transform, the projection into hyperbolic space makes it seem otherwise. That L2 financial analysis assumes Euclidean space while the underlying space is hyperbolic, where small does not mean small.

How many innovations, discontinuous ones, have been killed off by this projection? Uncountably many discontinuous innovations have died at the hands of small numbers. Few put those inventions through the stage-gated innovation process because the numbers were small. The inventors that used different stage gates pushed on without worrying about the eventual numbers succeeded wildly. But, these days, the VCs insist on the orthodox analysis, typical of the consumer commodity markets, that nobody hits one out of the ballpark and pays for the rest. The VCs hardly invest at all and insist on the immediate installation of the orthodoxy. This leads us to stasis and much replication of likes.

I see these geometry changes as smooth just as I see the Poisson to normal to high sigma normals as smooth. I haven’t read about differential geometry, but I know it exists. Yet, there is no such thing as differential statistics. We are stuck in data. We can use Monte Carlo Markov Chains (MCMC) to generate data to fit some hypothetical distribution from which we would build something to fit and test fitness towards that hypothetical distribution. But, in sampling that would be unethical or frowned upon. Then again, I’m not a statistician, so it just seems that way to me.

I discussed geometry change in Geometry and numerous other posts. But, in hunting up things for this post, I ran across this figure. I usually looked at the two-dimensional view of the underlying geometries. So this three-dimensional view is interesting. Resize each geometry as necessary and put them inside each other. The smallest would be the hyperbolic geometry. The largest geometry, the end containment would be the spherical geometry. That would express the geometries differentially in the order that they would occur in the technology adoption lifecycle (TALC) working from the inside out. Risk diminishes in this order as well.

In the above figure, I’ve correlated the TALC with the geometries. I’ve left the technical enthusiasts where Moore put them, rather than in my underlying infrastructural layer below the x-axis. I’ve omitted much of Moore’s TALC elements focusing on those placing the geometries. The early adopters are part of their vertical. Each early adopter owns their hyperbola, shown in black, and seeds the Euclidean of their vertical, shown in red, or normal of the vertical (not shown). There would be six early adopter/verticals rather than just the two I’ve drawn. The thick black line represents the aggregation of the verticals needed before one enters the tornado, a narrow phase at the beginning of the horizontal. The center of the Euclidean cylinder is the mean of the aggregate normal representing the entire TALC, aka category born by that particular TALC. The early phases of the TALC occur before the mean of the TALC. The late phases start immediately after the mean of the talk.

The Euclidean shown is the nascent seed of the eventual spherical. Where the Euclidean is realized is at a sigma of one. I used to say six, but I’ll go with one for now. Once the sigma is larger than one, the geometry is spherical and tending to more so as the sigmas increase.

From the risk point of view, it is said that innovation is risky. Sure discontinuous innovation (hyperbolic) has more risk than continuous (Euclidean) and commodity continuous (spherical) less risk. Quantifying risk, the hyperbolic geometry gives us an evolution towards a singular success. That singular success takes us to the Euclidean geometry. Further data collection takes us to the higher sigma normals, the spherical space of multiple pathways to numerous successes. The latter, the replications, being hardly risky at all.

Nesting these geometries reveal gaps (-) and surplusses (+).

## The Donut/Torus Again

In an earlier post, I characterized the overlap of distributions used in statistical inference as a donut, as a torus, and later as a ring cyclide. I looked at a figure that described a torus as having positive and negative curvature.

So the torus exhibits all three geometries. Those geometries transition through the Euclidean.

The underlying distributions lay on the torus as well. The standard normal has a sigma of one. The commodity normal has a sigma greater than one. The saddle and peaks refer to components of a hyperbolic saddle. The statistical process proceeds from the Poisson to the standard normal to the commodity normal. On a torus, the saddle points and peaks are concurrent and highly parallel.

Enjoy.