Stuff Left Out

My review of math has me learning things that I was never taught before. There were concepts left for later, if later ever arrived.

We had functions. At times we had no solutions. Later, they become functions without the right number of roots in the reals, as some of the solutions were in the space of a left out concept, complex roots. For a while, after a deeper understanding of complex roots, I’d go looking for the unit circle, but what was that unit circle centered on? Just this week, I found it centered on c, a variable in those equations like ax2+bx+c. So that open question comes to a close, but is that enough of an answer? I don’t know yet.

We had trigonometry, because we had trigonometric functions. We have the unit circle and an equation, notice that it’s not a function, because as a function it would fail the vertical line test. When a function violates the vertical line test, we create a collection of functions covering the same values in a way that does not violate the vertical line test. Trigonometry does that for us without bothering to explain the left out concept, manifolds.

One of Alexander Bogomolny’s (@CutTheKnotMath) tweets linked to a post about a theorem from high school geometry class about how a triangle with it’s base as a diameter of a circle and a vertex on the circle had angles that added up to 180 degrees. Yes, I remembered that. I hadn’t thought about it in decades. But, there was something left out. That only happens in a Euclidean geometry.

Well, I doodled in my graphics tools. I came to hypothesize about where the vertex would be if it were inside the circle, but not on it, and the base of the triangle was on the diameter. It would be in a hyperbolic geometry was my quick answer; outside the circle, same base, spherical. Those being intuitive ideas. A trapezoid can be formed in the spherical case using the intersections of the circle with the triangle. That leave us with the angles adding up to more than 180 degrees.

I continued to doodle. I ended up with the bases being above or below the diameter, and the vertex on the circle. Above the diameter, the geometry was hyperbolic; below, spherical. I got to a point using arc length were I could draw proper hyperbolic triangles. With the spherical geometries there were manifolds.

The point of all this concern about geometries involves how linear analyses break down in non-Euclidean spaces. The geometry of the Poisson games that describe the bowling allies of the technology adoption lifecycle (TALC) is hyperbolic. The verticals and the horizontals of the TALC approach the normal. Those normals at say six sigma give us a Euclidean geometry. Beyond six sigma, we get a spherical geometry, and with it a redundancy in our modeling–twelve models all of them correct, pick one.

We’ve looked at circles for decades without anyone saying we were constrained to the Euclidean. We’ve looked at circles without seeing that we were changing our geometries without thinking about it. We left out our geometries, geometries that inform us as to our non-linarites that put us at risk (hyperbolic) or give us too much confidence (spherical). We also left out the transitions between geometries.

The circle also illustrates why asking customers about their use cases and cognitive models, or even observing, doesn’t get us far enough into what stuff was left out. Back in school, it was the stuff that was left out that made for plateaus, stalls, stucks, and brick walls that impeded us until someone dropped a hint that led to the insight that dissolved the impedances and brought us to a higher level of performance, or a more complete cognitive model.


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