## Geometry

I was looking for the parameters of an eclipse earlier in the week. I ended up Wikipedia looking at the definition of Eccentricity. The parameter of interest is eccentricity. Right away eccentricity breaks down into four cases: circle (e=0), ellipse (0<e<1), parabola (e=1), and hyperbola (e>1). Notice that this aligns itself with the geometry of the space itself. Relative to the sum of the angles in a triangle we have three cases: hyperbolic (<180), Euclidean (=180), and spherical (>180). Notice also that this aligns itself with the definition of probabilities, as 0 ≤ p ≤ 1. And, footprints of distributions tie into eccentricity: normal as a circle, and Poisson as an ellipse. The distributions also tie into machine learning: Poisson giving us rule enforcement, and Gaussian (normal) giving us rule enforcement. Then, there is Ito processes: n = 0 giving us the Markov chain, n > 0 giving us an Ito process. The Markov chain is a special case of the Ito process. The holes in these associations is probably due to my having been exposed to that math yet. Everything in math is tied to everything else in Math.

I don’t have a correlation between the parabola and anything else. I’ll have to think about this single case.

The failures of a given innovation is excused by faulting innovation. But management as an idea was extended to innovation. Management as an idea was exclusive of innovation when Sloan created management. Nobody says management failed when an innovation fails. Christensen makes the case that managers excelling at management failed when their companies were disrupted. Ultimately what this boils down to is place, under a distribution in a specific geometry. I will finish this post talking about place, but I need to get back to eccentricity and geometry first.

In the Wikipedia post on eccentricity, there was an animation linking circles with ellipses, parabolas, and hyperbolas. Watch it several times, because I going to ask you to image the animation happening in a different order.

The animation begin with the circle. A blue dot represents the center of that circle. That dot goes on to represent the foci of the ellipse, the parabola, and the hyperbola. You can watch the dot move in each frame of the animation.

So now we can think about it in terms of the technology adoption lifecycle(TALC), or the processes organized by the lifecycle. We’ll start simply here. It will get messy as we go deeper. Start with a Poisson game. That’s when we are looking for those B2B early adopters in the TALC. That’s the second phase, the one adjacent to the technical enthusiasts.

A series of Poisson distributions generate a single Poisson distribution whose foot print is an ellipse. The major axis of the ellipse shows us a Markov process as the major axis grows. The major axis is a vector. We start with this Poisson distribution, because we are using a game-theoretic game to represent a game of unknown population, a Poisson game. You can play these games as Gaussian games, but my intuition is to go with discovery learning. Keep in mind that I’m talking about a discontinuous innovation here. Continuous innovations happen elsewhere in the TALC.

Now, this Poisson distribution starts off as a single infinite histogram, aka a point, in other words as a tiny circle. Markov chains are composed of Poisson distributions of arcs, whose pre-choice probabilities are taken from normal distributions of the nodes, small distributions. The Poisson would be external, while the normal would be internal.

We are representing the company and its customer base, as opposed to its prospect base as a Poisson distribution. Over time, that Poisson distribution tends to the normal. The ellipse gets longer and wider. The ellipse fits inside a rectangle that eventually becomes a square at which point the ellipse becomes a circle. The eccentricity changes from something between zero and one becoming zero. I’ve seen this in financial results of companies selling products to foster the adoption of discontinuous innovation. I trust this to be reliable.

The circle represents the vertical. The bowling ally is a collection of approaches to different verticals. The Poisson distributions of those approaches to their verticals point to their respective verticals, aka they walk to their vertical.  Arriving at the chasm is the event that correlates with the onset of the normal. The onset of the normal is also the onset of Euclidean space.

The circle goes on to represent the horizontal market. Consider it to be six sigma wide at the post tornado. Once it is larger than six sigma the geometry is spherical. The standard b-school case analysis becomes very reliable in spherical space. But, my focus is on why that same analysis fails us prior to the chasm. I hypothesize that the space prior to the Euclidean is hyperbolic.  We’ll go back to animation again, but this time I’ll capture the frames.

The animation ends with the hyperbola. Businesses don’t end with the hyperbola. They end in a spherical geometry usually with a black swan that makes their distribution contract. A category begins with a gap. Consider the space looking outward to the foci to be the gap.

I was going to show that the research front changed and call that period the research effort. But, the animation didn’t support that. The directrices moved instead. They do approach each other, but never converge. distance from one foci to the nearest directrix is equal to the eccentricity, which will be larger than one.

I’m going with the hyperbola, as it is unfamiliar and weird enough to lead to things like taxicab geometry where you can’t go straight there, instead having to stay on the grid. In the other geometries you can go straight there. I imagine linear algebra can make the hyperbolic linear, but I haven’t gotten to that math yet.

The time research takes would happen on a z-axis. The search that is research would happen on the surface of the research front. Notice I didn’t use the term R&D. Research gets us our technology and our s-curve. Products foster adoption of the technology. Technology is adopted. Products are sold.

Once the directrices have converged to their minimum separation the weak signal is emitted and the Poisson games begin. I had to draw the figure myself, because the ellipse was too large since in the animation the ellipse starts with a circle. The hyperbola in the figure is there to show the system before the directrices converged. The big bang here is the signed contract with the B2B early adopter. We grow from nothing starting here.

As an aside, Levy flights happen at the find you’re underlying technology phase, aka before the technical enthusiast phase of the TALC.

Now, we’ll go back to the notion of place. In the animation, the blue dots that represent the origin and the foci moving across the geometries. In the TALC, a normal of normal, discontinuous technologies undergo adoption from left to right starting at the far left. All other types of innovation start in the random-access sense somewhere to the left, aka in a different place. Starting at the left means being a monopolist or exiting the category. Starting to the right means competing on promo-spend dollars against fast followers and other look alikes. Those are different places. Samsung will never be Apple even if they hire Steve Jobs. Different places. Different times. Different pathways.

I’ll talk about place in a later post. Tweets about design and brand drive me nuts. They are phase specific–place specific.