Archive for February, 2016

More on Geometry

February 6, 2016

A few days ago, I dropped into B&N for a quick browse through the math section. There wasn’t much new there, so off to the business section. There was a new book about innovation, no I didn’t write down a citation, innovation in the sense of it being a matter of the orthodoxy, aka nothing new in the book. It mentioned the need for collaborations between companies should create more value than the sum of the individual part, aka synergy. A former CEO of ITT settled this synergy thing. He called it a myth.

Tonight, I came across another of Alexander Bogomolny’s (@CutTheKnotMath) tweets. This one showed how a cyclic quadrilateral or two triangles sharing a base would give rise to a line between the opposite vertexes, which in turn gives rise to a point E. See his figure.

I look at the figure and see two organizations, the triangles, sharing bits at the base. Those triangles represent decision trees. The base of such a triangle would represent the outcome of  the current collection of decisions, which I’ve called the NOW line. The past is back towards the root of the decision tree, or the vertex of the triangle opposite the base.

It gave me a tool to apply towards this issue of synergy. To get that synergy, the triangles would position themselves on a base line where the bases of the individual triangles would overlap where they gave rise to those synergistic bits. But, they only overlap for a few bits, not all of them, as in that cyclic quadrilateral. I built some models in GeoGebra. I found the models surprising. I’m not a sophisticated user yet, so there are too many hidden lines.

I was asking those geometry questions that I mentioned a few posts about where I drew many figures about what circumstances give rise to non-Euclidean geometries. So as I played with my GeoGebra models, I was always asking where the diameter was, and that was not something GeoGebra does at the click of a button. It does let you draw a circumcircular sector, which looks like a pie with a slice removed, and draw a midpoint of the line opposite a given vertex. That was enough to give me a simple way of seeing the underlying geometry of a triangle. When half the pie is removed, a line between the two points on the circumference is the diameter of the circle, so the triangle is Euclidean. I may have said that a triangle is always Euclidean in earlier posts, but I can see how that I wrong. To be Euclidean, the base of the triangle has to be on a diameter of the circle. A figure will clear this up.

I discussed my hypothesis in the previous post.

Three Geometries - Intuitive

The hypothesis was messy. I had triangles down as being locally Euclidean and globally possibly otherwise.

With the circumcircular sectors, the complications go away.

Three Geometries via Circumcircular Sectors

The new model is so much simpler.

I went on to look at two triangles that were not competitors. I looked at that synergy.

Potential Synergy

The red line represents the shared bits. The yellow shows the potential synergy. The gains from synergy, like the gains from M&As, shows up in the analysis, but rarely in the actuals.

I went on playing with this. I was amazed how decisions far away could have population effects, and functionality effects. This even if they don’t compete on the same vectors of differentiation. But these effects are economic once the other organization is outside your category (macro). We only compete within a category (micro).

Population Effects

In this figure, the populations overlap in the outliers. The triangles don’t overlap. They are not direct competitors. The do not share a vector of differentiation. Point A is not on line DE.

The circles represent their populations. The relative population scale is way off in this figure. The late firm should have a population as much as 10 times larger than the early firm.

The problem with modeling two firms, placing them in space relative to each other, means doing a lot work on coordinate systems or using me tensors. I started drawing a grid. I’ll get that done and look for more things to discover in this model. Enjoy!